## Tags - Part of: [[Science]] [[Formal science]] [[Natural science]] [[Omnidisciplionary]] - Related: - Includes: [[Applied mathematics]], [[Mathematical theory of artificial intelligence]] - Additional: ## Definitions - Area of [[knowledge]] that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes - [[Philosophy]] constrained by strict [[Symbol|symbolic]] [[formal]] [[Definition|definitions]] and rules ([[Logic]] studies those rules) studying arbitrary relations between arbitrary things or inner structure of things ## Technical summaries - Mathematics is a [[field]] of study that discovers and organizes [[method|methods]], [[theory|theories]] and [[theorem|theorems]] that are developed and proved for the needs of [[natural science|empirical sciences]] and mathematics itself. ## Main resources - [Mathematics - Wikipedia](https://en.wikipedia.org/wiki/Mathematics) <iframe src="https://en.wikipedia.org/wiki/Mathematics" allow="fullscreen" allowfullscreen="" style="height:100%;width:100%; aspect-ratio: 16 / 5; "></iframe> - [best mathematics books - Hledat Googlem](https://www.google.com/search?q=best+mathematics+books&sca_esv=e14f95cbc2b145ff&sca_upv=1&sxsrf=ADLYWIKpJ7WJou9L8iP333En8nqJBhZ9jQ%3A1727603368562&ei=qCL5ZqTtIfqKxc8P44Pt6AI&ved=0ahUKEwik4eGI8OeIAxV6RfEDHeNBGy0Q4dUDCA8&uact=5&oq=best+mathematics+books&gs_lp=Egxnd3Mtd2l6LXNlcnAiFmJlc3QgbWF0aGVtYXRpY3MgYm9va3MyBhAAGAcYHjIGEAAYBxgeMgYQABgHGB4yBhAAGAcYHjIGEAAYBxgeMgYQABgHGB4yBhAAGAcYHjIGEAAYBxgeMgYQABgHGB4yBhAAGAcYHkiNG1DBFFi8GnABeAGQAQCYAWqgAbMEqgEDNS4xuAEDyAEA-AEBmAIFoAKxA8ICChAAGLADGNYEGEfCAg0QABiABBiwAxhDGIoFwgIOEAAYsAMY5AIY1gTYAQHCAhMQLhiABBiwAxhDGMgDGIoF2AEBwgIHECMYsAIYJ5gDAIgGAZAGE7oGBggBEAEYCZIHAzQuMaAHxTQ&sclient=gws-wiz-serp) [best math books - Hledat Googlem](https://www.google.com/search?q=best+math+books&sca_esv=e14f95cbc2b145ff&sca_upv=1&sxsrf=ADLYWIK6X5bq9aZV2FcqXWuP6PyBIhb85Q%3A1727603381319&ei=tSL5Zr6UE-aXxc8PgraH0Ao&ved=0ahUKEwj-wuyO8OeIAxXmS_EDHQLbAaoQ4dUDCA8&uact=5&oq=best+math+books&gs_lp=Egxnd3Mtd2l6LXNlcnAiD2Jlc3QgbWF0aCBib29rczIGEAAYBxgeMgYQABgHGB4yBhAAGAcYHjIGEAAYBxgeMgYQABgHGB4yBhAAGAcYHjIGEAAYBxgeMgYQABgHGB4yBhAAGAcYHjIGEAAYBxgeSJUSUIEHWOwOcAF4AZABAJgBa6ABpAWqAQM0LjO4AQPIAQD4AQGYAgigAv0FwgIKEAAYsAMY1gQYR8ICDRAAGIAEGLADGEMYigXCAg4QABiwAxjkAhjWBNgBAcICExAuGIAEGLADGEMYyAMYigXYAQHCAgcQIxixAhgnwgIHECMYsAIYJ5gDAIgGAZAGELoGBggBEAEYCZIHAzIuNqAHrz8&sclient=gws-wiz-serp) - [[Math sorcerer]]: [Learn Mathematics from START to FINISH (2nd Edition) - YouTube](https://www.youtube.com/watch?v=didXE0HkSC8&pp=ygUubWF0aCBzb3JjZXJlciBtYXRoZW1hdGljcyBmcm9tIHN0YXJ0IHRvIGZpbmlzaA%3D%3D), [One Math Book For Every Math Subject - YouTube](https://www.youtube.com/watch?v=-mfaMbraEkU) - [Search | MIT OpenCourseWare | Free Online Course Materials](https://ocw.mit.edu/search/?d=Mathematics) - [Im honestly considering buying a membership for a math website because the free ones honestly teach me better than my teacher : r/matheducation](https://www.reddit.com/r/matheducation/comments/1foupc6/im_honestly_considering_buying_a_membership_for_a/) - [Math Academy](https://www.mathacademy.com/) ## Landscapes - There are many areas of mathematics, which include [[number theory]] (the study of [[numbers]]), [[algebra]] (the study of [[formula|formulas]] and related [[structure|structures]]), [[geometry]] (the study of [[shape|shapes]] and [[space|spaces]] that contain them), [[analysis]] (the study of [[continuous]] [[change|changes]]), and [[set theory]] (presently used as a [[foundation]] for all mathematics). - [[9112ec394e713c3a8023c6f0cc7bd040_MD5.jpeg|Open: Pasted image 20240418030623.png]] ![[9112ec394e713c3a8023c6f0cc7bd040_MD5.jpeg]] - [The Map of Mathematics - YouTube](https://www.youtube.com/watch?v=OmJ-4B-mS-Y&t=1s) <iframe title="The Map of Mathematics" src="https://www.youtube.com/embed/OmJ-4B-mS-Y?feature=oembed" height="113" width="200" allowfullscreen="" allow="fullscreen" style="aspect-ratio: 1.76991 / 1; width: 100%; height: 100%;"></iframe> - [Outline of mathematics - Wikipedia](https://en.wikipedia.org/wiki/Outline_of_mathematics) - <iframe src="https://en.wikipedia.org/wiki/Outline_of_mathematics" allow="fullscreen" allowfullscreen="" style="height:100%;width:100%; aspect-ratio: 16 / 5; "></iframe> - [Mathematics - Wikipedia](https://en.wikipedia.org/wiki/Mathematics) - [[Foundations of mathematics]] - [[Logic]] - [[Set Theory]] - [[Category Theory]] - [[Number theory]] - [[Algebra]] - [[Linear algebra]] - [[Group theory]] - [[Geometry]] - [[Topology]] - [[Calculus]] and [[Mathematical analysis]] - [[Discrete mathematics]] - [[Graph theory]] - [[Statistics]], [[Probability theory]] - [[Computer science]] - [[Theory of computation]] - Unity - [[Algebraic Geometry]] - [[Algebraic Topology]] - [[Differential Topology]] - [[Applied mathematics]] - [[Computational mathematics]] - [[Metamathematics]] - [mathematics in nLab](https://ncatlab.org/nlab/show/Mathematics) - [Mathematics | /sci/ Wiki | Fandom](https://4chan-science.fandom.com/wiki/Mathematics) - [[Physics]] - Artificial Intelligence x Mathematics ## Resources Stanford: mathematics https://www.youtube.com/playlist?list=PL4sA4oztWx15EtO2F-UGxPdClH39zdUeL https://www.youtube.com/playlist?list=PLzVGLyH55UdtlzZ33h29sQWvXxRu33nu8 https://www.youtube.com/playlist?list=PLMm0d6XoIQ0Zk5l__Rwqwj6BGqcQDYl0o applied linear algebra [Stanford ENGR108: Introduction to Applied Linear Algebra —Vectors, Matrices, and Least Squares - YouTube](https://www.youtube.com/playlist?list=PLoROMvodv4rMz-WbFQtNUsUElIh2cPmN9) [Linear Algebra Review Andrew Ng - Stanford University - YouTube](https://www.youtube.com/playlist?list=PL2qEL_7r0QISg3wu4D_j9xRJodZsfjBEu) [Gilbert Strang lectures on Linear Algebra (MIT) - YouTube](https://www.youtube.com/playlist?list=PL49CF3715CB9EF31D) probability [Stanford CS109 Introduction to Probability for Computer Scientists I 2022 I Chris Piech - YouTube](https://www.youtube.com/playlist?list=PLoROMvodv4rOpr_A7B9SriE_iZmkanvUg) [Combinatorics](https://chrispiech.github.io/probabilityForComputerScientists/en/part1/combinatorics/) https://web.stanford.edu/~mossr/pdf/p4cs.pdf [GitHub - mossr/machine_learning_book: Stanford's CS229 Machine Learning lecture notes compiled into a Tufte-style textbook](https://github.com/mossr/machine_learning_book/tree/master) [CS109](https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/) [CS109 | Final Exam](https://web.stanford.edu/class/archive/cs/cs109/cs109.1202/exams/final.html) computer science mathematics https://www.youtube.com/playlist?list=PLB7540DEDD482705B MIT: linear algebra in machine learning [MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, Spring 2018 - YouTube](https://www.youtube.com/playlist?list=PLUl4u3cNGP63oMNUHXqIUcrkS2PivhN3k) mathematics https://www.youtube.com/playlist?list=PLm0X8hqw4lIotQB9ep0MKVkT2447Lk2ue calculus [MIT 18.01 Single Variable Calculus, Fall 2006 - YouTube](https://www.youtube.com/playlist?list=PL590CCC2BC5AF3BC1) [MIT 18.02 Multivariable Calculus, Fall 2007 - YouTube](https://www.youtube.com/playlist?list=PL4C4C8A7D06566F38) linear algebra https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8 mathematics for computer science https://www.youtube.com/playlist?list=PLB7540DEDD482705B real analysis https://www.youtube.com/playlist?list=PLUl4u3cNGP61O7HkcF7UImpM0cR_L2gSw differential equations https://www.youtube.com/playlist?list=PLEC88901EBADDD980 https://www.youtube.com/playlist?list=PLUl4u3cNGP63oTpyxCMLKt_JmB0WtSZfG https://www.youtube.com/playlist?list=PL64BDFBDA2AF24F7E mathematics for engineering https://www.youtube.com/playlist?list=PL3A13781649466805 discrete mathematics https://www.youtube.com/playlist?list=PLWoMOTP6TgzLFi5fCmQBuRPb0WX4y_s7U graph theory and combinatorics https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX statistics https://www.youtube.com/playlist?list=PLUl4u3cNGP60uVBMaoNERc6knT_MgPKS0 mathematical modelling https://www.youtube.com/playlist?list=PLFFA35EF8CECBA074 set theory https://www.youtube.com/playlist?list=PLuiPz6iU5SQ_3Gubdqa1JHBvM0GBFcIV0 Harvard: abstract algebra and group theory https://www.youtube.com/playlist?list=PLzUeAPxtWcqzr80lS25FrzMn7a36BuXhj applied mathematics https://www.youtube.com/playlist?list=PL43IQ71lgJytIqhiJ6v5lNswFKeQ9952K statistics and probability [Statistics 110: Probability - YouTube](https://www.youtube.com/playlist?list=PL2SOU6wwxB0uwwH80KTQ6ht66KWxbzTIo) [Math 110 Videos - YouTube](https://www.youtube.com/playlist?list=PLLoOcJCfXeBuLjR4K3VMpU-ryEgeCzecN) advanced algorithms https://www.youtube.com/playlist?list=PL2SOU6wwxB0uP4rJgf5ayhHWgw7akUWSf Berkeley: discrete mathematics https://www.youtube.com/playlist?list=PLaVBOvvdB5ctaLM6AmkUaODhd4JhyP_zC https://www.youtube.com/playlist?list=PLu0nzW8Es1x0Ivn-757Za_ps090FJxOPd multivariable calculus https://www.youtube.com/playlist?list=PLaLOVNqqD-2GcoO8CLvCbprz2J0_1uaoZ number theory https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8 linear algebra and differential equations https://www.youtube.com/playlist?list=PLShth7hrtLHO2U1XkrI6ZgMyuPHDxRcob calculus https://www.youtube.com/playlist?list=PLShth7hrtLHPz41qo1XlGZRNl9pcVVTfj discrete mathematics and probability theory https://www.youtube.com/playlist?list=PLzAv_uHZw7dTI2e0F8-lxxOWV9zXMzwNE set theory and logic https://www.youtube.com/playlist?list=PLjJhPCaCziSQyON7NLc8Ac8ibdm6_iDQf algebraic geometry https://www.youtube.com/playlist?list=PL8yHsr3EFj53j51FG6wCbQKjBgpjKa5PX Princeton: advanced mathematics https://www.youtube.com/playlist?list=PLMKvcf1goyEtjS64FE5J7EXfqgkkSQYxM calculus https://www.youtube.com/playlist?list=PLGqzsq0erqU7h6_bpE-CgJp4iX5aRju28 linear algebra https://www.youtube.com/playlist?list=PLGqzsq0erqU7w7ZrTZ-pWWk4-AOkiGEGp Yale: math mornings https://www.youtube.com/playlist?list=PLqHnHG5X2PXBVZsf_rvAwGnUgZ-mGdqCy quantitative finance https://www.youtube.com/playlist?list=PL3F00F1C2D402D45C applied mathematics https://www.youtube.com/playlist?list=PL16161DDEB6FE40FF game theory https://www.youtube.com/playlist?list=PL6EF60E1027E1A10B music theory https://www.youtube.com/playlist?list=PL9LXrs9vCXK56qtyK4qcqwHrbf0em_81r Oxford: mathematics https://www.youtube.com/playlist?list=PLF_dLj7JZ2et6-T_AHlHOMFVHJ6bAayqy linear algebra https://www.youtube.com/playlist?list=PLMCRxGutHqfnC5QNWo9DAsU_QlXK5JoDX calculus https://www.youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S student lectures uni math https://www.youtube.com/playlist?list=PL4d5ZtfQonW1xKVEtYJd1iu9m52ATG7SV https://www.youtube.com/playlist?list=PL4d5ZtfQonW3dT0mRS3M-4bppBulIRyZG probability https://www.youtube.com/playlist?list=PL4d5ZtfQonW0B3qW24yAj1u1SuOvgKfP5 networks https://www.youtube.com/playlist?list=PL4d5ZtfQonW0MsGE4Pn12rxUprPXB4_VS ETH Zurich: data science https://www.youtube.com/playlist?list=PLiud-28tsatIKUitdoH3EEUZL-9i516IL math of machine learning https://www.youtube.com/playlist?list=PLiud-28tsatL0MbfJFQQS7MYkrFrujCYp systems dynamics and complexity https://www.youtube.com/playlist?list=PLaLOVNqqD-2EcroLJ0X_0Bb230JyMk4fS institute for advanced study 3500 mathematics lectures https://www.youtube.com/playlist?list=PLCA9C279868C62EB1 Mathematical statistics [Mathematical Statistics - YouTube](https://www.youtube.com/playlist?list=PLLyj1Zd4UWrOk5-wIki_oOxHJnNj0_437) [Mathematical Statistics (2024) NEW! - YouTube](https://www.youtube.com/playlist?list=PLLyj1Zd4UWrPZH-fknPLak0tlUpUISBZR) [Mathematical Statistics I (2020. Spring) - YouTube](https://www.youtube.com/playlist?list=PLpM_znWrmN6hpfqW1Rt1Luac-HUiAsj8_) Bayesian statistics [Bayesian Statistics - YouTube](https://www.youtube.com/playlist?list=PLvcbYUQ5t0UEkf2NUEo7XSsyVTyeEk3Gq) Statquest statistics [Statistics Fundamentals - YouTube](https://www.youtube.com/playlist?list=PLblh5JKOoLUK0FLuzwntyYI10UQFUhsY9) [GitHub - llSourcell/learn_math_fast: This is the Curriculum for "How to Learn Mathematics Fast" By Siraj Raval on Youtube](https://github.com/llSourcell/learn_math_fast)[GitHub - Developer-Y/math-science-video-lectures: List of Science courses with video lectures](https://github.com/Developer-Y/math-science-video-lectures) [GitHub - openlists/MathStatsResources](https://github.com/openlists/MathStatsResources) [GitHub - rossant/awesome-math: A curated list of awesome mathematics resources](https://github.com/rossant/awesome-math) [This is a collection of resources of mathematics for engineering students · GitHub](https://gist.github.com/yewalenikhil65/da90ca0c46af36c935b8123593ff0e9b) [GitHub - HimoriK/modern-math-collection: A super math collection of resources to study as painlessly as possible](https://github.com/HimoriK/modern-math-collection) [GitHub - atkirtland/awesome-computational-geometry: A curated list of awesome computational geometry visualizations, frameworks, and resources](https://github.com/atkirtland/awesome-computational-geometry#readme) [GitHub - benedekrozemberczki/awesome-graph-classification: A collection of important graph embedding, classification and representation learning papers with implementations.](https://github.com/benedekrozemberczki/awesome-graph-classification) [Courses | Brilliant](https://brilliant.org/courses/)[Courses | Brilliant](https://brilliant.org/courses/) [[nonAI mathcode long important]] ## Brainstorming Mathematics. The purest language. The language of formal patterns. The language of generalization. The language of order. The language of unification. The language of deflation. The language of existence. The language in which the universe is written in. The language in which all scales of reality operate in. I love it. I wish I could grasp all the mathematics of our reality, from the most fundamental to all the emergent, all at once, in a single thought, beyond this limited brain, trying to understand and predict itself and it's environment. Everything is mathematics Math is axioms and composite structures and theorems derived from them that help us predict and control the physical world reliably All possible decompositions of all possible decompositions of all possible mathematical objects Memorization is the first step towards generalization Is space an objective container or relational between objects? Math and physics is so magical in how such an explosion of so many equations, properties, theorems etc. pop out of just few axioms, definitions, and assumptions playing together (set theory, linear algebra, calculus, algebra, probability, conservation laws, symmetries, newton's equations, schrodinger equation, statistical mechanics,...) The fact that some mathematical theorems work in the first place is often so unintuitive It's still weird that multiplicating and adding numbers together can compress information and generalize so well in deep learning We should speak in mathematics only to minimize ambiguity Approximating differentiable curvefitted solution approximating all functions using grokked fourier series algorithm? Fourier series approximating any differential curvefitted solution? Duality? Taylor series approximations? Spline interpolation? Gaussian mixture models? Support vector machines? Decision trees? Random forests? Wavelets? General universal approximators of arbitrary functions? Generalized approximation theorem? Space of all possible general universal approximators? Inject all math text books into my blood stream in their purest form Writing x from scratch is the best way to learn anything in any field where it's possible, mathy or nonmathy One of my favorite ways of learning math with language models is prompting them to go step by step using examples through the various mathematical equations transforming data ## Deep dives - [3Blue1Brown](https://www.youtube.com/@3blue1brown) - [Summer of math exposition 2](https://www.youtube.com/hashtag/some2), [Summer of math exposition 3](https://www.youtube.com/hashtag/some3) - [List of unsolved problems in mathematics - Wikipedia](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics) <iframe src="https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics" allow="fullscreen" allowfullscreen="" style="height:100%;width:100%; aspect-ratio: 16 / 5; "></iframe> ## Brain storming My goal is to create a large visual map that contains as much information about mathematics as possible by showing the most important mathematical structures, definitions, and equations, with as little explanatory text as possible. I want the highest possible percentage of the map to consist primarily of mathematical symbols, all on one giant readable poster! I haven't seen anything like that yet. There are some maps, tables, lists, wikis, etc., on mathematics that inspire me, but I haven't seen anything like a visual map of tons of definitions and equations from mainly the foundations of math, pure math, and applied math: theoretical physics, systems theory, mathematical biology, AI, and other mathematically applied sciences and engineering fields that I find most important. Often they are too general or too specific in areas where I don't want them to be. There is, for example, a math map by [Domain of Science](<https://www.youtube.com/watch?v=OmJ-4B-mS-Y>), or [physics](<https://www.youtube.com/watch?v=ZihywtixUYo>) (he has [more](<https://www.google.com/search?sca_esv=f032846a98b531f9&sxsrf=ACQVn0976HXiiNvRPJyyV5C4j7DIBC8eyQ:1709279314434&q=map+of+physics&tbm=vid>)), [Mathematopia](<https://tomrocksmaths.com/2020/12/21/mathematopia-the-adventure-map-of-mathematics/>), [geometric representation of mathematics](<https://imgur.com/Tgd6HmA>), [http://srln.se/mapthematics.pdf](http://srln.se/mapthematics.pdf), by [Zooga](<https://www.reddit.com/r/math/comments/2av79v/map_of_mathematistan_source_in_comments/>), [this Langlands beauty](https://bastian.rieck.me/blog/2020/langlands/), or [here are a few listed in math stackexchange](<https://math.stackexchange.com/questions/124709/mind-maps-of-advanced-mathematics-and-various-branches-thereof>), or [google search finds some others](https://www.google.com/search?sca_esv=6416b2a2bca84fa5&sxsrf=ACQVn08EYgLRVx_d0OEctey6oKUsAtsrOg:1709276455770&q=map+of+mathematics&tbm=isch&source=lnms&sa=X&ved=2ahUKEwjUgeD_vtKEAxXS0AIHHaQOBFoQ0pQJegQICxAB&biw=1920&bih=878&dpr=1). Plus Peak Math is building a large visual interactive [map](https://www.peakmath.org/peakmath-landscape). There are also various wikis and lists: [Wikipedia](<https://en.wikipedia.org/wiki/Outline_of_science#Branches_of_science>) ([Mathematics](https://en.wikipedia.org/wiki/Mathematics): [category](https://en.wikipedia.org/wiki/Category:Mathematics), [outline](https://en.wikipedia.org/wiki/Outline_of_mathematics), [portal](<https://en.wikipedia.org/wiki/Portal:Mathematics>), [list of topics](https://en.wikipedia.org/wiki/Lists_of_mathematics_topics), [areas](https://en.wikipedia.org/wiki/Template:Areas_of_mathematics), [https://en.wikipedia.org/wiki/Category:Fields_of_mathematics](https://en.wikipedia.org/wiki/Category:Fields_of_mathematics), or [physics](https://en.wikipedia.org/wiki/Physics): [category](https://en.wikipedia.org/wiki/Category:Physics), [outline](https://en.wikipedia.org/wiki/Outline_of_physics), [portal](https://en.wikipedia.org/wiki/Portal:Physics)), [Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Main_Page), [Wolfram Math World](https://mathworld.wolfram.com/), [Mathematics Subject Classification](https://zbmath.org/classification/)([on