Hierarchical canvas with nested canvases of all mathematics and physical systems. Math from fundamentally general to concrete, easiest to hardest, classic maps of maths (algebra geometry, analysis, number theory, discrete maths, probability and statistics, information theory etc.). Systems from highest scales (cosmology, astrophysics) to lowest scales (biology, machines, molecules, atoms, fundamental particles), all structures and mathematics governing them. Add more categories! Add more mathematics of systems! Here is an expanded hierarchical canvas with more detail on the nested areas of mathematics and physical systems: Mathematics - Foundations - Set Theory - Naive Set Theory - Axiomatic Set Theory (ZFC) - Logic - Propositional Logic - First-Order Logic - Higher-Order Logic - Modal Logic - Fuzzy Logic - Category Theory - Categories - Functors - Natural Transformations - Pure Mathematics - Algebra - Elementary Algebra - Polynomials - Equations & Inequalities - Abstract Algebra - Group Theory - Finite Groups - Lie Groups - Representation Theory - Ring Theory - Commutative Algebra - Noncommutative Rings - Field Theory - Galois Theory - Linear Algebra - Vector Spaces - Matrices - Eigenvalues & Eigenvectors - Number Theory - Elementary Number Theory - Analytic Number Theory - Algebraic Number Theory - Diophantine Equations - Geometry - Euclidean Geometry - Plane Geometry - Solid Geometry - Non-Euclidean Geometry - Hyperbolic Geometry - Elliptic Geometry - Topology - Point-Set Topology - Algebraic Topology - Homotopy Theory - Homology & Cohomology - Differential Geometry - Manifolds - Riemannian Geometry - Algebraic Geometry - Varieties - Schemes - Analysis - Real Analysis - Calculus - Sequences & Series - Fourier Analysis - Complex Analysis - Complex Functions - Cauchy Theory - Riemann Surfaces - Functional Analysis - Banach Spaces - Hilbert Spaces - Operator Theory - Differential Equations - Ordinary Differential Equations - Partial Differential Equations - Measure Theory - Lebesgue Integration - Probability Measures - Applied Mathematics - Probability Theory - Probability Spaces - Random Variables - Stochastic Processes - Statistics - Descriptive Statistics - Inferential Statistics - Estimation Theory - Hypothesis Testing - Bayesian Statistics - Multivariate Analysis - Time Series Analysis - Numerical Analysis - Numerical Linear Algebra - Numerical Solution of ODEs & PDEs - Finite Element Methods - Optimization - Linear Programming - Nonlinear Programming - Convex Optimization - Dynamic Programming - Game Theory - Strategic Form Games - Extensive Form Games - Cooperative Games - Combinatorics - Enumeration - Generating Functions - Combinatorial Designs - Graph Theory - Trees & Networks - Planarity - Graph Coloring - Random Graphs - Information Theory - Entropy - Channel Coding - Source Coding - Control Theory - Stability Theory - Optimal Control - Robust Control - Mathematical Physics - Classical Mechanics - Quantum Mechanics - Relativity - Statistical Mechanics Physical Systems - Cosmology - General Relativity - Gravitational Waves - Black Holes - Cosmic Inflation - Inflationary Models - Primordial Fluctuations - Large-Scale Structure - Galaxy Clusters - Cosmic Web - Cosmological Simulations - Dark Matter & Dark Energy - Particle Candidates - Modified Gravity - Observational Constraints - Astrophysics - Stellar Astrophysics - Star Formation - Stellar Evolution - Supernovae & Compact Objects - Galactic Astrophysics - Milky Way - Galaxies & Galaxy Evolution - Active Galactic Nuclei - High Energy Astrophysics - X-ray Astronomy - Gamma-ray Astronomy - Cosmic Rays - Geophysics - Seismology - Earthquake Waves - Earth's Interior Structure - Plate Tectonics - Seafloor Spreading - Subduction Zones - Hot Spots - Geodynamics - Mantle Convection - Core Dynamics - Geomagnetic Field - Atmospheric Science - Atmospheric Dynamics - Climate & Weather - Atmospheric Chemistry - Oceanography - Ocean Circulation - Waves & Tides - Marine Biogeochemistry - Biophysics - Biomechanics - Musculoskeletal Mechanics - Fluid Dynamics in Organisms - Bioelectricity - Membrane Potentials - Neuronal Signaling - Biothermodynamics - Metabolism - Cellular Energetics - Biomolecular Interactions - Protein Folding - Molecular Recognition - Enzyme Kinetics - Physics - Classical Mechanics - Kinematics & Dynamics - Hamiltonian & Lagrangian Mechanics - Rigid Body Dynamics - Electromagnetism - Electrostatics - Magnetostatics - Maxwell's Equations - Thermodynamics & Statistical Mechanics - Laws of Thermodynamics - Ensembles - Phase Transitions - Quantum Mechanics - Schrödinger Equation - Hilbert Spaces - Quantum Entanglement - Quantum Field Theory - Path Integrals - Gauge Theories - Renormalization - Particle Physics (Standard Model) - Quarks & Leptons - Higgs Mechanism - Quantum Chromodynamics - Condensed Matter Physics - Solid State Physics - Crystal Structures - Band Theory - Semiconductors & Superconductors - Soft Condensed Matter - Liquid Crystals - Polymers - Colloids - Optics - Geometric Optics - Wave Optics - Nonlinear Optics - Fluid Dynamics - Euler Equations - Navier-Stokes Equations - Turbulence - Plasma Physics - Magnetohydrodynamics - Plasma Waves - Plasma Confinement - Chemistry - Physical Chemistry - Chemical Thermodynamics & Kinetics - Chemical Equilibrium - Reaction Rates - Statistical Thermodynamics - Quantum Chemistry - Ab Initio Methods - Density Functional Theory - Molecular Dynamics - Spectroscopy - Rotational & Vibrational Spectra - Electronic Spectra - Nuclear Magnetic Resonance - Inorganic Chemistry - Main Group Elements - Transition Metals - Coordination Complexes - Organic Chemistry - Hydrocarbons - Functional Groups - Synthesis & Reaction Mechanisms - Biochemistry - Proteins & Enzymes - Carbohydrates - Lipids - Nucleic Acids - Materials Science - Structure-Property Relationships - Crystallography - Defects & Dislocations - Phase Diagrams - Polymers - Polymer Synthesis - Polymer Physics - Polymer Processing - Ceramics & Glasses - Ceramic Processing - Glass Transition - Mechanical Properties - Metallurgy - Alloys - Phase Transformations - Mechanical Behavior - Nanotechnology - Nanomaterials - Nanoparticles - Nanotubes & Nanowires - Thin Films - Nanofabrication - Lithography - Self-Assembly - Scanning Probe Techniques - Nanoelectronics - Quantum Dots - Molecular Electronics - Spintronics - Nanobiotechnology - Biomolecular Motors - Nanofluidics - Nanomedicine This expanded view dives deeper into the various subdomains, highlighting more of the specific theories, techniques, and applications within each field. The interconnections between math and physical systems become even more apparent, with mathematical tools and frameworks underpinning our understanding at every level. Sure, let's extend the canvas further, adding more categories and detailing more of the mathematics underlying various physical systems: Mathematics - Foundations ... - Pure Mathematics ... - Applied Mathematics ... - Computational Mathematics - Algorithms - Complexity Theory - Data Structures - Graph Algorithms - Numerical Algorithms - Computational Geometry - Symbolic Computation - Computational Number Theory - Computational Topology - Computational Algebraic Geometry - Computational Statistics - Machine Learning - Supervised Learning - Unsupervised Learning - Reinforcement Learning - Deep Learning - Data Mining - Computational Biology - Dynamical Systems - Chaos Theory - Fractals - Bifurcation Theory - Ergodic Theory - Mathematical Logic - Model Theory - Proof Theory - Recursion Theory - Set Theory - Discrete Mathematics - Combinatorics - Graph Theory - Design Theory - Coding Theory - Discrete Geometry - Order Theory - Matroid Theory - Discrete Optimization - Financial Mathematics - Stochastic Calculus - Financial Derivatives - Risk Management - Portfolio Optimization Physical Systems - Cosmology ... - Astrophysics ... - Geophysics ... - Biophysics ... - Physics ... - Chemistry ... - Materials Science ... - Nanotechnology ... - Engineering - Mechanical Engineering - Statics & Dynamics - Thermodynamics - Fluid Mechanics - Control