wiki](https://en.wikipedia.org/wiki/Mathematics_Subject_Classification)), [Math Fandom](https://math.fandom.com/wiki/Math_Wiki), [Mathematics Atlas](https://web.archive.org/web/20150429140457/http://www.math.niu.edu/%7Erusin/known-math/welcome.html), [Awesome Math](https://github.com/rossant/awesome-math), [this tree](https://imgur.com/d8KqaFx), etc., but these are not visual maps as they are wikis, and in some ways, they are either insufficiently specialized or too detailed and include all sorts of extra text around the equations. I want to have as little of that extra text as possible, packing as many mathematical symbols as possible into the map. I also like to prompt AIs and try to extract concepts, equations, associations around various fields and topics through prompts like "write a gigantic list of all subfields in math/physics," "write a gigantic list of the most important structures and equations used in this subfield of physics or mathematics," etc., and then I look them up. Or [The Princeton Companion to Mathematics](<https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809>)([pdf](<https://sites.math.rutgers.edu/~zeilberg/akherim/PCM.pdf>)), [https://www.amazon.com/Princeton-Companion-Applied-Mathematics/dp/0691150397?ref=d6k_applink_bb_dls&dplnkId=352f8fc3-ee97-4716-817a-e8feea9cd8c2](https://www.amazon.com/Princeton-Companion-Applied-Mathematics/dp/0691150397?ref=d6k_applink_bb_dls&dplnkId=352f8fc3-ee97-4716-817a-e8feea9cd8c2) looks interesting, or there's also [Mathematical Promenade](https://arxiv.org/abs/1612.06373). Or there's also [ProofWiki](https://proofwiki.org/wiki/Category:Proofs), but that's mainly for proofs, and I primarily want to compress the results as much as possible, so as many resulting definitions, equations, and various connections fit into as little space as possible. Or Quanta Magazine has a map on some [math](<https://mathmap.quantamagazine.org/map/>) and [physics](<https://www.quantamagazine.org/theories-of-everything-mapped-20150803/>). Wiki also has a nice [theoretical physics](<https://en.wikipedia.org/wiki/Theoretical_physics>) and [mathematical physics](https://en.wikipedia.org/wiki/Mathematical_physics) or [https://en.wikipedia.org/wiki/Mathematical_and_theoretical_biology](https://en.wikipedia.org/wiki/Mathematical_and_theoretical_biology), trillion [AI theory of mathematics](https://arxiv.org/abs/2106.10165) (principles of deep learning theory, statistical learning theory), [free energy principle](https://arxiv.org/abs/2201.06387),... Dynamical systems, systems theory,... There's also [nLab](https://ncatlab.org/nlab/show/HomePage) ([math](https://ncatlab.org/nlab/show/mathematics), [physics](https://ncatlab.org/nlab/show/higher+category+theory+and+physics)) but that's mostly magic from the category theory fanatics, which I only want to have as part of my map. Category theory brilliantly allows for connecting individual mathematical universes ([from Math3ma](https://www.math3ma.com/blog/what-is-category-theory-anyway), [from Southwell](https://www.youtube.com/playlist?list=PLCTMeyjMKRkoS699U0OJ3ymr3r01sI08l)). This guy has a nice list of [more specific books on math subfields](https://www.reddit.com/r/math/comments/kqnfn5/suggestions_for_starting_a_personal_library/gi9k4gj/?context=3). I'd love to go through all of it in the tiniest details and learn everything, but I'd need infinite time. ## Written by AI (may include factually incorrect information) #### Map of mathematics 1 # The Gigantic Map of Mathematics Mathematics is a vast, interconnected web of concepts, theories, and applications that span numerous fields and subfields. This map aims to provide a comprehensive overview of the major areas of mathematics, illustrating their interrelations and key components. --- ## 1. **Foundations of Mathematics** - **Logic** - Propositional Logic - Predicate Logic - Modal Logic - Intuitionistic Logic - Proof Theory - Model Theory - Recursion Theory (Computability) - Non-Classical Logics - **Set Theory** - Naive Set Theory - Axiomatic Set Theory (ZFC) - Ordinals and Cardinals - Continuum Hypothesis - Large Cardinals - Descriptive Set Theory - Forcing and Independence Proofs - **Category Theory** - Categories, Functors, Natural Transformations - Limits and Colimits - Adjunctions - Monads and Comonads - Higher Category Theory - Topos Theory --- ## 2. **Algebra** - **Elementary Algebra** - Variables and Expressions - Equations and Inequalities - Polynomials and Factoring - Rational Expressions - **Linear Algebra** - Vectors and Vector Spaces - Matrices and Determinants - Systems of Linear Equations - Eigenvalues and Eigenvectors - Inner Product Spaces - **Abstract Algebra** - Group Theory - Finite Groups - Infinite Groups - Symmetry Groups - Lie Groups - Ring Theory - Commutative Rings - Ideals and Quotients - Noetherian Rings - Field Theory - Field Extensions - Galois Theory - Module Theory - Modules over Rings - Homological Algebra - Algebraic Structures - Quasigroups and Loops - Lattices and Boolean Algebras - **Representation Theory** - Group Representations - Character Theory - Representations of Lie Algebras - **Universal Algebra** - Algebraic Structures and Homomorphisms - Varieties and Equational Classes --- ## 3. **Number Theory** - **Elementary Number Theory** - Divisibility and Primes - Congruences - Diophantine Equations - Arithmetic Functions - **Analytic Number Theory** - Prime Number Theorem - Riemann Zeta Function - L-functions - Modular Forms - **Algebraic Number Theory** - Number Fields - Rings of Integers - Class Field Theory - Arithmetic of Elliptic Curves - **Transcendental Number Theory** - Liouville Numbers - Algebraic Independence - **Computational Number Theory** - Primality Testing - Integer Factorization - Cryptographic Applications --- ## 4. **Geometry** - **Euclidean Geometry** - Points, Lines, and Planes - Angles and Triangles - Circles and Conic Sections - **Non-Euclidean Geometry** - Hyperbolic Geometry - Elliptic Geometry - Spherical Geometry - **Differential Geometry** - Manifolds - Riemannian Geometry - Tensor Calculus - Geodesics and Curvature - **Algebraic Geometry** - Affine and Projective Varieties - Schemes and Sheaves - Intersection Theory - Computational Algebraic Geometry - **Complex Geometry** - Complex Manifolds - Hodge Theory - **Convex Geometry** - Convex Sets and Functions - Brunn-Minkowski Theory - **Discrete and Computational Geometry** - Polyhedra - Tiling and Packing - Voronoi Diagrams - Computational Algorithms --- ## 5. **Topology** - **General Topology (Point-Set Topology)** - Topological Spaces - Continuity and Homeomorphisms - Compactness and Connectedness - Metric Spaces - **Algebraic Topology** - Homotopy Theory - Fundamental Group and Covering Spaces - Homology and Cohomology - Higher Homotopy Groups - Fiber Bundles and Characteristic Classes - **Differential Topology** - Smooth Manifolds - Morse Theory - Transversality - **Geometric Topology** - Knot Theory - 3-Manifolds and 4-Manifolds - Low-Dimensional Topology - **Topological Dynamics** - Dynamical Systems - Ergodic Theory - Topological Entropy --- ## 6. **Analysis** - **Real Analysis** - Sequences and Series - Continuity and Differentiation - Integration (Riemann and Lebesgue) - Measure Theory - Functional Analysis - **Complex Analysis** - Analytic Functions - Cauchy's Theorem and Integral Formula - Laurent Series and Residues - Conformal Mapping - **Functional Analysis** - Banach and Hilbert Spaces - Linear Operators - Spectral Theory - Banach Algebras - **Harmonic Analysis** - Fourier Series and Transforms - Plancherel and Parseval Theorems - Wavelets - **Operator Theory** - Self-Adjoint Operators - Unbounded Operators - C*-Algebras and von Neumann Algebras - **Nonstandard Analysis** - Hyperreal Numbers - Infinitesimals - **Numerical Analysis** - Numerical Solutions to Equations - Numerical Integration and Differentiation - Numerical Linear Algebra - Error Analysis and Stability --- ## 7. **Differential Equations** - **Ordinary Differential Equations (ODEs)** - Linear and Nonlinear ODEs - Existence and Uniqueness Theorems - Stability and Phase Plane Analysis - **Partial Differential Equations (PDEs)** - Classification: Elliptic, Parabolic, Hyperbolic - Methods of Solution - Sobolev Spaces - Applications in Physics and Engineering - **Dynamical Systems** - Continuous and Discrete Systems - Chaos Theory - Bifurcation Theory - Hamiltonian and Lagrangian Mechanics --- ## 8. **Probability and Statistics** - **Probability Theory** - Probability Spaces and Measures - Random Variables and Expectations - Laws of Large Numbers - Central Limit Theorem - Stochastic Processes - Markov Chains - Martingales - Brownian Motion - **Mathematical Statistics** - Statistical Inference - Estimation and Hypothesis Testing - Regression Analysis - Nonparametric Methods - **Statistical Mechanics** - Thermodynamics and Entropy - Phase Transitions - **Information Theory** - Entropy and Information Measures - Coding Theory - Data Compression - **Actuarial Science** - Risk Theory - Life Contingencies - Financial Mathematics --- ## 9. **Discrete Mathematics** - **Combinatorics** - Enumerative Combinatorics - Combinatorial Designs - Graph Theory - Trees and Connectivity - Planarity and Graph Coloring - Network Flows - **Algorithm Theory** - Complexity Classes (P, NP, NP-Complete) - Computational Complexity - Cryptography - **Discrete Structures** - Lattices and Boolean Algebras - Finite State Machines - Automata Theory - **Matroid Theory** - Independence and Circuits - Duality --- ## 10. **Applied Mathematics** - **Mathematical Modeling** - Differential Equations in Modeling - Simulation and Computational Models - **Optimization** - Linear Programming - Nonlinear Optimization - Convex Analysis - Game Theory - **Control Theory** - Feedback Systems - Optimal Control - Stability Theory - **Computational Mathematics** - Numerical Methods - Scientific Computing - Computational Geometry - **Mathematical Physics** - Quantum Mechanics - Schrödinger Equation - Operator Algebras - General Relativity - Differential Geometry in Spacetime - String Theory - Higher-Dimensional Manifolds - **Mathematical Biology** - Population Dynamics - Epidemiology Models - Bioinformatics - **Financial Mathematics** - Stochastic Calculus - Derivatives Pricing - Risk Management --- ## 11. **Interdisciplinary Fields** - **Cryptography** - Public-Key Cryptography - Elliptic Curve Cryptography - Cryptographic Protocols - **Computer Science Theory** - Algorithms and Data Structures - Formal Languages and Automata - Computational Complexity - **Quantum Computing** - Quantum Algorithms - Quantum Information Theory - **Topological Data Analysis** - Persistent Homology - Computational Topology - **Mathematical Psychology** - Psychometrics - Cognitive Modeling --- ## 12. **Advanced and Specialized Topics** - **Homological Algebra** - Chain Complexes - Derived Categories - Ext and Tor Functors - **K-Theory** - Algebraic K-Theory - Topological K-Theory - **Noncommutative Geometry** - Operator Algebras - Quantum Groups - **Higher Category Theory** - ∞-Categories - Derived Algebraic Geometry - **Topos Theory** - Grothendieck Topoi - Sheaf Theory --- ## 13. **Emerging and Modern Areas** - **Artificial Intelligence and Machine Learning** - Statistical Learning Theory - Neural Networks - Optimization in High Dimensions - **Computational Topology** - Algorithms for Topological Computation - Applications in Data Analysis - **Mathematics of Networks** - Network Theory - Complex Systems - **Mathematical Aspects of Quantum Field Theory** - Renormalization - Gauge Theories - Anomalies - **Mirror Symmetry and String Theory** - Calabi-Yau Manifolds - Gromov-Witten Invariants --- ## 14. **Historical and Philosophical Aspects** - **History of Mathematics** - Development of Mathematical Concepts - Biographies of Mathematicians - **Philosophy of Mathematics** - Foundations and Paradoxes - Mathematical Platonism vs. Formalism - Constructivism --- ## 15. **Mathematics Education** - **Pedagogical Methods** - Curriculum Development - Educational Technology - **Mathematical Literacy** - Public Understanding of Mathematics - Outreach and Communication --- This map is by no means exhaustive, but it provides a broad overview of the rich and diverse world of mathematics. Each of these areas can be further subdivided into more specialized topics, and the connections between them often lead to new and exciting fields of study. Mathematics is continually evolving, with new theories and applications emerging as researchers explore the frontiers of knowledge. --- **Note:** Mathematics is highly interconnected. Many concepts in one area rely on or influence those in another. For example, algebraic topology uses tools from algebra to solve topological problems, while functional analysis combines elements of analysis and linear algebra. As you delve deeper into each field, you'll discover these fascinating interdependencies. #### Map of mathematics 2 I. Pure Mathematics A. Algebra 1. Abstract Algebra a. Group Theory - Lie Groups - Representation Theory b. Ring Theory c. Field Theory - Galois Theory d. Homological Algebra - Category Theory - K-Theory 2. Linear Algebra - Vector Spaces - Matrix Theory - Tensor Analysis 3. Number Theory - Analytic Number Theory - Algebraic Number Theory - Diophantine Equations 4. Combinatorics - Graph Theory - Enumerative Combinatorics - Algebraic Combinatorics 5. Algebraic Geometry - Schemes - Sheaf Theory - Cohomology Theories 6. Commutative Algebra - Ideal Theory - Homological Methods 7. Non-Commutative Algebra - Representation Theory - Hopf Algebras - Quantum Groups B. Analysis 1. Real Analysis - Measure Theory - Functional Analysis - Harmonic Analysis 2. Complex Analysis - Riemann Surfaces - Analytic Functions - Complex Dynamics 3. Functional Analysis a. Hilbert Spaces b. Banach Spaces c. Operator Theory d. Spectral Theory 4. Harmonic Analysis a. Fourier Analysis b. Wavelets c. Representation Theory 5. Differential Equations a. Ordinary Differential Equations - Dynamical Systems - Bifurcation Theory - Stability Theory b. Partial Differential Equations - Elliptic PDEs - Parabolic PDEs - Hyperbolic PDEs 6. Probability Theory - Stochastic Processes - Markov Chains - Martingales 7. Statistics - Estimation Theory - Hypothesis Testing - Regression Analysis C. Geometry and Topology 1. Euclidean Geometry 2. Non-Euclidean Geometry a. Hyperbolic Geometry b. Elliptic Geometry 3. Differential Geometry a. Riemannian Geometry b. Symplectic Geometry c. Poisson Geometry d. Kähler Geometry e. Finsler Geometry 4. Algebraic Topology a. Homotopy Theory b. Homology Theory c. Cohomology Theory d. K-Theory 5. Differential Topology a. Morse Theory b. Floer Homology c. Contact Topology 6. Geometric Topology a. Knot Theory b. 3-Manifolds c. 4-Manifolds 7. Lie Groups and Lie Algebras - Representation Theory - Structure Theory - Enveloping Algebras D. Logic and Foundations 1. Set Theory - Axiomatic Set Theory - Descriptive Set Theory - Large Cardinals 2. Model Theory - First-Order Logic - Stability Theory - o-Minimality 3. Proof Theory - Constructive Mathematics - Type Theory - Homotopy Type Theory 4. Computability Theory - Recursive Functions - Turing Machines - Complexity Theory 5. Category Theory - Topos Theory - Homological Algebra - Higher Category Theory II. Applied Mathematics A. Classical Mechanics 1. Newton's Laws 2. Lagrangian Mechanics - Euler-Lagrange Equations - Hamilton's Principle - Noether's Theorem 3. Hamiltonian Mechanics - Hamilton's Equations - Poisson Brackets - Liouville's Theorem B. Continuum Mechanics 1. Fluid Dynamics - Navier-Stokes Equations - Turbulence - Boundary Layer Theory 2. Elasticity Theory - Hooke's Law - Stress-Strain Relations - Plate and Shell Theory 3. Plasticity Theory - Yield Criteria - Flow Rules - Hardening Laws C. Relativity 1. Special Relativity - Lorentz Transformations - Minkowski Spacetime - Relativistic Mechanics 2. General Relativity - Einstein Field Equations - Schwarzschild Solution - Cosmological Models D. Quantum Mechanics 1. Schrödinger Equation - Wave Functions - Operators - Eigenvalues and Eigenfunctions 2. Heisenberg Uncertainty Principle 3. Dirac Equation - Spinors - Relativistic Quantum Mechanics 4. Quantum Field Theory - Feynman Diagrams - Renormalization - Gauge Theories E. Statistical Mechanics 1. Thermodynamics - Laws of Thermodynamics - Entropy - Free Energy 2. Kinetic Theory - Boltzmann Equation - H-Theorem - Transport Phenomena 3. Statistical Ensembles - Microcanonical Ensemble - Canonical Ensemble - Grand Canonical Ensemble F. Numerical Analysis 1. Finite Difference Methods 2. Finite Element Methods 3. Spectral Methods 4. Monte Carlo Methods 5. Optimization Algorithms G. Optimization 1. Linear Programming - Simplex Method - Duality Theory - Interior Point Methods 2. Nonlinear Programming - Gradient Methods - Newton's Method - Conjugate Gradient Methods 3. Variational Calculus - Euler-Lagrange Equations - Hamilton's Principle - Noether's Theorem 4. Optimal Control Theory - Pontryagin's Maximum Principle - Dynamic Programming - Hamilton-Jacobi-Bellman Equation H. Dynamical Systems 1. Chaos Theory - Lyapunov Exponents - Strange Attractors - Fractal Dimensions 2. Bifurcation Theory - Hopf Bifurcation - Saddle-Node Bifurcation - Pitchfork Bifurcation 3. Ergodic Theory - Ergodic Theorems - Mixing - Entropy III. Interdisciplinary Fields A. Mathematical Physics 1. Quantum Field Theory - Gauge Theories - Renormalization - Conformal Field Theory 2. String Theory - Superstring Theory - M-Theory - Dualities 3. Conformal Field Theory - Virasoro Algebra - Kac-Moody Algebras - Vertex Operator Algebras 4. Integrable Systems - Soliton Equations - Inverse Scattering Transform - Quantum Integrable Systems B. Mathematical Biology 1. Population Dynamics - Lotka-Volterra Equations - Predator-Prey Models - Evolutionary Game Theory 2. Epidemiology - SIR Models - Network Models - Stochastic Epidemic Models 3. Neuroscience - Hodgkin-Huxley Model - FitzHugh-Nagumo Model - Neural Networks 4. Bioinformatics - Sequence Alignment - Phylogenetics - Gene Expression Analysis C. Mathematical Finance 1. Stochastic Calculus - Itô Calculus - Stochastic Differential Equations - Feynman-Kac Formula 2. Option Pricing Theory - Black-Scholes Model - Binomial Option Pricing Model - Lévy Processes 3. Portfolio Optimization - Mean-Variance Analysis - Capital Asset Pricing Model - Arbitrage Pricing Theory 4. Risk Management - Value at Risk - Expected Shortfall - Copula Theory D. Mathematical Economics 1. Game Theory - Nash Equilibrium - Evolutionary Game Theory - Mechanism Design 2. Econometrics - Linear Regression - Time Series Analysis - Panel Data Analysis 3. General Equilibrium Theory - Arrow-Debreu Model - Welfare Theorems - Existence and Uniqueness 4. Mechanism Design - Auction Theory - Matching Theory - Social Choice Theory E. Mathematical Computer Science 1. Algorithms and Complexity - Analysis of Algorithms - Computational Complexity Theory - Approximation Algorithms 2. Cryptography - Public Key Cryptography - Elliptic Curve Cryptography - Quantum Cryptography 3. Machine Learning - Supervised Learning - Unsupervised Learning - Reinforcement Learning 4. Quantum Computing - Quantum Algorithms - Quantum Error Correction - Quantum Complexity Theory #### Map of proofs Mathematics serves as the foundational framework for understanding and advancing scientific knowledge. Below is an extensive map highlighting significant mathematical proofs and their profound impact across various scientific disciplines. --- #### *Group Theory* - **Lagrange's Theorem** - *States that the order of a subgroup divides the order of the entire group.* - **Applications:** Symmetry operations in quantum mechanics, crystallography, and particle physics. #### *Ring Theory* - **Fundamental Theorems of Ring Homomorphisms** - **Applications:** Algebraic coding theory, cryptography, and the study of polynomial rings in computer algebra systems. #### *Fundamental Theorem of Calculus* - **Links differentiation and integration, showing they are inverse processes.** - **Applications:** Formulates the basis for continuous change in physics, engineering, and economics. #### *Fourier Analysis* - **Convergence of Fourier Series** - **Applications:** Signal processing, heat transfer, quantum mechanics, and acoustics. #### *Differential Equations* - **Existence and Uniqueness Theorems (Picard-Lindelöf Theorem)** - **Applications:** Modeling natural phenomena in physics, biology, and engineering systems. #### *Pythagorean Theorem* - **Relates the lengths of the sides of a right-angled triangle.** - **Applications:** Fundamental in physics for resolving vector components and in engineering design. #### *Gauss-Bonnet Theorem* - **Connects geometry and topology by relating curvature to topology.** - **Applications:** General relativity and the study of spacetime curvature. #### *Euler Characteristic* - **A topological invariant representing a space's shape or structure.** - **Applications:** Network topology, molecular chemistry, and condensed matter physics. #### *Fermat's Last Theorem* - **States that there are no three positive integers \( a \), \( b \), and \( c \) that satisfy \( a^n + b^n = c^n \) for \( n > 2 \).** - **Applications:** Influences cryptographic algorithms and computational number theory. #### *Prime Number Theorem* - **Describes the asymptotic distribution of prime numbers.** - **Applications:** Cryptography, particularly in RSA encryption. #### *Central Limit Theorem* - **States that the sum of a large number of independent random variables tends toward a normal distribution.** - **Applications:** Statistical mechanics, error analysis in measurements, and financial modeling. #### *Law of Large Numbers* - **Describes the result of performing the same experiment many times.** - **Applications:** Quantum physics interpretations, reliability engineering, and risk assessment. #### *Gödel's Incompleteness Theorems* - **Demonstrate inherent limitations in every formal axiomatic system.** - **Applications:** Foundations of mathematics, computer science, and theories of computation. #### *Turing's Halting Problem* - **Proves that there is no general algorithm to solve the halting problem for all possible program-input pairs.** - **Applications:** Limits of computation, development of decidability theory, and complexity classes. #### *Graph Theory* - **Euler's Formula for Planar Graphs** - **Applications:** Network analysis, circuit design, and molecular biology. #### *NP-Completeness Theory* - **Cook-Levin Theorem** - **Applications:** Cryptography, algorithm design, and computational complexity. #### *Spectral Theorem* - **States that every normal operator on a finite-dimensional complex vector space is diagonalizable.** - **Applications:** Quantum mechanics (observables), vibration analysis, and image processing. #### *Cayley-Hamilton Theorem* - **Every square matrix satisfies its own characteristic equation.** - **Applications:** Control theory, electrical engineering, and system dynamics. #### *Hahn-Banach Theorem* - **Extension of bounded linear functionals.** - **Applications:** Quantum physics, optimization problems, and economics. #### *Banach-Tarski Paradox* - **A ball can be decomposed and reassembled into two identical copies.** - **Applications:** Theoretical implications in measure theory and the concept of infinity. #### *Ricci Flow and the Poincaré Conjecture* - **Grigori Perelman's proof using Ricci flow to solve a century-old problem.** - **Applications:** Topology of three-dimensional spaces, cosmology, and geometric analysis. #### *Riemannian Geometry* - **Studies smooth manifolds with Riemannian metrics.** - **Applications:** General relativity and the modeling of spacetime. #### *Cauchy's Integral Theorem* - **Fundamental theorem in complex analysis about the integration of holomorphic functions.** - **Applications:** Fluid dynamics, electromagnetic theory, and aerodynamics. #### *Riemann Hypothesis (Unproven)* - **Speculates about the zeros of the Riemann zeta function.** - **Potential Applications:** Prime number distribution, quantum chaos, and cryptography. #### *Cantor's Diagonal Argument* - **Demonstrates that real numbers are uncountable.** - **Applications:** Understanding different sizes of infinity, foundations of mathematics. #### *Zermelo-Fraenkel Axioms with the Axiom of Choice (ZFC)* - **Provide a standard form of axiomatic set theory.** - **Applications:** Basis for most of modern mathematics. #### *Noether's Theorem* - **States that every differentiable symmetry of the action of a physical system corresponds to a conservation law.** - **Applications:** Conservation of energy, momentum, and charge in physics. #### *Yang-Mills Theory* - **Framework for understanding the behavior of elementary particles using non-Abelian gauge theories.** - **Applications:** Standard Model of particle physics and quantum chromodynamics. #### *Yoneda Lemma* - **Fundamental result relating functors and natural transformations.** - **Applications:** Abstract algebra, topology, and theoretical computer science. #### *Lagrange Multipliers* - **Technique for finding local maxima and minima of functions subject to equality constraints.** - **Applications:** Economics (utility maximization), engineering design, and physics. #### *Newton-Raphson Method* - **Iterative method for finding successively better approximations to the roots of a real-valued function.** - **Applications:** Numerical solutions in engineering, physics simulations, and computational finance. --- This comprehensive map illustrates the profound connections between mathematical proofs and scientific advancements. Each theorem not only represents a pinnacle of mathematical thought but also serves as a crucial tool for scientists and engineers exploring the complexities of the natural world. #### Map of mathematics 2 1. Foundations a. Logic and Set Theory - Propositional Logic - First-Order Logic - Axiomatic Set Theory - Zermelo-Fraenkel Set Theory b. Mathematical Reasoning and Proof - Deductive Reasoning - Inductive Reasoning - Proof Techniques (Direct, Contradiction, Induction) c. Number Systems - Natural Numbers - Integers - Rational Numbers - Real Numbers - Complex Numbers 2. Algebra a. Elementary Algebra - Variables and Expressions - Equations and Inequalities - Systems of Equations b. Polynomials and Rational Expressions - Polynomial Arithmetic - Factoring - Rational Expressions and Functions c. Equations and Inequalities - Linear Equations and Inequalities - Quadratic Equations - Polynomial Equations - Exponential and Logarithmic Equations - Trigonometric Equations d. Functions - Function Notation and Graphing - Transformations of Functions - Inverse Functions - Polynomial Functions - Rational Functions - Exponential and Logarithmic Functions - Trigonometric Functions e. Matrices and Determinants - Matrix Operations - Determinants - Eigenvalues and Eigenvectors - Matrix Decompositions f. Abstract Algebra - Group Theory - Ring Theory - Field Theory - Galois Theory - Representation Theory g. Linear Algebra - Vector Spaces - Linear Transformations - Inner Product Spaces - Orthogonality - Least Squares Approximation h. Boolean Algebra - Logical Operations - Truth Tables - Minimization of Boolean Functions - Applications in Computer Science 3. Geometry a. Euclidean Geometry - Points, Lines, and Planes - Angles and Triangles - Congruence and Similarity - Circles and Spheres - Polygons and Polyhedra b. Analytic Geometry - Coordinate Systems - Distance and Midpoint Formulas - Equations of Lines and Planes - Conic Sections - Parametric Equations - Vector Geometry c. Non-Euclidean Geometries - Hyperbolic Geometry - Elliptic Geometry - Projective Geometry d. Trigonometry - Trigonometric Ratios - Trigonometric Identities - Solving Triangles - Trigonometric Functions and Graphs e. Differential Geometry - Curves and Surfaces - Tangent Spaces - Riemannian Geometry - Geodesics - Curvature f. Algebraic Geometry - Affine Varieties - Projective Varieties - Schemes - Cohomology - Intersection Theory g. Topology - Topological Spaces - Continuity and Homeomorphisms - Connectedness and Compactness - Homotopy and Fundamental Groups - Homology and Cohomology - Knot Theory 4. Calculus and Analysis a. Limits and Continuity - Limit Definition and Properties - Continuity - Intermediate Value Theorem b. Derivatives and Differentiation - Definition of the Derivative - Differentiation Rules - Implicit Differentiation - Higher-Order Derivatives - Applications of Derivatives c. Integrals and Integration - Definite and Indefinite Integrals - Fundamental Theorem of Calculus - Integration Techniques - Improper Integrals - Applications of Integration d. Sequences and Series - Convergence of Sequences - Series and Convergence Tests - Power Series - Taylor Series e. Multivariable Calculus - Partial Derivatives - Gradients, Divergence, and Curl - Multiple Integrals - Change of Variables - Line and Surface Integrals f. Vector Calculus - Vector Fields - Green's Theorem - Stokes' Theorem - Divergence Theorem g. Differential Equations - First-Order Differential Equations - Second-Order Linear Differential Equations - Laplace Transforms - Systems of Differential Equations - Partial Differential Equations h. Real Analysis - The Real Number System - Sequences and Series of Functions - Metric Spaces - Continuity and Differentiability - Riemann Integration - Lebesgue Integration i. Complex Analysis - Complex Numbers and Functions - Analytic Functions - Cauchy's Theorem and Integral Formula - Laurent Series and Residues - Conformal Mappings j. Functional Analysis - Normed Vector Spaces - Hilbert Spaces - Banach Spaces - Linear Operators - Spectral Theory k. Measure Theory - Measurable Spaces and Functions - Measures and Integrals - Lp Spaces - Radon-Nikodym Theorem - Fubini's Theorem 5. Probability and Statistics a. Probability Theory - Probability Axioms - Conditional Probability - Independence - Random Variables - Expectation and Variance - Moment Generating Functions b. Combinatorics - Permutations and Combinations - Binomial Coefficients - Inclusion-Exclusion Principle - Generating Functions - Recurrence Relations c. Discrete Probability Distributions - Bernoulli and Binomial Distributions - Geometric and Negative Binomial Distributions - Hypergeometric Distribution - Poisson Distribution d. Continuous Probability Distributions - Uniform Distribution - Normal Distribution - Exponential Distribution - Gamma and Beta Distributions - Chi-Square, t, and F Distributions e. Statistical Inference - Point Estimation - Interval Estimation - Hypothesis Testing - Likelihood Ratio Tests - Nonparametric Methods f. Regression Analysis - Simple Linear Regression - Multiple Linear Regression - Nonlinear Regression - Logistic Regression - Time Series Analysis g. Bayesian Statistics - Bayes' Theorem - Prior and Posterior Distributions - Bayesian Inference - Markov Chain Monte Carlo Methods - Bayesian Networks h. Stochastic Processes - Markov Chains - Poisson Processes - Brownian Motion - Martingales - Stochastic Calculus 6. Discrete Mathematics a. Graph Theory - Graphs and Digraphs - Connectivity - Trees and Spanning Trees - Eulerian and Hamiltonian Graphs - Planar Graphs - Graph Coloring - Matchings and Coverings b. Combinatorics - Pigeonhole Principle - Ramsey Theory - Latin Squares - Block Designs - Polya Enumeration c. Number Theory - Divisibility and Prime Numbers - Congruences - Diophantine Equations - Quadratic Reciprocity - Continued Fractions - Elliptic Curves d. Cryptography - Classical Cryptosystems - Public-Key Cryptography - RSA Cryptosystem - Discrete Logarithms - Elliptic Curve Cryptography e. Game Theory - Two-Person Zero-Sum Games - Nash Equilibrium - Cooperative Games - Evolutionary Game Theory - Mechanism Design f. Computational Complexity - Time Complexity - P and NP Problems - NP-Completeness - Approximation Algorithms - Randomized Algorithms 7. Applied Mathematics a. Mathematical Physics - Classical Mechanics - Lagrangian and Hamiltonian Mechanics - Noether's Theorem - Rigid Body Dynamics - Celestial Mechanics - Quantum Mechanics - Schrödinger Equation - Hilbert Spaces and Operators - Angular Momentum and Spin - Perturbation Theory - Quantum Field Theory - Relativity - Special Relativity - General Relativity - Differential Geometry of Spacetime - Cosmology - Thermodynamics - Laws of Thermodynamics - Statistical Mechanics - Ensemble Theory - Phase Transitions - Non-Equilibrium Thermodynamics b. Fluid Dynamics - Navier-Stokes Equations - Inviscid Flow - Viscous Flow - Turbulence - Boundary Layer Theory - Computational Fluid Dynamics c. Partial Differential Equations - Classification of PDEs - Separation of Variables - Fourier Series and Transforms - Green's Functions - Finite Difference and Finite Element Methods d. Numerical Analysis - Error Analysis - Interpolation and Approximation - Numerical Differentiation and Integration - Numerical Linear Algebra - Numerical Solution of ODEs and PDEs e. Optimization - Linear Programming - Nonlinear Programming - Convex Optimization - Variational Methods - Optimal Control Theory f. Control Theory - Feedback Control Systems - Stability Analysis - Controllability and Observability - Optimal Control - Robust Control g. Mathematical Biology - Population Dynamics - Epidemiology - Biochemical Kinetics - Physiological Modeling - Neural Networks h. Mathematical Finance - Portfolio Theory - Options Pricing - Black-Scholes Model - Stochastic Calculus in Finance - Risk Management i. Operations Research - Linear Programming - Network Flow Problems - Integer Programming - Dynamic Programming - Queuing Theory - Inventory Theory - Decision Analysis 8. Foundations of Mathematics a. Mathematical Logic - Propositional and First-Order Logic - Completeness and Incompleteness Theorems - Model Theory - Recursion Theory b. Set Theory - Axioms of Set Theory - Ordinal and Cardinal Numbers - Continuum Hypothesis - Large Cardinals - Forcing and Independence Results c. Category Theory - Categories and Functors - Natural Transformations - Limits and Colimits - Adjoint Functors - Topoi d. Proof Theory - Formal Systems - Sequent Calculus - Cut Elimination - Proof Normalization - Ordinal Analysis e. Computability Theory - Turing Machines - Recursive Functions - Church-Turing Thesis - Unsolvable Problems - Degrees of Unsolvability f. Philosophy of Mathematics - Mathematical Platonism - Intuitionism and Constructivism - Formalism - Structuralism - Nominalism 9. History of Mathematics a. Ancient Mathematics - Egyptian and Babylonian Mathematics - Greek Mathematics (Pythagoras, Euclid, Archimedes) - Chinese Mathematics - Indian Mathematics b. Medieval Mathematics - Islamic Mathematics - European Mathematics - Fibonacci and the Hindu-Arabic Numeral System c. Early Modern Mathematics - The Renaissance and the Rise of Algebra - The Development of Analytic Geometry - The Invention of Calculus (Newton and Leibniz) d. 19th Century Mathematics - The Rigorous Foundations of Analysis - The Emergence of Non-Euclidean Geometry - The Development of Abstract Algebra - The Birth of Set Theory and Mathematical Logic e. 20th Century Mathematics - The Crisis in the Foundations of Mathematics - The Rise of Topology and Functional Analysis - The Emergence of Computer Science and Discrete Mathematics - The Development of Chaos Theory and Fractal Geometry f. Contemporary Mathematics - New Developments in Pure Mathematics (e.g., Langlands Program, Monstrous Moonshine) - Advances in Applied Mathematics (e.g., Compressed Sensing, Topological Data Analysis) - The Impact of Computers on Mathematical Research - Open Problems and Conjectures (e.g., Riemann Hypothesis, P vs. NP) ##### Map of all mathematics expanded 1. Foundations a. Logic and Set Theory - Propositional Logic - Syntax and Semantics - Logical Connectives - Truth Tables - Tautologies and Contradictions - Logical Equivalence - Deduction and Inference Rules - First-Order Logic - Syntax and Semantics - Quantifiers - Models and Interpretations - Validity and Satisfiability - Completeness and Compactness Theorems - Löwenheim-Skolem Theorems - Axiomatic Set Theory - Cantor's Naive Set Theory - Russell's Paradox - Zermelo-Fraenkel Set Theory (ZFC) - Axiom of Choice - Ordinal and Cardinal Numbers - Transfinite Induction and Recursion - Alternative Set Theories - Von Neumann-Bernays-Gödel Set Theory (NBG) - Morse-Kelley Set Theory (MK) - New Foundations (NF) b. Mathematical Reasoning and Proof - Deductive Reasoning - Modus Ponens and Modus Tollens - Hypothetical Syllogism - Disjunctive Syllogism - Constructive Dilemma - Reductio ad Absurdum - Inductive Reasoning - Generalization - Analogy - Causal Inference - Probablistic Reasoning - Bayesian Inference - Proof Techniques - Direct Proof - Proof by Contradiction - Proof by Contrapositive - Proof by Induction - Mathematical Induction - Strong Induction - Transfinite Induction - Proof by Cases - Nonconstructive Proofs c. Number Systems - Natural Numbers - Peano Axioms - Arithmetic Operations - Ordering and Inequalities - Prime Numbers and Factorization - Diophantine Equations - Integers - Construction from Natural Numbers - Arithmetic Operations - Divisibility and Greatest Common Divisors - Fundamental Theorem of Arithmetic - Linear Diophantine Equations - Rational Numbers - Construction as Quotients of Integers - Arithmetic Operations - Decimal Expansions - Continued Fractions - Farey Sequences - Real Numbers - Dedekind Cuts - Cauchy Sequences - Completeness and Archimedes' Axiom - Decimal Expansions - Cantor's Theorem and Uncountability - Complex Numbers - Definition and Arithmetic Operations - Polar Form and Euler's Formula - Roots of Unity - Fundamental Theorem of Algebra - Riemann Sphere - Quaternions - Definition and Arithmetic Operations - Conjugation and Norm - Quaternionic Exponential and Logarithm - Rotations in 3D Space - Octonions - Definition and Arithmetic Operations - Non-Associativity and Alternative Algebra - Automorphism Group and Triality - Applications in String Theory and Exceptional Lie Groups - p-adic Numbers - p-adic Valuations and Absolute Values - Hensel's Lemma - p-adic Expansions - p-adic Analysis and Integration - Applications in Number Theory and Cryptography - Surreal Numbers - Conway's Construction - Arithmetic Operations and Ordering - Transfinite Numbers and Infinitesimals - Surreal Analysis and Integration 2. Algebra a. Elementary Algebra - Variables and Expressions - Constants and Variables - Algebraic Expressions - Evaluating Expressions - Simplifying Expressions - Translating Verbal Descriptions into Algebraic Expressions - Equations and Inequalities - Solving Linear Equations - Solving Linear Inequalities - Absolute Value Equations and Inequalities - Literal Equations - Formulas and Applications - Systems of Equations - Solving Systems of Linear Equations - Graphical Method - Substitution Method - Elimination Method - Solving Systems of Nonlinear Equations - Applications and Word Problems b. Polynomials and Rational Expressions - Polynomial Arithmetic - Addition and Subtraction of Polynomials - Multiplication of Polynomials - Division of Polynomials (Long Division and Synthetic Division) - Polynomial Identities - Factoring Polynomials - Greatest Common Factor - Grouping - Difference of Squares - Sum and Difference of Cubes - General Trinomials - Factoring by Substitution - Rational Expressions and Functions - Simplifying Rational Expressions - Multiplication and Division of Rational Expressions - Addition and Subtraction of Rational Expressions - Complex Fractions - Solving Rational Equations - Applications and Word Problems c. Equations and Inequalities - Linear Equations and Inequalities - Solving Linear Equations - Solving Linear Inequalities - Absolute Value Equations and Inequalities - Compound Inequalities - Linear Equations and Inequalities in Two Variables - Quadratic Equations - Solving by Factoring - Completing the Square - Quadratic Formula - Discriminant and Nature of Roots - Solving Quadratic Inequalities - Polynomial Equations - Rational Root Theorem - Descartes' Rule of Signs - Bounds on Real Roots - Solving Cubic and Quartic Equations - Fundamental Theorem of Algebra - Exponential and Logarithmic Equations - Properties of Exponential and Logarithmic Functions - Solving Exponential Equations - Solving Logarithmic Equations - Applications and Word Problems - Trigonometric Equations - Basic Trigonometric Equations - Equations Involving Trigonometric Identities - Multiple-Angle Equations - Parametric Equations - Applications and Word Problems d. Functions - Function Notation and Graphing - Definition of a Function - Function Notation - Domain and Range - Graphing Functions - Vertical Line Test - Transformations of Functions - Vertical and Horizontal Shifts - Reflections - Stretches and Compressions - Combining Transformations - Applications and Modeling - Inverse Functions - Definition and Properties - Finding Inverse Functions - Graphing Inverse Functions - One-to-One Functions - Applications and Word Problems - Polynomial Functions - Quadratic Functions - Cubic Functions - Higher-Degree Polynomial Functions - Graphing Polynomial Functions - Polynomial Inequalities - Rational Functions - Graphing Rational Functions - Asymptotes (Vertical, Horizontal, and Oblique) - Holes and Points of Discontinuity - Solving Rational Inequalities - Applications and Word Problems - Exponential and Logarithmic Functions - Exponential Functions and Graphs - Logarithmic Functions and Graphs - Properties of Logarithms - Exponential and Logarithmic Equations - Applications (Growth and Decay, Compound Interest, etc.) - Trigonometric Functions - Radian Measure - Unit Circle and Trigonometric Functions - Graphs of Trigonometric Functions - Inverse Trigonometric Functions - Trigonometric Identities - Applications (Harmonic Motion, Waves, etc.) e. Matrices and Determinants - Matrix Operations - Matrix Addition and Subtraction - Scalar Multiplication - Matrix Multiplication - Transpose of a Matrix - Matrix Powers - Determinants - Definition and Properties - Cofactor Expansion - Laplace Expansion - Cramer's Rule - Applications (Area, Volume, etc.) - Inverse Matrices - Definition and Properties - Finding Inverse Matrices - Gaussian Elimination - Gauss-Jordan Elimination - Applications (Systems of Linear Equations, Cryptography, etc.) - Matrix Factorizations - LU Decomposition - QR Decomposition - Singular Value Decomposition (SVD) - Eigendecomposition - Applications (Least Squares, Principal Component Analysis, etc.) - Matrix Equations - Linear Matrix Equations - Sylvester Equation - Lyapunov Equation - Riccati Equation - Applications (Control Theory, Signal Processing, etc.) f. Abstract Algebra - Group Theory - Definition and Axioms - Examples of Groups - Subgroups and Lagrange's Theorem - Cosets and Quotient Groups - Homomorphisms and Isomorphisms - Normal Subgroups and Factor Groups - Fundamental Theorem of Finite Abelian Groups - Sylow Theorems - Applications (Symmetry, Cryptography, Coding Theory, etc.) - Ring Theory - Definition and Axioms - Examples of Rings - Subrings and Ideals - Quotient Rings - Homomorphisms and Isomorphisms - Prime and Maximal Ideals - Euclidean Domains and Principal Ideal Domains - Unique Factorization Domains - Applications (Number Theory, Algebraic Geometry, etc.) - Field Theory - Definition and Axioms - Examples of Fields - Field Extensions - Algebraic and Transcendental Extensions - Splitting Fields - Finite Fields - Applications (Coding Theory, Cryptography, etc.) - Galois Theory - Fundamental Theorem of Galois Theory - Galois Groups - Solvability by Radicals - Insolvability of the Quintic - Applications (Constructible Numbers, Origami, etc.) - Module Theory - Definition and Axioms - Examples of Modules - Submodules and Quotient Modules - Homomorphisms and Isomorphisms - Free Modules and Projective Modules - Injective Modules and Flat Modules - Tensor Products - Applications (Representation Theory, Homological Algebra, etc.) - Representation Theory - Group Representations - Character Theory - Irreducible Representations - Schur's Lemma - Decomposition of Representations - Induced Representations - Applications (Quantum Mechanics, Harmonic Analysis, etc.) - Lie Algebras - Definition and Axioms - Examples of Lie Algebras - Subalgebras and Ideals - Homomorphisms and Isomorphisms - Solvable and Nilpotent Lie Algebras - Semisimple Lie Algebras and Root Systems - Representations of Lie Algebras - Applications (Differential Geometry, Quantum Field Theory, etc.) g. Linear Algebra - Vector Spaces - Definition and Axioms - Examples of Vector Spaces - Subspaces - Linear Combinations and Spans - Linear Independence and Dependence - Bases and Dimension - Coordinates and Change of Basis - Linear Transformations - Definition and Properties - Kernel and Image - Matrix Representations - Composition of Linear Transformations - Invertible Linear Transformations - Similarity and Diagonalization - Applications (Computer Graphics, Quantum Mechanics, etc.) - Eigenvalues and Eigenvectors - Definition and Properties - Characteristic Polynomial - Eigenspaces and Geometric Multiplicity - Diagonalization and Spectral Theorem - Cayley-Hamilton Theorem - Applications (Dynamical Systems, Markov Chains, etc.) - Inner Product Spaces - Definition and Axioms - Examples of Inner Product Spaces - Norm and Distance - Orthogonality and Orthonormal Bases - Gram-Schmidt Orthogonalization - Orthogonal Complements and Projections - Applications (Quantum Mechanics, Signal Processing, etc.) - Singular Value Decomposition (SVD) - Definition and Properties - Existence and Uniqueness - Compact SVD - Truncated SVD - Applications (Data Compression, Recommender Systems, etc.) - Tensor Algebra - Definition and Properties - Tensor Products - Symmetric and Antisymmetric Tensors - Tensor Fields - Covariant and Contravariant Tensors - Applications (Continuum Mechanics, General Relativity, etc.) h. Boolean Algebra - Logical Operations - Conjunction (AND) - Disjunction (OR) - Negation (NOT) - Implication and Equivalence - Exclusive OR (XOR) - Truth Tables - Construction and Interpretation - Tautologies and Contradictions - Logical Equivalence - Functional Completeness - Canonical Forms - Minterms and Maxterms - Sum-of-Products (SOP) Form - Product-of-Sums (POS) Form - Conversion between Forms - Minimization of Boolean Functions - Karnaugh Maps - Quine-McCluskey Algorithm - Espresso Algorithm - applications (Circuit Design, Switching Theory, etc.) - Boolean Algebra and Set Theory - Isomorphism between Boolean Algebras and Sets - Venn Diagrams - De Morgan's Laws - Power Set and Cartesian Product - Applications in Computer Science - Digital Logic Design - Switching Circuits - Finite State Machines - Coding Theory and Error Correction 3. Geometry a. Euclidean Geometry - Points, Lines, and Planes - Definitions and Axioms - Incidence and Betweenness - Segments and Rays - Halfplanes and Angles - Vertical Angles and Linear Pairs - Triangles - Classification of Triangles - Congruence and Similarity - Triangle Inequality - Medians, Altitudes, and Angle Bisectors - Pythagorean Theorem and Its Converse - Special Triangles (Isosceles, Equilateral, 30-60-90, 45-45-90) - Polygons - Definition and Properties - Classification of Polygons - Diagonals and Convexity - Interior and Exterior Angles - Regular Polygons and Their Properties - Area Formulas for Polygons - Circles - Definition and Properties - Central Angles and Inscribed Angles - Chords, Secants, and Tangents - Inscribed and Circumscribed Polygons - Arc Length and Sector Area - Equation of a Circle - Solid Geometry - Lines and Planes in Space - Dihedral and Polyhedral Angles - Polyhedra (Prisms, Pyramids, Platonic Solids) - Cylinders, Cones, and Spheres - Surface Area and Volume Formulas - Constructions and Loci - Basic Constructions with Straightedge and Compass - Constructible Numbers and Lengths - Geometric Loci and Their Equations - Applications (Linkages, Origami, etc.) b. Analytic Geometry - Coordinate Systems - Cartesian Coordinate System - Polar Coordinate System - Cylindrical and Spherical Coordinate Systems - Conversion between Coordinate Systems - Distance and Midpoint Formulas - Distance between Two Points - Midpoint of a Line Segment - Distance from a Point to a Line - Perpendicular Distance between Lines - Equations of Lines and Planes - Slope-Intercept Form - Point-Slope Form - General Form - Parametric Equations - Vector Equations - Angles between Lines and Planes - Distances between Points, Lines, and Planes - Conic Sections - Circle - Parabola - Ellipse - Hyperbola - Eccentricity and Focal Properties - Tangents and Normals - Quadric Surfaces - Ellipsoid - Hyperboloid of One and Two Sheets - Paraboloid (Elliptic and Hyperbolic) - Cone and Cylinder - Canonical Forms and Classification - Coordinate Transformations - Translation - Rotation - Reflection - Scaling - Shear - Composite Transformations - Applications (Computer Graphics, Robotics, etc.) c. Non-Euclidean Geometries - Hyperbolic Geometry - Hilbert's Axioms and Saccheri Quadrilaterals - Models of Hyperbolic Geometry (Poincaré Disk, Upper Half-Plane) - Hyperbolic Lines and Angles - Hyperbolic Trigonometry - Hyperbolic Area and Gauss-Bonnet Theorem - Applications (Special Relativity, Complex Analysis, etc.) - Elliptic Geometry - Spherical Geometry - Great Circles and Geodesics - Spherical Trigonometry - Spherical Excess and Area Formula - Antipodal Points and Identification - Applications (Navigation, Astronomy, etc.) - Projective Geometry - Projective Plane and Projective Space - Homogeneous Coordinates - Duality and Principle of Duality - Cross-Ratio and Harmonic Conjugates - Desargues' Theorem and Pappus' Theorem - Projective Transformations - Applications (Perspective Drawing, Camera Models, etc.) d. Trigonometry - Trigonometric Ratios - Sine, Cosine, and Tangent - Reciprocal Ratios (Cosecant, Secant, Cotangent) - Trigonometric Functions of Special Angles - Reference Angles and Angle Measure - Trigonometric Identities - Fundamental Identities - Cofunction Identities - Even-Odd Identities - Sum and Difference Formulas - Double-Angle and Half-Angle Formulas - Power-Reduction Formulas - Product-to-Sum and Sum-to-Product Formulas - Inverse Trigonometric Functions - Definitions and Domains - Graphing Inverse Trigonometric Functions - Identities Involving Inverse Trigonometric Functions - Solving Equations with Inverse Trigonometric Functions - Applications of Trigonometry - Right Triangle Problems - Oblique Triangle Problems (Law of Sines, Law of Cosines) - Harmonic Motion and Waves - Navigation and Surveying - Optics and Acoustics e. Differential Geometry - Curves in the Plane and Space - Parametric Curves - Arc Length and Reparameterization - Curvature and Torsion - Frenet-Serret Formulas - Osculating Circle and Osculating Plane - Surfaces - Parametric Surfaces - Tangent Plane and Normal Vector - First and Second Fundamental Forms - Principal Curvatures and Gaussian Curvature - Minimal Surfaces - Geodesics and Geodesic Curvature - Riemannian Geometry - Riemannian Metric and Riemannian Manifolds - Christoffel Symbols and Levi-Civita Connection - Parallel Transport and Geodesic Equation - Riemann Curvature Tensor - Ricci Curvature and Scalar Curvature - Einstein Field Equations and General Relativity - Symplectic Geometry - Symplectic Manifolds and Symplectic Forms - Hamiltonian Mechanics - Lagrangian Submanifolds - Moment Maps and Symplectic Reduction - Poisson Structures and Integrable Systems - Applications (Classical Mechanics, Quantum Mechanics, etc.) f. Algebraic Geometry - Affine Varieties - Algebraic Sets and Zariski Topology - Hilbert's Nullstellensatz - Regular Functions and Coordinate Rings - Morphisms of Affine Varieties - Dimension and Tangent Spaces - Projective Varieties - Projective Space and Homogeneous Coordinates - Projective Closure and Projective Varieties - Veronese Embedding and Segre Embedding - Grassmannians and Flag Varieties - Blow-ups and Resolutions - Schemes - Affine Schemes and Spec Construction - Morphisms of Schemes - Fiber Products and Base Change - Separated and Proper Morphisms - Sheaves and Cohomology - Etale and Smooth Morphisms - Intersection Theory - Divisors and Line Bundles - Intersection Multiplicity - Bezout's Theorem - Riemann-Roch Theorem for Curves - Hirzebruch-Riemann-Roch Theorem - Chern Classes and Todd Classes - Elliptic Curves and Abelian Varieties - Weierstrass Equations and Group Law - Torsion Points and Mordell-Weil Theorem - Isogenies and Endomorphism Rings - Jacobians and Abel-Jacobi Map - Modular Forms and Modular Curves - Applications (Cryptography, Number Theory, etc.) g. Topology - Topological Spaces - Open and Closed Sets - Basis and Subbasis - Closure, Interior, and Boundary - Limit Points and Isolated Points - Hausdorff and Normal Spaces - Continuity and Homeomorphisms - Continuous Functions - Homeomorphisms and Topological Invariants - Topological Properties (Connectedness, Compactness, etc.) - Separation Axioms (T0, T1, T2, etc.) - Urysohn's Lemma and Tietze Extension Theorem - Fundamental Group and Covering Spaces - Homotopy and Homotopy Equivalence - Fundamental Group and Homomorphisms - Covering Spaces and Lifting Properties - Universal Covering Space - Applications (Knot Theory, 3-Manifolds, etc.) - Homology and Cohomology - Simplicial and Singular Homology - Homology Groups and Betti Numbers - Euler Characteristic - Cohomology Groups and Cup Product - Poincaré Duality and Lefschetz Duality - de Rham Cohomology and Hodge Theory - Algebraic Topology - Fundamental Group and Higher Homotopy Groups - Fibrations and Homotopy Fiber - Spectral Sequences and Serre Spectral Sequence - Eilenberg-MacLane Spaces and Cohomology Operations - Characteristic Classes (Stiefel-Whitney, Chern, Pontryagin) - K-Theory and Index Theorems - Low-Dimensional Topology - Surfaces and Classification Theorem - Seifert Surfaces and Knots - Knot Invariants (Linking Number, Alexander Polynomial, etc.) - 3-Manifolds and Heegaard Splittings - Dehn Surgery and Lickorish-Wallace Theorem - Geometrization Theorem and Thurston's Work 4. Calculus and Analysis a. Limits and Continuity - Limit of a Function - Definition and Intuitive Understanding - One-Sided Limits - Infinite Limits and Limits at Infinity - Limit Laws and Computation Techniques - Squeeze Theorem and Indeterminate Forms - Continuity - Definition and Types of Discontinuities - Properties of Continuous Functions - Intermediate Value Theorem - Extreme Value Theorem - Uniform Continuity - Sequences and Series - Convergence of Sequences - Monotone Sequences and Bounded Sequences - Cauchy Sequences and Completeness - Convergence of Series (Geometric, p-series, etc.) - Convergence Tests (Ratio, Root, Integral, etc.) - Absolute and Conditional Convergence b. Derivatives and Differentiation - Definition of the Derivative - Tangent Lines and Rates of Change - Definition as a Limit - One-Sided Derivatives - Higher-Order Derivatives - Notation (Leibniz, Lagrange, Newton) - Differentiation Rules - Constant Rule and Power Rule - Product Rule and Quotient Rule - Chain Rule - Implicit Differentiation - Logarithmic Differentiation - Derivatives of Inverse Functions - Applications of Derivatives - Related Rates - Linear Approximation and Differentials - Newton's Method - L'Hôpital's Rule and Indeterminate Forms - Optimization Problems - Monotonicity and Concavity - Curve Sketching c. Integrals and Integration - Indefinite Integrals - Antiderivatives and Integration Constants - Basic Integration Rules (Power Rule, Constant Multiple Rule, etc.) - Integration by Substitution (u-Substitution) - Integration by Parts - Trigonometric Integrals - Integration of Rational Functions (Partial Fractions) - Definite Integrals - Riemann Sums and Definition of the Definite Integral - Properties of Definite Integrals - Fundamental Theorem of Calculus - Techniques for Evaluating Definite Integrals - Improper Integrals (Unbounded Intervals, Unbounded Functions) - Applications of Integration - Area between Curves - Volumes of Solids (Slicing, Shells, Disks/Washers) - Arc Length and Surface Area - Work, Hydrostatic Pressure, and Centroids - Probability Density Functions and Cumulative Distribution Functions d. Sequences and Series - Sequences - Definition and Notation - Limit of a Sequence - Bounded, Monotone, and Cauchy Sequences - Convergence Theorems - Subsequences and Bolzano-Weierstrass Theorem - Series - Definition and Notation - Convergence and Divergence of Series - Geometric Series and p-Series - Convergence Tests (Ratio, Root, Integral, Comparison, etc.) - Absolute and Conditional Convergence - Rearrangements of Series - Power Series - Definition and Radius of Convergence - Differentiation and Integration of Power Series - Taylor Series and Maclaurin Series - Binomial Series - Applications (Approximation, Solving Differential Equations, etc.) e. Multivariable Calculus - Functions of Several Variables - Graphs and Level Curves/Surfaces - Limits and Continuity - Partial Derivatives - Tangent Planes and Linear Approximation - Directional Derivatives and Gradients - Higher-Order Partial Derivatives - Multiple Integrals - Double Integrals and Iterated Integrals - Change of Variables and Jacobians - Triple Integrals and Applications (Volume, Mass, Moments, etc.) - Fubini's Theorem - Improper Multiple Integrals - Vector Calculus - Parametric Curves and Arc Length - Vector Fields and Line Integrals - Conservative Vector Fields and Independence of Path - Green's Theorem - Surface Integrals and Flux - Divergence Theorem (Gauss' Theorem) - Stokes' Theorem f. Differential Equations - First-Order Differential Equations - Separable Equations - Linear Equations - Exact Equations and Integrating Factors - Bernoulli Equations - Homogeneous Equations - Applications (Population Growth, Cooling/Heating, Circuits, etc.) - Second-Order Linear Differential Equations - Homogeneous Equations with Constant Coefficients - Nonhomogeneous Equations (Method of Undetermined Coefficients, Variation of Parameters) - Cauchy-Euler Equations - Power Series Solutions - Applications (Harmonic Oscillators, RLC Circuits, Beams, etc.) - Laplace Transforms - Definition and Properties - Inverse Laplace Transforms - Solving Initial Value Problems - Convolutions and Integral Equations - Discontinuous and Periodic Functions - Applications (Control Theory, Signal Processing, etc.) - Systems of Differential Equations - Linear Systems and Matrices - Eigenvalues and Eigenvectors - Phase Portraits and Stability - Nonlinear Systems and Linearization - Applications (Predator-Prey Models, Chemical Reactions, etc.) - Partial Differential Equations - Classification (Elliptic, Parabolic, Hyperbolic) - Separation of Variables - Fourier Series and Sturm-Liouville Problems - Heat Equation - Wave Equation - Laplace's Equation and Harmonic Functions - Applications (Diffusion, Vibrations, Electrostatics, etc.) g. Real Analysis - The Real Number System - Axioms and Properties - Completeness and Dedekind Cuts - Supremum and Infimum - Archimedean Property - Density of Rationals and Irrationals - Sequences and Series - Convergence and Divergence - Cauchy Sequences and Completeness - Monotone Convergence Theorem - Bolzano-Weierstrass Theorem - Limsup and Liminf - Double Sequences and Iterated Limits - Continuity and Uniform Continuity - Continuity and Properties - Intermediate Value Theorem - Extreme Value Theorem - Uniform Continuity and Heine-Cantor Theorem - Lipschitz Continuity - Continuous Functions on Compact Sets - Differentiation - Definition and Properties - Mean Value Theorems (Rolle's, Lagrange, Cauchy) - L'Hôpital's Rule - Taylor's Theorem and Taylor Series - Implicit and Inverse Function Theorems - Differentiability and Continuity - Riemann Integration - Partitions and Riemann Sums - Riemann Integrability - Properties of Riemann Integrals - Fundamental Theorem of Calculus - Improper Integrals - Lebesgue's Criterion for Riemann Integrability - Sequences and Series of Functions - Pointwise and Uniform Convergence - Uniform Convergence and Continuity - Uniform Convergence and Integration - Uniform Convergence and Differentiation - Power Series and Radius of Convergence - Metric Spaces - Definition and Examples - Open and Closed Sets - Convergence and Completeness - Continuity and Homeomorphisms - Compact Sets and Heine-Borel Theorem - Connected Sets and Continuous Functions h. Complex Analysis - Complex Numbers and Functions - Algebraic and Geometric Representations - Polar Form and Euler's Formula - Powers and Roots - Complex Functions and Mappings - Limits and Continuity - Analytic Functions - Differentiability and Cauchy-Riemann Equations - Harmonic Functions - Exponential, Trigonometric, and Logarithmic Functions - Complex Integration and Contour Integrals - Cauchy's Integral Theorem and Formula - Liouville's Theorem and Fundamental Theorem of Algebra - Series Representations - Taylor Series and Maclaurin Series - Laurent Series - Zeros and Poles - Residue Theorem and Residue Calculus - Evaluation of Real Integrals - Conformal Mappings - Definition and Properties - Möbius Transformations - Schwarz-Christoffel Mappings - Riemann Mapping Theorem - Applications (Fluid Dynamics, Electrostatics, etc.) - Harmonic Functions - Definition and Properties - Poisson Integral Formula - Dirichlet Problem - Maximum Principle - Harnack's Inequality - Applications (Potential Theory, Heat Conduction, etc.) i. Functional Analysis - Normed Vector Spaces - Definition and Examples - Banach Spaces and Completeness - Equivalent Norms - Finite-Dimensional Normed Spaces - Compactness and Finite Dimension - Inner Product Spaces - Definition and Examples - Hilbert Spaces and Completeness - Orthonormal Bases and Parseval's Identity - Orthogonal Complements and Projections - Riesz Representation Theorem - Adjoint Operators and Self-Adjoint Operators - Linear Operators - Bounded and Unbounded Operators - Operator Norms and Continuity - Compact Operators - Spectrum and Resolvent - Fredholm Alternative - Spectral Theorem for Compact Self-Adjoint Operators - Banach Algebras - Definition and Examples - Spectrum and Spectral Radius - Holomorphic Functional Calculus - Gelfand Transform - C*-Algebras and von Neumann Algebras - Topological Vector Spaces - Definition and Examples - Locally Convex Spaces - Weak Topologies - Hahn-Banach Theorem and Separating Hyperplanes - Krein-Milman Theorem - Banach-Alaoglu Theorem j. Measure Theory - Measurable Spaces and Functions - Sigma-Algebras and Measurable Sets - Generated Sigma-Algebras and Borel Sets - Measurable Functions - Simple Functions and Approximation - Littlewood's Three Principles - Measures and Integrals - Definition and Properties of Measures - Lebesgue Measure and Lebesgue Integral - Monotone Convergence Theorem - Fatou's Lemma and Dominated Convergence Theorem - Fubini's Theorem and Tonelli's Theorem - Absolute Continuity and Radon-Nikodym Theorem - Lp Spaces - Definition and Examples - Hölder's Inequality and Minkowski's Inequality - Completeness and Banach Space Structure - Dual Spaces and Riesz Representation Theorem - Hilbert Spaces and Riesz-Fischer Theorem - Signed Measures and Complex Measures - Hahn Decomposition Theorem - Jordan Decomposition Theorem - Total Variation and Absolute Continuity - Lebesgue Decomposition Theorem - Radon-Nikodym Derivatives - Differentiation and Integration - Functions of Bounded Variation - Absolutely Continuous Functions - Fundamental Theorem of Calculus for Lebesgue Integrals - Lebesgue Differentiation Theorem - Vitali Covering Lemma - Applications (Probability Theory, Fourier Analysis, etc.) 5. Probability and Statistics a. Probability Theory - Probability Axioms - Sample Spaces and Events - Axioms of Probability - Probabiloity of Complements and Unions - Inclusion-Exclusion Principle - Continuity of Probability Measures - Conditional Probability and Independence - Definition of Conditional Probability - Multiplication Rule and Total Probability - Bayes' Theorem - Independent Events - Conditional Independence - Applications (Medical Testing, Legal Cases, etc.) - Random Variables - Discrete and Continuous Random Variables - Cumulative Distribution Functions - Probability Mass Functions and Probability Density Functions - Joint, Marginal, and Conditional Distributions - Functions of Random Variables - Transformations of Random Variables - Expectation and Variance - Definition and Properties of Expectation - Linearity of Expectation - Variance and Standard Deviation - Covariance and Correlation - Conditional Expectation - Moment Generating Functions and Characteristic Functions - Limit Theorems - Markov's Inequality and Chebyshev's Inequality - Weak Law of Large Numbers - Strong Law of Large Numbers - Central Limit Theorem - Applications (Polling, Quality Control, etc.) b. Combinatorics - Counting Principles - Multiplication Principle - Addition Principle - Pigeonhole Principle - Inclusion-Exclusion Principle - Binomial Coefficients and Pascal's Triangle - Permutations and Combinations - Definition and Notation - Permutations with and without Repetition - Combinations with and without Repetition - Binomial Theorem and Combinatorial Proofs - Catalan Numbers and Stirling Numbers - Generating Functions - Ordinary Generating Functions - Exponential Generating Functions - Solving Recurrence Relations - Partition Functions - Polya's Enumeration Theorem - Graph Theory and Enumeration - Paths and Cycles - Trees and Spanning Trees - Matchings and Perfect Matchings - Coloring Problems - Ramsey Theory and Extremal Graph Theory c. Discrete Probability Distributions - Bernoulli and Binomial Distributions - Bernoulli Trials and Success Probabilities - Probability Mass Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Normal Approximation to Binomial - Applications (Coin Flips, Defective Items, etc.) - Geometric and Negative Binomial Distributions - Probability Mass Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Memoryless Property and Waiting Times - Applications (Runs of Success, Quality Control, etc.) - Hypergeometric Distribution - Probability Mass Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Relationship to Binomial Distribution - Applications (Card Games, Sampling without Replacement, etc.) - Poisson Distribution - Probability Mass Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Relationship to Binomial Distribution - Applications (Rare Events, Traffic Flow, etc.) d. Continuous Probability Distributions - Uniform Distribution - Probability Density Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Applications (Random Number Generation, Buffon's Needle, etc.) - Normal Distribution - Probability Density Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Standard Normal Distribution and Z-scores - Central Limit Theorem and Normal Approximation - Applications (IQ Scores, Measurement Errors, etc.) - Exponential Distribution - Probability Density Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Memoryless Property and Waiting Times - Applications (Radioactive Decay, Queuing Theory, etc.) - Gamma and Beta Distributions - Probability Density Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Relationship to Other Distributions - Applications (Waiting Times, Bayesian Inference, etc.) - Chi-Square, t, and F Distributions - Probability Density Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Relationship to Normal Distribution - Applications (Hypothesis Testing, Confidence Intervals, ANOVA, etc.) e. Statistical Inference - Point Estimation - Estimators and Their Properties (Unbiasedness, Consistency, Efficiency) - Method of Moments Estimation - Maximum Likelihood Estimation - Bayesian Estimation - Sufficient Statistic and Rao-Blackwell Theorem - Interval Estimation - Confidence Intervals - Pivotal Quantities and Confidence Intervals - Large-Sample Confidence Intervals - Bayesian Credible Intervals - Applications (Opinion Polls, Quality Control, etc.) - Hypothesis Testing - Null and Alternative Hypotheses - Type I and Type II Errors - Test Statistics and p-Values - Power of a Test and Sample Size Determination - Neyman-Pearson Lemma and Likelihood Ratio Tests - Applications (Clinical Trials, A/B Testing, etc.) - Nonparametric Methods - Sign Test and Wilcoxon Signed-Rank Test - Mann-Whitney U Test - Kruskal-Wallis Test - Spearman's Rank Correlation Coefficient - Kolmogorov-Smirnov Test - Bootstrap and Permutation Tests f. Regression Analysis - Simple Linear Regression - Least Squares Estimation - Correlation Coefficient and Coefficient of Determination - Hypothesis Tests and Confidence Intervals for Regression Parameters - Prediction and Prediction Intervals - Assumptions and Diagnostics (Residual Plots, QQ Plots, etc.) - Multiple Linear Regression - Matrix Notation and Estimation - Partial Correlation and Adjusted R-Squared - Multicollinearity and Variable Selection - Dummy Variables and Categorical Predictors - Interactions and Polynomial Regression - Nonlinear Regression - Transformations and Linearization - Nonlinear Least Squares Estimation - Gauss-Newton and Levenberg-Marquardt Algorithms - Model Selection and Akaike Information Criterion (AIC) - Applications (Growth Curves, Dose-Response Models, etc.) - Logistic Regression - Odds Ratios and Logit Transformation - Maximum Likelihood Estimation - Interpretation of Coefficients - Receiver Operating Characteristic (ROC) Curves - Applications (Medical Diagnosis, Credit Scoring, etc.) - Time Series Analysis - Stationarity and Autocorrelation - Autoregressive (AR) Models - Moving Average (MA) Models - Autoregressive Moving Average (ARMA) Models - Autoregressive Integrated Moving Average (ARIMA) Models - Forecasting and Model Selection g. Bayesian Statistics - Bayes' Theorem - Prior and Posterior Probabilities - Likelihood Functions - Marginal Likelihood and Model Evidence - Bayesian Updating and Sequential Learning - Prior and Posterior Distributions - Conjugate Priors and Exponential Families - Noninformative Priors - Posterior Summaries (Mean, Median, Credible Intervals) - Posterior Predictive Distribution - Bayesian Hypothesis Testing and Bayes Factors - Bayesian Inference - Bayesian Point Estimation - Bayesian Interval Estimation - Bayesian Hypothesis Testing - Bayesian Model Selection and Bayesian Information Criterion (BIC) - Empirical Bayes Methods - Markov Chain Monte Carlo (MCMC) Methods - Metropolis-Hastings Algorithm - Gibbs Sampling - Convergence Diagnostics and Effective Sample Size - Hamiltonian Monte Carlo and No-U-Turn Sampler - Variational Inference and Mean Field Approximation - Bayesian Networks - Directed Acyclic Graphs and Conditional Independence - Inference in Bayesian Networks - Parameter Learning and Structure Learning - Hidden Markov Models and Dynamic Bayesian Networks - Applications (Medical Diagnosis, Spam Filtering, etc.) h. Stochastic Processes - Markov Chains - State Space and Transition Probabilities - Chapman-Kolmogorov Equations - Stationary Distribution and Limiting Behavior - Ergodicity and Mixing Times - Absorption Probabilities and First Passage Times - Applications (PageRank, Gambler's Ruin, etc.) - Poisson Processes - Counting Processes and Arrival Times - Homogeneous and Nonhomogeneous Poisson Processes - Superposition and Thinning of Poisson Processes - Conditional Poisson Processes and Cox Processes - Applications (Queuing Theory, Reliability, etc.) - Brownian Motion - Definition and Properties - Wiener Process and Diffusion Processes - Stochastic Integrals and Ito's Lemma - Stochastic Differential Equations - Geometric Brownian Motion and Black-Scholes Model - Applications (Financial Mathematics, Physics, etc.) - Martingales - Definition and Examples - Martingale Transforms and Martingale Inequalities - Optional Stopping Theorem - Martingale Convergence Theorems - Martingale Representation Theorem - Applications (Gambling, Optimal Stopping, etc.) - Stochastic Calculus - Stochastic Integrals and Ito's Lemma - Stochastic Differential Equations - Feynman-Kac Formula and Kolmogorov Equations - Girsanov's Theorem and Change of Measure - Applications (Mathematical Finance, Filtering Theory, etc.) 6. Discrete Mathematics a. Graph Theory - Graphs and Digraphs - Definition and Terminology - Adjacency and Incidence Matrices - Subgraphs and Graph Isomorphism - Degree Sequences and Handshaking Lemma - Bipartite Graphs and Complete Graphs - Connectivity - Paths and Cycles - Connected Components and Strongly Connected Components - Bridges and Articulation Points - Menger's Theorem and Disjoint Paths - Graph Reconstruction and Whitney's Theorem - Trees and Spanning Trees - Definition and Properties - Minimum Spanning Trees (Kruskal's and Prim's Algorithms) - Cayley's Formula and Prüfer Code - Arboricity and Nash-Williams Theorem - Applications (Network Design, Phylogenetic Trees, etc.) - Matchings and Coverings - Definition and Examples - Hall's Marriage Theorem - König's Theorem and Edge Coloring - Vertex Covers and Independent Sets - Perfect Matchings and Tutte's Theorem - Graph Coloring - Chromatic Number and Chromatic Polynomial - Greedy Coloring and Brooks' Theorem - Four Color Theorem and Heawood Conjecture - Graph Homomorphisms and Hadwiger's Conjecture - Applications (Scheduling, Register Allocation, etc.) - Planar Graphs - Definition and Characterization - Euler's Formula and Kuratowski's Theorem - Dual Graphs and Geometric Duality - Graph Minors and Wagner's Theorem - Applications (Circuit Design, Facility Location, etc.) b. Enumerative Combinatorics - Generating Functions - Ordinary Generating Functions - Exponential Generating Functions - Partition Functions and Gaussian Polynomials - Dirichlet Generating Functions - Applications (Fibonacci Numbers, Catalan Numbers, etc.) - Recurrence Relations - Linear Recurrences and Characteristic Polynomials - Solving Recurrences using Generating Functions - Catalan Numbers and Motzkin Numbers - q-Analogs and q-Binomial Coefficients - Applications (Lattice Paths, Permutation Patterns, etc.) - Polya Enumeration - Burnside's Lemma and Cauchy-Frobenius Lemma - Cycle Index and Pattern Inventory - Polya's Enumeration Formula - de Bruijn Sequences and Universal Cycles - Applications (Chemical Isomers, Necklace Problem, etc.) - Combinatorial Designs - Block Designs and Incidence Matrices - Balanced Incomplete Block Designs (BIBD) - Steiner Systems and Finite Projective Planes - Latin Squares and Orthogonal Arrays - Hadamard Matrices and Combinatorial Matrix Theory - Extremal Combinatorics - Ramsey Numbers and Ramsey's Theorem - Turán's Theorem and Forbidden Subgraphs - Szemerédi's Regularity Lemma - Erdős-Ko-Rado Theorem and Intersecting Families - Combinatorial Nullstellensatz and Additive Combinatorics c. Number Theory - Divisibility and Prime Numbers - Division Algorithm and Greatest Common Divisor - Fundamental Theorem of Arithmetic - Distribution of Prime Numbers and Prime Number Theorem - Primality Testing and Factorization Algorithms - Twin Primes and Goldbach's Conjecture - Congruences - Modular Arithmetic and Residue Classes - Linear Congruences and Chinese Remainder Theorem - Fermat's Little Theorem and Euler's Theorem - Quadratic Residues and Legendre Symbol - Primitive Roots and Discrete Logarithms - Diophantine Equations - Linear Diophantine Equations and Bezout's Identity - Pell's Equation and Continued Fractions - Pythagorean Triples and Fermat's Last Theorem - Elliptic Curves and Mordell's Theorem - Diophantine Approximation and Transcendental Numbers - Quadratic Forms and Lattices - Binary Quadratic Forms and Discriminants - Representation of Integers by Quadratic Forms - Minkowski's Convex Body Theorem - Lattice Basis Reduction and LLL Algorithm - Cryptographic Applications and Lattice-Based Cryptography - Algebraic Number Theory - Number Fields and Ring of Integers - Ideals and Unique Factorization - Norm, Trace, and Discriminant - Class Groups and Class Number Formula - Cyclotomic Fields and Fermat's Last Theorem - Zeta Functions and L-Functions d. Cryptography - Classical Cryptography - Substitution Ciphers and Frequency Analysis - Transposition Ciphers and Permutations - Vigenère Cipher and Kasiski Examination - One-Time Pad and Perfect Secrecy - Cryptanalysis Techniques and Ciphertext-Only Attacks - Symmetric-Key Cryptography - Block Ciphers and Stream Ciphers - Data Encryption Standard (DES) and Advanced Encryption Standard (AES) - Modes of Operation (ECB, CBC, CTR, etc.) - Message Authentication Codes (MAC) and Hash Functions - Key Exchange Protocols and Diffie-Hellman Protocol - Public-Key Cryptography - RSA Cryptosystem and Modular Exponentiation - Elliptic Curve Cryptography (ECC) - Digital Signatures and Public Key Infrastructure (PKI) - Homomorphic Encryption and Fully Homomorphic Encryption - Post-Quantum Cryptography and Lattice-Based Cryptography - Cryptographic Protocols - Zero-Knowledge Proofs and Interactive Proof Systems - Oblivious Transfer and Secure Multiparty Computation - Blind Signatures and Digital Cash - Threshold Cryptography and Secret Sharing - Blockchain and Cryptocurrency (Bitcoin, Ethereum, etc.) - Cryptanalysis and Security - Linear and Differential Cryptanalysis - Side-Channel Attacks and Fault Attacks - Quantum Cryptanalysis and Shor's Algorithm - Provable Security and Random Oracle Model - Cryptographic Hardness Assumptions and Complexity Theory e. Game Theory - Two-Person Zero-Sum Games - Payoff Matrices and Minimax Theorem - Solving Games using Linear Programming - Mixed Strategies and Nash Equilibrium - Poker and Bluffing - Matching Pennies and Rock-Paper-Scissors - Non-Cooperative Games - Normal Form and Extensive Form Games - Dominant Strategies and Iterated Elimination - Nash Equilibrium and Best Response Dynamics - Cournot and Bertrand Competition - Prisoner's Dilemma and Tragedy of the Commons - Cooperative Games - Coalitional Games and Characteristic Functions - Core and Shapley Value - Bargaining Theory and Axiomatic Bargaining Solutions - Matching Markets and Stable Matchings - Voting Games and Power Indices - Evolutionary Game Theory - Replicator Dynamics and Evolutionarily Stable Strategies - Hawk-Dove Game and War of Attrition - Evolutionary Stability and Convergence - Evolutionary Graphs and Spatial Games - Applications in Biology, Economics, and Social Sciences - Mechanism Design - Social Choice Theory and Arrow's Impossibility Theorem - Vickrey-Clarke-Groves (VCG) Mechanism - Revelation Principle and Incentive Compatibility - Auctions and Optimal Auction Design - Algorithmic Game Theory and Price of Anarchy f. Computational Complexity - Time Complexity - Big-O Notation and Asymptotic Analysis - Polynomial-Time Algorithms and Tractability - Exponential-Time Algorithms and Intractability - Time Hierarchy Theorem and Deterministic Time Complexity - Randomized Algorithms and Probabilistic Time Complexity - Space Complexity - Turing Machines and Space Complexity Classes - PSPACE and NPSPACE - Savitch's Theorem and PSPACE = NPSPACE - Space Hierarchy Theorem and Deterministic Space Complexity - Randomized Space Complexity and RL and BPL - Complexity Classes - P and NP - co-NP and Complementation - NP-Completeness and Cook-Levin Theorem - EXPTIME and NEXPTIME - Polynomial Hierarchy and PH = PSPACE - Reducibility and Completeness - Many-One Reducibility and Turing Reducibility - NP-Completeness and Karp Reductions - PSPACE-Completeness and Quantified Boolean Formulas - #P-Completeness and Counting Problems - Unconditional Lower Bounds and Relativization - Advanced Topics - Interactive Proofs and IP = PSPACE - Probabilistically Checkable Proofs and PCP Theorem - Complexity of Approximation and Hardness of Approximation - Parameterized Complexity and Fixed-Parameter Tractability - Quantum Complexity and BQP and QMA 7. Applied Mathematics a. Mathematical Physics - Classical Mechanics - Lagrangian Mechanics - Generalized Coordinates and Lagrangian - Euler-Lagrange Equations - Symmetries and Conservation Laws (Noether's Theorem) - Variational Principles and Hamilton's Principle - Applications (Pendulum, Double Pendulum, etc.) - Hamiltonian Mechanics - Phase Space and Hamiltonian Function - Hamilton's Equations - Canonical Transformations and Generating Functions - Poisson Brackets and Liouville's Theorem - Applications (Harmonic Oscillator, Kepler Problem, etc.) - Rigid Body Dynamics - Angular Velocity and Inertia Tensor - Euler's Equations and Euler Angles - Precession and Nutation - Stability and Rotating Frames - Applications (Gyroscopes, Spacecraft Attitude Control, etc.) - Celestial Mechanics - Two-Body Problem and Kepler's Laws - Orbital Elements and Orbital Perturbations - N-Body Problem and Restricted Three-Body Problem - Lagrange Points and Stability - Applications (Satellite Orbits, Interplanetary Missions, etc.) - Quantum Mechanics - Schrödinger Equation - Wavefunction and Probability Interpretation - Time-Dependent and Time-Independent Schrödinger Equations - Stationary States and Energy Eigenvalues - Boundary Conditions and Normalization - Applications (Particle in a Box, Harmonic Oscillator, etc.) - Hilbert Spaces and Operators - Hilbert Space Formulation of Quantum Mechanics - Linear Operators and Observables - Hermitian Operators and Spectral Theorem - Commutators and Uncertainty Principle - Spectral Measures and Projection-Valued Measures - Angular Momentum and Spin - Orbital Angular Momentum and Spherical Harmonics - Spin Angular Momentum and Pauli Matrices - Addition of Angular Momentum and Clebsch-Gordan Coefficients - Wigner-Eckart Theorem and Selection Rules - Applications (Hydrogen Atom, Zeeman Effect, etc.) - Perturbation Theory - Time-Independent Perturbation Theory (Rayleigh-Schrödinger) - Degenerate Perturbation Theory - Time-Dependent Perturbation Theory (Dyson Series) - Fermi's Golden Rule and Transition Rates - Applications (Stark Effect, Fine Structure, etc.) - Quantum Field Theory - Second Quantization and Fock Space - Creation and Annihilation Operators - Canonical Commutation Relations and Anticommutation Relations - Feynman Diagrams and Perturbative Expansion - Renormalization and Regularization - Applications (Quantum Electrodynamics, Standard Model, etc.) - Relativity - Special Relativity - Lorentz Transformations and Minkowski Spacetime - Relativistic Kinematics and Four-Vectors - Relativistic Dynamics and Four-Force - Relativistic Energy and Mass-Energy Equivalence - Applications (Particle Colliders, GPS, etc.) - General Relativity - Principle of Equivalence and Curved Spacetime - Metric Tensor and Geodesics - Einstein Field Equations - Schwarzschild Solution and Black Holes - Cosmological Models and Friedmann Equations - Differential Geometry of Spacetime - Manifolds and Coordinate Charts - Tangent Spaces and Cotangent Spaces - Covariant Derivative and Parallel Transport - Curvature Tensor and Ricci Tensor - Variational Formulation and Einstein-Hilbert Action - Cosmology - Hubble's Law and Expansion of the Universe - Big Bang Theory and Cosmic Microwave Background - Dark Matter and Dark Energy - Inflation and Primordial Perturbations - Large-Scale Structure and Cosmic Web - Thermodynamics - Laws of Thermodynamics - Zeroth Law and Thermal Equilibrium - First Law and Conservation of Energy - Second Law and Entropy - Third Law and Absolute Zero Temperature - Applications (Heat Engines, Refrigerators, etc.) - Statistical Mechanics - Microstates and Macrostates - Boltzmann Distribution and Partition Function - Canonical Ensemble and Grand Canonical Ensemble - Fermi-Dirac and Bose-Einstein Statistics - Applications (Ideal Gas, Blackbody Radiation, etc.) - Ensemble Theory - Microcanonical Ensemble and Ergodicity - Canonical Ensemble and Free Energy - Grand Canonical Ensemble and Chemical Potential - Gibbs Ensemble and Equivalence of Ensembles - Applications (Phase Transitions, Critical Phenomena, etc.) - Phase Transitions - First-Order and Second-Order Phase Transitions - Landau Theory and Order Parameters - Critical Exponents and Universality Classes - Renormalization Group and Fixed Points - Applications (Ising Model, Superconductivity, etc.) - Non-Equilibrium Thermodynamics - Linear Response Theory and Green-Kubo Relations - Onsager Reciprocal Relations and Fluctuation-Dissipation Theorem - Nonlinear Dynamics and Pattern Formation - Stochastic Thermodynamics and Fluctuation Theorems - Applications (Transport Phenomena, Biological Systems, etc.) b. Fluid Dynamics - Navier-Stokes Equations - Conservation of Mass and Continuity Equation - Conservation of Momentum and Momentum Equation - Conservation of Energy and Energy Equation - Boundary Conditions and Initial Conditions - Dimensionless Numbers (Reynolds, Mach, Prandtl, etc.) - Inviscid Flow - Euler Equations and Bernoulli's Principle - Potential Flow Theory and Stream Function - Circulation and Vorticity - Kelvin's Circulation Theorem and Helmholtz Theorems - Applications (Airfoil Theory, Conformal Mapping, etc.) - Viscous Flow - Boundary Layer Theory and Prandtl's Equations - Laminar and Turbulent Boundary Layers - Blasius Solution and Falkner-Skan Equation - Separation and Stall - Applications (Drag Reduction, Heat Transfer, etc.) - Turbulence - Reynolds-Averaged Navier-Stokes (RANS) Equations - Turbulence Models (k-ε, k-ω, etc.) - Large Eddy Simulation (LES) and Subgrid-Scale Models - Direct Numerical Simulation (DNS) and Kolmogorov Microscales - Applications (Atmospheric Boundary Layer, Jet Engines, etc.) - Computational Fluid Dynamics (CFD) - Finite Difference Methods - Finite Volume Methods - Finite Element Methods - Spectral Methods - Mesh Generation and Adaptive Mesh Refinement - Applications (Aerodynamics, Hydrodynamics, etc.) c. Partial Differential Equations (PDEs) - Classification of PDEs - Linear and Nonlinear PDEs - Elliptic, Parabolic, and Hyperbolic PDEs - Order and Characteristics - Well-Posed Problems and Existence of Solutions - Separation of Variables - Sturm-Liouville Problems and Eigenfunction Expansions - Fourier Series and Generalized Fourier Series - Laplace's Equation and Harmonic Functions - Applications (Vibrating Strings, Heat Conduction, etc.) - Fourier Transforms and Integral Transforms - Fourier Transform and Inverse Fourier Transform - Convolution and Parseval's Theorem - Laplace Transform and Inverse Laplace Transform - Applications (Signal Processing, Control Theory, etc.) - Green's Functions - Definition and Properties of Green's Functions - Dirac Delta Function and Fundamental Solutions - Boundary Value Problems and Integral Equations - Applications (Electrostatics, Quantum Mechanics, etc.) - Variational Methods - Weak Formulations and Test Functions - Galerkin Method and Ritz Method - Finite Element Method and Basis Functions - Applications (Structural Mechanics, Fluid Dynamics, etc.) d. Numerical Analysis - Error Analysis - Truncation Error and Rounding Error - Absolute Error and Relative Error - Stability and Conditioning - Convergence and Order of Convergence - Interpolation and Approximation - Polynomial Interpolation and Lagrange Interpolation - Spline Interpolation and Bezier Curves - Least Squares Approximation and Orthogonal Polynomials - Rational Approximation and Pade Approximants - Applications (Curve Fitting, Data Compression, etc.) - Numerical Differentiation and Integration - Finite Difference Formulas - Richardson Extrapolation and Romberg Integration - Gaussian Quadrature and Orthogonal Polynomials - Adaptive Quadrature and Singularities - Applications (Numerical Solutions of ODEs and PDEs, etc.) - Numerical Linear Algebra - Gaussian Elimination and LU Decomposition - Cholesky Decomposition and Positive Definite Matrices - QR Decomposition and Least Squares Problems - Singular Value Decomposition and Pseudoinverse - Iterative Methods (Jacobi, Gauss-Seidel, Conjugate Gradient, etc.) - Numerical Solutions of ODEs and PDEs - Euler Methods and Runge-Kutta Methods - Multistep Methods and Predictor-Corrector Methods - Finite Difference Methods for PDEs - Finite Element Methods for PDEs - Spectral Methods and Pseudospectral Methods e. Optimization - Linear Programming - Formulation of Linear Programs - Simplex Method and Tableau Form - Duality Theory and Dual Simplex Method - Sensitivity Analysis and Shadow Prices - Applications (Resource Allocation, Transportation, etc.) - Nonlinear Programming - Unconstrained Optimization and Gradient Methods - Constrained Optimization and Lagrange Multipliers - Karush-Kuhn-Tucker (KKT) Conditions - Quadratic Programming and Second-Order Conditions - Applications (Portfolio Optimization, Control Systems, etc.) - Convex Optimization - Convex Sets and Convex Functions - Jensen's Inequality and Epigraphs - Convex Duality and Fenchel Conjugate - Subgradients and Optimality Conditions - Applications (Machine Learning, Signal Processing, etc.) - Integer Programming - Formulation of Integer Programs - Branch-and-Bound Method - Cutting Plane Methods - Lagrangian Relaxation and Benders Decomposition - Applications (Scheduling, Network Design, etc.) - Stochastic Optimization - Stochastic Programming and Recourse Problems - Chance Constraints and Probabilistic Constraints - Markov Decision Processes and Dynamic Programming - Reinforcement Learning and Q-Learning - Applications (Inventory Control, Financial Engineering, etc.) f. Control Theory - Linear Systems - State-Space Representation and Transfer Functions - Controllability and Observability - Stability Analysis and Routh-Hurwitz Criterion - Pole Placement and State Feedback - Observers and Output Feedback - Optimal Control - Variational Approach and Euler-Lagrange Equations - Pontryagin's Maximum Principle - Dynamic Programming and Hamilton-Jacobi-Bellman Equation - Linear Quadratic Regulator (LQR) and Riccati Equation - Applications (Robotics, Aerospace, etc.) - Robust Control - Uncertainty Modeling and Robustness Analysis - H-infinity Optimal Control - Structured Singular Value (μ) and μ-Synthesis - Linear Matrix Inequalities (LMIs) and Convex Optimization - Applications (Process Control, Automotive Systems, etc.) - Nonlinear Control - Lyapunov Stability Theory - Feedback Linearization and Input-Output Linearization - Sliding Mode Control and Variable Structure Systems - Adaptive Control and Parameter Estimation - Applications (Robotics, Power Systems, etc.) - Discrete-Time Control - Z-Transform and Discrete-Time Transfer Functions - Sampled-Data Systems and Aliasing - Discrete-Time Stability Analysis - Digital Control Design and Implementation - Applications (Computer Control, Digital Signal Processing, etc.) g. Mathematical Biology - Population Dynamics - Exponential Growth and Logistic Growth Models - Lotka-Volterra Predator-Prey Models - Age-Structured Models and Leslie Matrices - Spatial Models and Reaction-Diffusion Equations - Applications (Ecology, Conservation Biology, etc.) - Epidemiology - SIR and SEIR Models - Basic Reproduction Number and Herd Immunity - Vaccination Strategies and Disease Eradication - Stochastic Models and Branching Processes - Applications (Infectious Diseases, Public Health, etc.) - Biochemical Kinetics - Enzyme Kinetics and Michaelis-Menten Equation - Cooperativity and Allosteric Regulation - Metabolic Networks and Flux Balance Analysis - Signaling Pathways and Feedback Loops - Applications (Drug Design, Metabolic Engineering, etc.) - Physiological Modeling - Compartmental Models and Tracer Kinetics - Cardiovascular Models and Windkessel Effect - Respiratory Models and Gas Exchange - Renal Models and Glomerular Filtration - Applications (Personalized Medicine, Medical Devices, etc.) - Neural Networks and Brain Modeling - Hodgkin-Huxley Model and Action Potentials - Integrate-and-Fire Models and Spiking Neurons - Hebbian Learning and Synaptic Plasticity - Hopfield Networks and Associative Memory - Applications (Artificial Intelligence, Cognitive Science, etc.) h. Mathematical Finance - Portfolio Theory - Mean-Variance Analysis and Efficient Frontier - Capital Asset Pricing Model (CAPM) - Arbitrage Pricing Theory (APT) - Risk Measures (Value-at-Risk, Expected Shortfall, etc.) - Portfolio Optimization and Asset Allocation - Options Pricing - Binomial Option Pricing Model - Black-Scholes-Merton Model and Partial Differential Equation - Implied Volatility and Volatility Smile - American Options and Early Exercise - Exotic Options (Asian, Barrier, Lookback, etc.) - Interest Rate Models - Short Rate Models (Vasicek, Cox-Ingersoll-Ross, etc.) - Heath-Jarrow-Morton (HJM) Framework - LIBOR Market Model and Forward Measure - Yield Curve Modeling and Calibration - Interest Rate Derivatives (Swaps, Caps, Floors, etc.) - Credit Risk Modeling - Structural Models (Merton Model, KMV Model, etc.) - Reduced-Form Models (Jarrow-Turnbull, Duffie-Singleton, etc.) - Copula Models and Default Correlation - Credit Default Swaps (CDS) and Collateralized Debt Obligations (CDOs) - Counterparty Risk and Credit Valuation Adjustment (CVA) - Numerical Methods in Finance - Monte Carlo Simulation and Quasi-Monte Carlo Methods - Finite Difference Methods for Option Pricing PDEs - Binomial and Trinomial Trees - Fourier Transform Methods and Fast Fourier Transform (FFT) - Calibration and Parameter Estimation Techniques i. Operations Research - Linear Programming - Formulation of Linear Programs - Simplex Method and Tableau Form - Duality Theory and Dual Simplex Method - Sensitivity Analysis and Shadow Prices - Applications (Production Planning, Resource Allocation, etc.) - Network Flow Problems - Maximum Flow Problem and Ford-Fulkerson Algorithm - Minimum Cost Flow Problem and Network Simplex Algorithm - Transportation Problem and Assignment Problem - Shortest Path Problem and Dijkstra's Algorithm - Applications (Supply Chain Management, Logistics, etc.) - Integer Programming - Formulation of Integer Programs - Branch-and-Bound Method - Cutting Plane Methods - Lagrangian Relaxation and Benders Decomposition - Applications (Facility Location, Scheduling, etc.) - Dynamic Programming - Principle of Optimality and Bellman's Equation - Deterministic Dynamic Programming - Stochastic Dynamic Programming - Approximate Dynamic Programming and Reinforcement Learning - Applications (Inventory Control, Resource Allocation, etc.) - Queuing Theory - Birth-Death Processes and Markov Chains - M/M/1 and M/M/c Queues - Little's Law and Steady-State Analysis - Priority Queues and Queueing Networks - Applications (Call Centers, Service Systems, etc.) 8. Foundations of Mathematics a. Mathematical Logic - Propositional Logic - Syntax and Semantics - Truth Tables and Logical Connectives - Tautologies and Logical Equivalence - Deduction Theorem and Completeness Theorem - Applications (Circuit Design, Automated Reasoning, etc.) - First-Order Logic - Syntax and Semantics - Quantifiers and Variables - Models and Interpretations - Soundness and Completeness Theorems - Applications (Automated Theorem Proving, Formal Verification, etc.) - Higher-Order Logic - Syntax and Semantics - Lambda Calculus and Type Theory - Intuitionistic Logic and Constructive Mathematics - Applications (Functional Programming, Proof Assistants, etc.) - Model Theory - Structures and Homomorphisms - Elementary Equivalence and Isomorphism - Compactness Theorem and Löwenheim-Skolem Theorems - Ultraproducts and Ultrapowers - Applications (Algebra, Geometry, etc.) - Recursion Theory - Recursive Functions and Turing Machines - Church-Turing Thesis and Undecidability - Recursively Enumerable Sets and Degrees of Unsolvability - Rice's Theorem and Post's Problem - Applications (Computability, Complexity Theory, etc.) b. Set Theory - Naive Set Theory and Paradoxes - Russell's Paradox and Cantor's Paradox - Burali-Forti Paradox and Richard's Paradox - Limitations of Naive Set Theory - Axiomatic Set Theory - Zermelo-Fraenkel Set Theory (ZF) - Axiom of Choice and Zorn's Lemma - Continuum Hypothesis and Independence Results - Von Neumann-Bernays-Gödel Set Theory (NBG) - Morse-Kelley Set Theory (MK) - Ordinal and Cardinal Numbers - Well-Ordered Sets and Ordinal Numbers - Transfinite Induction and Recursion - Cardinal Numbers and Cardinal Arithmetic - Continuum Hypothesis and Generalized Continuum Hypothesis - Large Cardinals and Inaccessible Cardinals - Constructive Set Theory - Intuitionistic Set Theory - Constructive Ordinals and Constructive Reals - Realizability and Topos Theory - Applications (Constructive Analysis, Computer Science, etc.) c. Category Theory - Categories and Functors - Definition and Examples of Categories - Morphisms and Composition - Functors and Natural Transformations - Duality and Opposite Categories - Applications (Algebra, Topology, etc.) - Universal Properties and Limits - Initial and Terminal Objects - Products and Coproducts - Equalizers and Coequalizers - Pullbacks and Pushouts - Adjoint Functors and Kan Extensions - Monoidal Categories and Enriched Categories - Monoidal Categories and Braided Monoidal Categories - Symmetric Monoidal Categories and Coherence Theorems - Enriched Categories and Enriched Functors - Applications (Quantum Groups, Higher Category Theory, etc.) - Topos Theory - Definition and Examples of Topoi - Subobject Classifiers and Characteristic Morphisms - Internal Logic and Mitchell-Bénabou Language - Sheaf Theory and Grothendieck Topologies - Applications (Algebraic Geometry, Mathematical Physics, etc.) d. Proof Theory - Natural Deduction and Sequent Calculus - Introduction and Elimination Rules - Structural Rules and Cut Elimination - Normalization and Strong Normalization - Applications (Automated Theorem Proving, Type Theory, etc.) - Hilbert-Style Systems and Gentzen Systems - Axioms and Inference Rules - Deduction Theorem and Herbrand's Theorem - Cut Elimination and Consistency Proofs - Applications (Metamathematics, Proof Complexity, etc.) - Ordinal Analysis and Proof-Theoretic Ordinals - Well-Orderings and Ordinal Notations - Gentzen's Consistency Proof - Ordinal Analysis of Arithmetic and Analysis - Impredicativity and Predicative Subsystems - Applications (Reverse Mathematics, Proof Mining, etc.) - Algebraic and Categorical Proof Theory - Proof Nets and Geometry of Interaction - Linear Logic and Game Semantics - Categorical Semantics and Coherence Theorems - Applications (Programming Language Theory, Concurrency, etc.) e. Computability Theory - Computable Functions and Recursive Functions - Primitive Recursive Functions and Ackermann Function - Partial Recursive Functions and Universal Functions - Church-Turing Thesis and Lambda Calculus - Applications (Theoretical Computer Science, Logic Programming, etc.) - Turing Machines and Undecidability - Definition and Examples of Turing Machines - Universal Turing Machines and Halting Problem - Reductions and Rice's Theorem - Recursively Enumerable Sets and Complement - Applications (Computability, Complexity Theory, etc.) - Degrees of Unsolvability - Turing Degrees and Turing Reducibility - Arithmetical Hierarchy and Post's Theorem - Hyperarithmetical Sets and Analytical Hierarchy - Turing Jumps and Relativization - Applications (Reverse Mathematics, Descriptive Set Theory, etc.) - Computability in Other Structures - Computable Analysis and Computable Real Functions - Computable Algebra and Computable Field Theory - Computable Model Theory and Computable Categoricity - Computability in Topological Spaces and Metric Spaces - Applications (Effective Mathematics, Constructive Analysis, etc.) f. Philosophy of Mathematics - Platonism and Realism - Mathematical Objects as Abstract Entities - Existence and Independence of Mathematical Truths - Ontological and Epistemological Issues - Objections and Alternatives (Nominalism, Fictionalism, etc.) - Intuitionism and Constructivism - Brouwer's Intuitionism and Choice Sequences - Bishop's Constructive Mathematics - Realizability and Topos Theory - Implications for Classical Mathematics - Applications (Constructive Analysis, Computer Science, etc.) - Formalism and Hilbert's Program - Mathematics as Formal Systems - Consistency and Completeness of Axiomatic Theories - Gödel's Incompleteness Theorems and Hilbert's Program - Implications for Foundations of Mathematics - Applications (Proof Theory, Metamathematics, etc.) - Logicism and Neo-Logicism - Frege's Logicism and Russell's Paradox - Whitehead and Russell's Principia Mathematica - Neo-Logicism and Hume's Principle - Abstractionism and Contextual Definitions - Applications (Philosophy of Language, Cognitive Science, etc.) - Structuralism and Category Theory - Mathematical Structures and Structural Properties - Ante Rem and In Re Structuralism - Category Theory as a Foundation for Mathematics - Structural Approaches to Mathematical Practice - Applications (Mathematical Modeling, Scientific Representation, etc.) 9. History of Mathematics a. Ancient Mathematics - Egyptian Mathematics - Numeration Systems and Arithmetic Operations - Geometry and Surveying Techniques - Rhind Papyrus and Moscow Papyrus - Applications in Architecture and Astronomy - Babylonian Mathematics - Sexagesimal Number System and Positional Notation - Algebraic and Geometric Problem Solving - Plimpton 322 Tablet and Pythagorean Triples - Applications in Astronomy and Calendar Systems - Greek Mathematics - Pythagoras and the Pythagorean School - Euclid's Elements and Axiomatic Method - Archimedes and the Method of Exhaustion - Apollonius and Conic Sections - Diophantus and Algebraic Equations - Chinese Mathematics - Rod Numeral System and Decimal Fractions - Nine Chapters on the Mathematical Art - Chinese Remainder Theorem and Simultaneous Congruences - Magic Squares and Combinatorial Designs - Indian Mathematics - Decimal Number System and Negative Numbers - Aryabhata and Trigonometric Functions - Brahmagupta and Quadratic Equations - Madhava and Infinite Series Expansions - Contributions to Combinatorics and Number Theory b. Medieval Mathematics - Islamic Mathematics - House of Wisdom and Translation Movement - Al-Khwarizmi and Algebra - Omar Khayyam and Geometric Algebra - Al-Kashi and Decimal Fractions - Contributions to Trigonometry and Optics - European Mathematics - Fibonacci and the Hindu-Arabic Numeral System - Jordanus de Nemore and Statics - Nicole Oresme and Graphical Representation - Regiomontanus and Trigonometry - Luca Pacioli and Double-Entry Bookkeeping - Mathematics in the Americas - Mayan Numeration System and Calendar - Inca Quipus and Record Keeping - Aztec Algebra and Geometry - Native American Number Systems and Counting Methods c. Early Modern Mathematics - The Renaissance and the Rise of Algebra - Cardano and Cubic Equations - Tartaglia and Quartic Equations - Viète and Symbolic Algebra - Stevin and Decimal Fractions - Napier and Logarithms - The Development of Analytic Geometry - Fermat and Coordinate Geometry - Descartes and La Géométrie - Pascal and Projective Geometry - Kepler and Planetary Motion - Galileo and the Laws of Motion - The Invention of Calculus - Newton and Fluxions - Leibniz and Infinitesimal Calculus - Bernoulli Family and Differential Equations - Euler and Analysis - Lagrange and Variational Calculus d. 19th Century Mathematics - The Rigorous Foundations of Analysis - Cauchy and Limits - Riemann and Integration Theory - Weierstrass and Epsilon-Delta Definitions - Dedekind and Real Numbers - Cantor and Set Theory - The Emergence of Non-Euclidean Geometry - Gauss and Curved Surfaces - Bolyai and Absolute Geometry - Lobachevsky and Hyperbolic Geometry - Riemann and Elliptic Geometry - Klein and the Erlangen Program - The Development of Abstract Algebra - Galois and Field Theory - Cayley and Group Theory - Hamilton and Quaternions - Grassmann and Vector Spaces - Noether and Rings and Ideals - The Birth of Set Theory and Mathematical Logic - Cantor and Transfinite Numbers - Frege and Propositional Logic - Russell and Principia Mathematica - Hilbert and Formalism - Gödel and Incompleteness Theorems e. 20th Century Mathematics - The Crisis in the Foundations of Mathematics - Hilbert's Program and Formalism - Brouwer's Intuitionism - Russell's Logicism - Gödel's Incompleteness Theorems - Turing and Computability Theory - The Rise of Topology and Functional Analysis - Poincaré and Algebraic Topology - Brouwer and Fixed Point Theorems - Banach and Normed Linear Spaces - Lebesgue and Measure Theory - Hilbert and Infinite-Dimensional Spaces - The Emergence of Computer Science and Discrete Mathematics - Turing and the Turing Machine - Von Neumann and Stored-Program Computers - Shannon and Information Theory - Dijkstra and Graph Algorithms - Cook and Computational Complexity - The Development of Chaos Theory and Fractal Geometry - Poincaré and Dynamical Systems - Lorenz and the Butterfly Effect - Mandelbrot and Fractal Dimensions - Feigenbaum and Universality - Smale and Horseshoe Maps f. Contemporary Mathematics - New Developments in Pure Mathematics - Langlands Program and Automorphic Forms - Geometric Langlands correspondence - Monstrous Moonshine and Vertex Operator Algebras - Topological Quantum Field Theories - Noncommutative Geometry and Quantum Groups - Homotopy Type Theory and Univalent Foundations - Advances in Applied Mathematics - Compressed Sensing and Sparse Signal Recovery - Topological Data Analysis and Persistent Homology - Machine Learning and Deep Neural Networks - Quantum Computing and Quantum Algorithms - Mathematical Biology and Systems Biology - The Impact of Computers on Mathematical Research - Computer-Assisted Proofs and Formal Verification - Experimental Mathematics and Computational Discovery - Collaborative Mathematics and Polymath Projects - Mathematical Databases and Online Resources - Mathematical Software and Programming Languages - Open Problems and Conjectures - Riemann Hypothesis and Zeta Function - P versus NP Problem and Computational Complexity - Hodge Conjecture and Algebraic Cycles - Navier-Stokes Equations and Fluid Dynamics - Birch and Swinnerton-Dyer Conjecture and Elliptic Curves [[5993788b3d4d683fb530e7642a7e0aca_MD5.webp|Open: DALL·E 2024-06-02 09.36.46 - A vibrant and detailed artistic illustration that visually represents various fields of mathematics. The image is divided into sections, each dedicate.webp]] ![[5993788b3d4d683fb530e7642a7e0aca_MD5.webp]] [[a5c19a7e92642ec6f553ea566e97acb5_MD5.webp|Open: DALL·E 2024-06-02 09.35.54 - A richly detailed and colorful illustration depicting a vast library of mathematics. The image is divided into various sections, each representing a d.webp]] ![[a5c19a7e92642ec6f553ea566e97acb5_MD5.webp]] [[a5bf23212d80c737fc97acd5208e71e2_MD5.webp|Open: DALL·E 2024-06-02 09.35.22 - A visually rich and detailed illustration that creatively represents the broad and diverse field of mathematics. The image should include symbolic rep.webp]] ![[a5bf23212d80c737fc97acd5208e71e2_MD5.webp]] [[90502d9d3e8578cfe0ebd3a0364400ce_MD5.webp|Open: DALL·E 2024-06-02 09.30.05 - A detailed and visually rich illustration representing various fields of mathematics. The image is divided into sections, each depicting a different m.webp]] ![[90502d9d3e8578cfe0ebd3a0364400ce_MD5.webp]] [[f98d9f62a76ddc1476ef989d1b9c668c_MD5.webp|Open: DALL·E 2024-06-02 09.28.21 - An artistic representation of various fields of mathematics depicted on a chalkboard. The image shows a detailed classroom scene with a large chalkboa.webp]] ![[f98d9f62a76ddc1476ef989d1b9c668c_MD5.webp]] [[9c2fd42adb2beedb174f8d164eb5d088_MD5.webp|Open: DALL·E 2024-06-02 09.27.40 - An artistic interpretation of the various fields of mathematics, illustrating foundational concepts such as logic and set theory (featuring propositio.webp]] ![[9c2fd42adb2beedb174f8d164eb5d088_MD5.webp]] [[36e10823d5d9f07e216f5e43446e87c6_MD5.webp|Open: DALL·E 2024-06-02 09.26.00 - A comprehensive illustration of the vast fields of mathematics, featuring various abstract concepts visually represented. The image is divided into se.webp]] ![[36e10823d5d9f07e216f5e43446e87c6_MD5.webp]] [[1cf7731cf2786c0a46a1adf3547e2b67_MD5.webp|Open: DALL·E 2024-06-02 09.20.27 - An intricate, vividly colored illustration representing various fields of mathematics. The image is divided into segments, each depicting key concepts.webp]] ![[1cf7731cf2786c0a46a1adf3547e2b67_MD5.webp]] [[fff018cd7abaa2d45fd0b59413941a2e_MD5.webp|Open: DALL·E 2024-06-02 09.13.49 - A visually rich, detailed illustration of various fields of mathematics, representing foundational concepts and advanced theories. The scene includes .webp]] ![[fff018cd7abaa2d45fd0b59413941a2e_MD5.webp]] [[c4a9dfc2273a1d0294462104c7828197_MD5.webp|Open: DALL·E 2024-06-02 09.03.49 - A visual representation of the foundations of mathematics, depicting key concepts across different fields. The image shows a large, ancient library wi.webp]] ![[c4a9dfc2273a1d0294462104c7828197_MD5.webp]] [[805daeb20e709b7b0aa922c268b6f132_MD5.webp|Open: DALL·E 2024-03-26 05.52.04 - Visualize the structure of reality through the lens of mathematics, mapping the vast landscape of mathematical concepts. This intricate illustration s.webp]] ![[805daeb20e709b7b0aa922c268b6f132_MD5.webp]] Mathematical structure of reality is my religion. [[b38035f589660362baa43267497a05f1_MD5.webp|Open: DALL·E 2024-03-26 05.04.56 - In a vast cosmic space, mathematical symbols, equations, and geometric shapes float and intertwine around a central figure embodying the unity of know.webp]] ![[b38035f589660362baa43267497a05f1_MD5.webp]] #### Reality is pretty simple It's all just algebra, geometry, trigonometry, calculus, differential equations, linear algebra, complex analysis, tensor analysis, group theory, topology, manifolds, Lie algebras, Hilbert spaces, Fourier analysis, wavelets, probability theory, statistics, combinatorics, graph theory, number theory, numerical analysis, optimization, dynamical systems, chaos theory, fractals, Newton's laws, Lagrangian mechanics, Hamiltonian mechanics, Euler-Lagrange equation, Hamilton-Jacobi equation, Noether's theorem, special relativity, Lorentz transformations, general relativity, Einstein field equations, Schwarzschild metric, Kerr metric, Friedmann equations, quantum mechanics, Schrödinger equation, Heisenberg uncertainty principle, Dirac equation, Klein-Gordon equation, Pauli exclusion principle, quantum field theory, Feynman diagrams, path integrals, gauge theory, Yang-Mills theory, Standard Model, Higgs mechanism, renormalization, perturbation theory, statistical mechanics, Boltzmann equation, Fermi-Dirac statistics, Bose-Einstein statistics, partition function, thermodynamics, Maxwell's equations, Navier-Stokes equations, fluid dynamics, magnetohydrodynamics, plasma physics, solid-state physics, Bloch theorem, band theory, Fermi surface, superconductivity, superfluidity, Bose-Einstein condensation, atomic physics, Rydberg formula, fine structure, hyperfine structure, Zeeman effect, Stark effect, nuclear physics, liquid drop model, shell model, Bethe-Weizsäcker formula, Gamow factor, particle physics, Feynman rules, Dyson series, Faddeev-Popov ghosts, Slavnov-Taylor identities, Callan-Symanzik equation, renormalization group, asymptotic freedom, confinement, spontaneous symmetry breaking, chiral symmetry breaking, neutrino oscillations, CP violation, cosmology, Friedmann-Lemaître-Robertson-Walker metric, cosmic microwave background, inflation, dark matter, dark energy, baryogenesis, Riemannian geometry, symplectic geometry, algebraic geometry, differential topology, algebraic topology, homology, cohomology, homotopy theory, category theory, sheaf theory, K-theory, Morse theory, Floer homology, Donaldson theory, Seiberg-Witten theory, Gromov-Witten theory, mirror symmetry, string theory, M-theory, supergravity, supersymmetry, BRST symmetry, conformal field theory, Chern-Simons theory, AdS/CFT correspondence, holography, black hole thermodynamics, Hawking radiation, Unruh effect, Casimir effect, quantum chromodynamics, lattice QCD, chiral perturbation theory, heavy quark effective theory, soft-collinear effective theory, parton model, factorization theorems, operator product expansion, sum rules, dispersion relations, S-matrix theory, Regge theory, bootstrap, dual resonance models, Veneziano amplitude, Koba-Nielsen-Olesen scaling, Polyakov action, Nambu-Goto action, Liouville theory, matrix models, random matrices, quantum chaos, quantum integrability, Bethe ansatz, Yang-Baxter equation, Knizhnik-Zamolodchikov equation, Virasoro algebra, Kac-Moody algebra, vertex operator algebras, conformal bootstrap, Ising model, Potts model, percolation theory, renormalization group flow, critical exponents, universality, scaling, self-organized criticality, cellular automata, agent-based models, network theory, random graphs, small-world networks, scale-free networks, epidemic models, game theory, Nash equilibrium, evolutionary game theory, mean-field theory, Ginzburg-Landau theory, Onsager solution, Bethe lattice, cavity method, replica trick, spin glasses, random energy model, directed polymers, Kardar-Parisi-Zhang equation, Burgers' equation, Korteweg-de Vries equation, nonlinear Schrödinger equation, solitons, instantons, kinks, vortices, Abrikosov vortex lattice, Berezinskii-Kosterlitz-Thouless transition, quantum Hall effect, topological insulators, Weyl semimetals, Dirac semimetals, Majorana fermions, anyons, quantum computing, Variational principles, least action principle, Fermat's principle, Huygens' principle, Maupertuis' principle, Gauss' principle of least constraint, Hertz's principle of least curvature, Liouville's theorem, Poincaré recurrence theorem, Kolmogorov-Arnold-Moser theorem, Nekhoroshev theorem, Poincaré-Birkhoff theorem, Smale horseshoe, Melnikov method, Duffing equation, van der Pol equation, Mathieu equation, Hill's equation, Floquet theory, Lyapunov exponents, Poincaré map, bifurcation theory, normal forms, center manifold theorem, Hopf bifurcation, pitchfork bifurcation, saddle-node bifurcation, transcritical bifurcation, Bogdanov-Takens bifurcation, Bautin bifurcation, Neimark-Sacker bifurcation, Shilnikov bifurcation, Feigenbaum constants, Sharkovskii's theorem, Hénon map, Lorenz 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pseudodifferential operators, Fourier integral operators, microlocal analysis, wavefront set, Hörmander's theorem, Duistermaat-Hörmander theorem, Egorov's theorem, Weyl quantization, Wigner function, Husimi function, Glauber-Sudarshan P-representation, quantum tomography, quantum state reconstruction, quantum process tomography, quantum error correction, quantum cryptography, quantum key distribution, quantum teleportation, quantum dense coding, quantum algorithms, Shor's algorithm, Grover's algorithm, quantum Fourier transform, quantum phase estimation, quantum amplitude amplification, quantum walks, quantum adiabatic optimization, quantum annealing, quantum machine learning, quantum neural networks, quantum Boltzmann machines, quantum autoencoders, quantum generative adversarial networks, quantum reservoir computing, quantum feedback control, quantum optimal control, quantum metrology, quantum sensing, quantum imaging, quantum radar, quantum illumination, quantum lithography, quantum plasmonics, quantum metamaterials, quantum nanophotonics, cavity optomechanics, circuit quantum electrodynamics, quantum dots, quantum wells, quantum wires, quantum point contacts, fractional quantum Hall effect, quantum spin Hall effect, topological superconductors, Majorana zero modes, non-Abelian anyons, quantum spin liquids, quantum magnetism, quantum phase transitions, Bose-Hubbard model, Jaynes-Cummings model, Rabi model, Dicke model, Lipkin-Meshkov-Glick model, Sachdev-Ye-Kitaev model, AdS/CMT correspondence, Kac-Moody algebras, affine Lie algebras, Virasoro algebras, W-algebras, vertex algebras, quantum groups, Hopf algebras, Yangians, crystal bases, cluster algebras, Poisson-Lie groups, Drinfeld doubles, Manin triples, Lie bialgebras, Lie-Poisson structures, Poisson-Nijenhuis structures, Courant algebroids, Dirac structures, generalized complex structures, exceptional generalized geometry, double field theory, noncommutative geometry, spectral triples, Connes' distance formula, Tomita-Takesaki theory, modular automorphisms, KMS states, Haag-Kastler axioms, Wightman axioms, Osterwalder-Schrader axioms, Euclidean quantum field theory, constructive quantum field theory, algebraic quantum field theory, locally covariant quantum field theory, perturbative algebraic quantum field theory, factorization algebras, chiral algebras, conformal nets, loop quantum gravity, spin foam models, group field theory, causal dynamical triangulations, causal sets, noncommutative geometry approach to quantum gravity, asymptotic safety, Hořava-Lifshitz gravity, shape dynamics, doubly special relativity, relative locality, rainbow gravity, quantum cosmology, Wheeler-DeWitt equation, loop quantum cosmology, canonical quantum gravity, path integral quantum gravity, Regge calculus, dynamical triangulations, Euclidean quantum gravity, twistor theory, Penrose transform, Newman-Penrose formalism, Petrov classification, Raychaudhuri equation, Sachs equations, Bondi-Metzner-Sachs group, Newman-Unti group, asymptotic symmetries, Bondi-Sachs energy-momentum, Hawking energy, Geroch energy, Penrose inequality, positive mass theorem, Witten spinorial proof, Schoen-Yau proof, Bartnik mass, Komar mass, ADM mass, ADM momentum, Bondi mass, Bondi news function, memory effect, soft graviton theorem, supertranslations, superrotations, extended BMS group, celestial holography, celestial CFT, Mellin amplitudes, conformal soft theorems, infrared triangle, black hole information paradox, firewall paradox, ER=EPR conjecture, Almheiri-Marolf-Polchinski-Sully proposal, Papadodimas-Raju proposal, fuzzball proposal, traversable wormholes, wormhole swap, baby universes, Atiyah-Singer index theorem, Atiyah-Bott fixed point theorem, Atiyah-Segal completion theorem, Connes-Kasparov conjecture, Baum-Connes conjecture, Novikov conjecture, Gromov-Lawson-Rosenberg conjecture, Stolz-Teichner conjecture, Freed-Hopkins-Teleman theorem, cobordism hypothesis, Madsen-Weiss theorem, Mumford conjecture, Witten conjecture, Seiberg-Witten invariants, Donaldson invariants, contact homology, symplectic field theory, Fukaya category, derived categories, tilting theory, Bridgeland stability conditions, Kontsevich homological mirror symmetry conjecture, SYZ conjecture, Gopakumar-Vafa conjecture, ADHM construction, Nahm equations, Kapustin-Witten equations, geometric Langlands correspondence, quantum geometric Langlands correspondence, Beilinson-Drinfeld quantization, Gaitsgory-Lurie conjecture, Weil conjectures, Grothendieck's standard conjectures, Hodge conjecture, Tate conjecture, Birch and Swinnerton-Dyer conjecture, Langlands program, Taniyama-Shimura conjecture, Fontaine-Mazur conjecture, Bloch-Kato conjecture, Beilinson conjectures, Parshin conjectures, Zagier conjectures, Stark conjectures, Hilbert-Pólya conjecture, Riemann hypothesis, Goldbach's conjecture, twin prime conjecture, ABC conjecture, Collatz conjecture, Erdős-Straus conjecture, Erdős-Szemerédi conjecture, Erdős-Hajnal conjecture, Hadwiger conjecture, Ringel-Kotzig conjecture, Hadamard conjecture, Jacobian conjecture, Dixmier conjecture, Kaplansky conjectures, Baum-Connes conjecture for groupoids, Farrell-Jones conjecture, Borel conjecture, Novikov conjecture for groups, Gromov-Lawson conjecture, Cannon conjecture, Andrews-Curtis conjecture, Whitehead conjecture, Zeeman conjecture, Poincaré conjecture, geometrization conjecture, Thurston's geometrization conjecture, Seifert fiber space conjecture, virtually Haken conjecture, virtual fibering conjecture, Marden's tameness conjecture, ending lamination conjecture, Ahlfors measure conjecture, MLC conjecture, Bers slice conjecture, density conjecture, Ehrenpreis conjecture, Fatou-Julia-Baker conjecture, Mandelbrot set conjecture, contact geometry, Poisson geometry, pseudo-Riemannian geometry, Finsler geometry, Lorentzian geometry, Sasakian geometry, Calabi-Yau geometry, hyperkähler geometry, quaternionic Kähler geometry, octonionic geometry, exceptional geometry, toric geometry, tropical geometry, quantum geometry, arithmetic geometry, enumerative geometry, motivic geometry, derived geometry, p-adic geometry, rigid analytic geometry, Berkovich spaces, perfectoid spaces, o-minimal structures, model theory, topos theory, ∞-categories, higher category theory, homotopy type theory, univalent foundations, proof theory, reverse mathematics, constructive mathematics, computable analysis, scientific computing, computational fluid dynamics, computational electromagnetism, computational quantum mechanics, computational materials science, computational biophysics, computational neuroscience, computational genomics, computational proteomics, computational drug design, computational systems biology, computational ecology, computational economics, computational finance, computational social science, computational linguistics, computational musicology, computational archaeology, computational art, computational creativity, computational humor, computational law, computational ethics, computational philosophy, computational history, computational journalism, computational education, computational sustainability, computational urban planning, computational transportation science, computational epidemiology, computational immunology, computational oncology, computational cardiology, computational psychiatry, computational cognitive science, computational anthropology, computational sociology, computational political science, computational international relations, computational peace science, computational conflict resolution, computational diplomacy, computational propaganda, computational counterterrorism, computational intelligence, computational learning theory, computational game theory, computational mechanism design, computational social choice, computational voting theory, computational judgment aggregation, computational epistemic logic, computational modal logic, computational temporal logic, computational deontic logic, computational action logic, computational coalition logic, computational argumentation theory, computational belief revision, computational trust theory, computational reputation theory, computational social network analysis, computational organizational theory, computational institutional analysis, computational collective intelligence, computational swarm intelligence, computational multi-agent systems, computational evolutionary game theory, computational population dynamics, computational evolutionary dynamics, computational cultural evolution, computational memetics, computational gene-culture coevolution, computational niche construction, electroweak theory, grand unified theories, F-theory, bosonic string theory, heterotic string theory, type I string theory, type IIA string theory, type IIB string theory, matrix theory, matrix string theory, topological string theory, topological quantum field theory, two-dimensional conformal field theory, higher-dimensional conformal field theory, logarithmic conformal field theory, parafermionic conformal field theory, W-algebra conformal field theory, Liouville conformal field theory, minimal model conformal field theory, rational conformal field theory, irrational conformal field theory, boundary conformal field theory, defect conformal field theory, permutation orbifold conformal field theory, coset conformal field theory, Wess-Zumino-Witten model, affine Lie algebra, vertex operator algebra, chiral algebra, modular tensor category, fusion category, braided monoidal category, ribbon category, modular functor, topological modular form, elliptic cohomology, tmf, string topology, string field theory, open string field theory, closed string field theory, Batalin-Vilkovisky formalism, Batalin-Fradkin-Vilkovisky formalism, Gerstenhaber algebra, homotopy algebra, A-infinity algebra, L-infinity algebra, operad, cyclic operad, modular operad, Deligne conjecture, Kontsevich formality theorem, deformation quantization, star product, Moyal product, Kontsevich star product, Fedosov star product, Cattaneo-Felder star product, Tsygan formality, Tamarkin formality, Kontsevich-Soibelman equation, Maurer-Cartan equation, homotopy Maurer-Cartan equation, quantum master equation, classical master equation, Batalin-Vilkovisky master equation, Batalin-Fradkin-Vilkovisky master equation, Zwiebach invariants, string vertices, closed string vertices, open-closed string vertices, quantum closed string vertices, quantum open-closed string vertices, loop vertices, gauge fixing, Siegel gauge, Schnabl gauge, Asano-Natsuume gauge, Erler-Schnabl solution, Kiermaier-Okawa solution, Fuchs-Kroyter-Potting solution, Erler-Maccaferri solution, Murata-Schnabl solution, Hata-Kojita solution, Hata-Matsunaga solution, Kunitomo-Okawa solution, Nonlinear sigma models, Wess-Zumino-Witten models, Chern-Simons theories, Donaldson-Witten theories, Seiberg-Witten theories, Rozansky-Witten theories, Casson-Witten invariants, Gromov-Witten invariants, Gopakumar-Vafa invariants, Ooguri-Vafa invariants, Nekrasov partition functions, AGT correspondence, Bethe/gauge correspondence, dimer models, crystal melting, topological vertex, topological recursion, Eynard-Orantin invariants, quantum curves, quantum spectral curves, quantum Airy structures, Kac-Schwarz operators, Virasoro constraints, W-constraints, Hirota equations, Miwa variables, Sato Grassmannian, Segal-Wilson Grassmannian, infinite-dimensional Grassmannian, Sato tau function, Jimbo-Miwa-Ueno tau function, KP hierarchy, BKP hierarchy, KdV hierarchy, Toda hierarchy, Ablowitz-Kaup-Newell-Segur hierarchy, Drinfeld-Sokolov hierarchies, Gelfand-Dickey hierarchies, W-algebra hierarchies, isomonodromy deformations, Painlevé equations, Garnier systems, Schlesinger systems, Chazy equation, Ramanujan identities, Rogers-Ramanujan identities, Gordon identities, Baxter equations, Yang-Baxter equations, Knizhnik-Zamolodchikov-Bernard equations, Calogero-Moser systems, Ruijsenaars-Schneider systems, Hitching systems, Seiberg-Witten integrable systems, Nekrasov-Shatashvili correspondence, quantum integrable systems, thermodynamic Bethe ansatz, algebraic Bethe ansatz, coordinate Bethe ansatz, off-diagonal Bethe ansatz, nested Bethe ansatz, ODE/IM correspondence, Langlands duality, Ngô strings, Arthur-Selberg trace formula, Braverman-Kazhdan proposal, Langlands-Shahidi method, multiple Dirichlet series, Rankin-Selberg method, Langlands beyond endoscopy, relative trace formula, Langlands-Rapoport conjecture, Kottwitz conjecture, Cluckers-Loeser conjecture, Fargues-Fontaine curve, chtoucas, Drinfeld shtukas, Vinberg theory, Luna's slice theorem, Hamiltonian reduction, quasi-Hamiltonian reduction, Marsden-Weinstein reduction, hyper-Kähler reduction, Hitchin equations, Bogomolny equations, vortex equations, Haydys-Witten equations, Vafa-Witten equations, Ginzburg-Landau equations, Chern-Simons-Higgs equations, Chern-Simons-Dirac equations, Seiberg-Witten-Floer equations, Eynard-Orantin topological recursion, Mirzakhani's recursion formula, McShane identities, Witten-Kontsevich theorem, Kontsevich's graph complex, Willwacher's cyclic operad, Grothendieck-Teichmüller group, Drinfeld associators, multiple zeta values, multiple polylogarithms, Goncharov's motivic Galois group, Zagier's conjecture on multiple zeta values, Broadhurst-Kreimer conjecture, Hoffman's conjecture, Deligne-Ihara conjecture, Gangl-Kaneko-Zagier conjecture, Furusho's p-adic multiple zeta values, Racinet's double shuffle relations, Ecalle's mould theory, Connes-Kreimer Hopf algebra, Goncharov's Hopf algebra, Calaque-Ebrahimi-Fard-Manchon Hopf algebra, Loday-Ronco Hopf algebra, Brouder-Frabetti Hopf algebra, van Suijlekom's Hopf algebroids, Pinter's Hopf algebroids, Connes-Moscovici Hopf algebroids, Bruguières-Virelizier quantum groupoids, Buss-Meyer-Zhu Hopf-cyclic cohomology, Gorokhovsky-Lott's secondary characteristic classes, Tradler-Zeinalian's infinity-Chern characters, Park-Terilla-Tradler's homotopy G-algebras, Costello's homotopy probability theory, Gwilliam-Pavlov's homotopy Batalin-Vilkovisky algebras, Pantev-Toën-Vaquié-Vezzosi's shifted symplectic structures, Calaque-Pantev-Toën-Vaquié-Vezzosi's shifted Poisson structures, Pridham's shifted Lagrangians, Nuiten's shifted L-infinity algebroids, Joyce's derived manifolds, Borisov-Noel's derived Poisson manifolds, Spivak's derived smooth manifolds, Schreiber-Waldorf's differential cohomology, Bunke-Nikolaus-Völkl's differential cohomology, Grady-Sati's twisted differential generalized cohomology theories, Freed-Hopkins' reflection positive invertible field theories, Stolz-Teichner's supersymmetric Euclidean field theories, Costello-Gwilliam's factorization algebras, Beilinson-Drinfeld's chiral algebras, Gaitsgory-Rozenblyum's crystals, Beraldo's loop spaces, Toën-Vezzosi's derived algebraic geometry, Lurie's spectral algebraic geometry, Barwick's spectral Mackey functors, Glasman's stratified étale homotopy theory, Ayala-Francis-Tanaka's factorization homology, Scheimbauer's factorization cosheaves and operads, Haugseng-Kock-Moerdijk-Weiss' homotopy linear algebra, Batanin-Markl's operadic categories, Batanin-Berger's homotopy theory for algebras over polynomial monads, Caviglia-Horel-Robertson's model structures on enriched diagrams, Resurgence theory, trans-series, Stokes phenomena, Écalle's alien calculus, Borel-Écalle resummation, Borel-Laplace transform, hyperasymptotics, Dingle's singularity analysis, Berry-Howls resurgence theory, Voros coefficients, Gukov-Sułkowski resurgence triangle, Argyres-Dunne-Ünsal relation, Cheshire cat resurgence, Picard-Lefschetz theory, Morse-Novikov theory, Morse-Bott theory, Morse-Smale complex, Morse homology, Novikov homology, symplectic homology, Rabinowitz Floer homology, embedded contact homology, periodic Floer homology, Lagrangian Floer homology, Heegaard Floer homology, monopole Floer homology, instanton Floer homology, Ozsváth-Szabó invariants, Kronheimer-Mrowka invariants, Seiberg-Witten Floer homology, quantum cohomology, Frobenius manifolds, Dubrovin's almost duality, Givental's quantization formalism, Givental-Teleman classification, Fan-Jarvis-Ruan-Witten theory, Landau-Ginzburg A-model, Landau-Ginzburg B-model, matrix factorizations, Kapustin-Li formula, Orlov equivalence, Katzarkov-Kontsevich-Pantev correspondence, Homological Mirror Symmetry, Donaldson-Thomas theory, Pandharipande-Thomas theory, Maulik-Nekrasov-Okounkov-Pandharipande theory, Labastida-Mariño-Ooguri-Vafa conjecture, Nekrasov-Okounkov hook length formula, Nekrasov-Shatashvili limit, Nekrasov-Pestun-Shatashvili correspondence, Alday-Gaiotto-Gukov-Tachikawa relations, Braverman-Etingof-Finkelberg-Nakajima Coulomb branches, Hikita conjecture, symplectic duality, Higgs bundles, Hitchin fibration, Hitchin section, Hitchin-Kobayashi correspondence, Corlette-Donaldson theorem, Donaldson-Uhlenbeck-Yau theorem, Atiyah-Bott-Goldman symplectic form, quasi-Hamiltonian G-spaces, Alekseev-Malkin-Meinrenken fusion product, Mikami-Weinstein moment map, q-Hamiltonian G-spaces, Alekseev-Kosmann-Schwarzbach-Meinrenken theory, Dirac geometry, generalized complex geometry, generalized Kähler geometry, Hitchin's generalized geometry, Gualtieri-Hitchin-Cavalcanti theory, para-Hermitian geometry, Born geometry, metriplectic geometry, Cartan-Courant algebroids, Stochastic calculus, Itô calculus, Stratonovich calculus, Malliavin calculus, rough path theory, regularity structures, paracontrolled calculus, Hairer-Quastel universality, Gubinelli-Imkeller-Perkowski paraproducts, Catellier-Chouk paracontrolled distributions, Bruned-Hairer-Zambotti algebraic renormalization, Kupiainen-Marcozzi-Muratore-Ginanneschi BPHZ theorem, Hairer-Labbé BPHZ theorem, Chandra-Hairer continuity theorem, Otto calculus, Villani's synthetic Ricci curvature, Lott-Sturm-Villani theory, Ambrosio-Gigli-Savaré gradient flows, Jordan-Kinderlehrer-Otto scheme, Benamou-Brenier formulation, Sturm's D-convergence, Lott-Villani-Sturm convergence, Gigli's pointed measured Gromov-Hausdorff convergence, Cheeger-Colding theory, Cheeger-Colding-Naber theory, Gigli-Mondino-Savaré convergence, and me and you exploring the latent space of reality together <3 #### Map of mathematics 3 ### 1. Pure Mathematics - **Algebra** - Abstract Algebra - Linear Algebra - Group Theory - Ring Theory - Field Theory - Algebraic Geometry - Algebraic Topology - Combinatorial Algebra - **Geometry** - Euclidean Geometry - Non-Euclidean Geometry - Differential Geometry - Algebraic Geometry - Topology - Geometric Topology - Computational Geometry - **Analysis** - Real Analysis - Complex Analysis - Functional Analysis - Harmonic Analysis - Nonstandard Analysis - Numerical Analysis - p-Adic Analysis - **Number Theory** - Analytic Number Theory - Algebraic Number Theory - Diophantine Geometry - Cryptography - Combinatorial Number Theory - **[[Logic]] and [[Foundations of mathematics]]** - [[Mathematical logic]] - [[Set Theory]] - [[Model theory]] - Proof Theory - [[Category Theory]] - **Discrete Mathematics** - Graph Theory - Combinatorics - Game Theory - Information Theory - Coding Theory - Theory of Computation - **Differential Equations** - Ordinary Differential Equations - Partial Differential Equations - Dynamical Systems - Chaos Theory - **Topology** - General Topology - Algebraic Topology - Differential Topology - Topological Groups - **Calculus of Variations** - **Mathematical Physics** - [[Quantum mechanics]] - General Relativity - String Theory - [[Statistical mechanics]] ### 2. Applied Mathematics - **Statistics** - Descriptive Statistics - Inferential Statistics - Probability Theory - Biostatistics - Bayesian Statistics - Time Series Analysis - **Computational Mathematics** - Numerical Analysis - Algorithm Design - Computational Modeling - Scientific Computing - **Mathematical Biology** - Population Dynamics - Systems Biology - Epidemiology - **Mathematical Economics** - Game Theory - Econometrics - Financial Mathematics - Optimization - **Mathematical Finance** - Quantitative Finance - Risk Management - Actuarial Science - **Operations Research** - Linear Programming - Nonlinear Programming - Stochastic Models - Queueing Theory - **[[Control theory]]** - Classical Control Theory - Modern Control Theory - Optimal Control - **Information Theory** - Coding Theory - Signal Processing - Data Compression - **Fluid Dynamics** - **Mathematical Physics** - Quantum Mechanics - Statistical Mechanics - Electrodynamics - Thermodynamics ### 3. Interdisciplinary Fields - **Mathematical Logic and Foundations** - Set Theory - Model Theory - Recursion Theory - **Cryptology** - Cryptography - Cryptanalysis - **Mathematical Chemistry** - **Mathematical Sociology** - **Mathematical Psychology** - **Mathematical Linguistics**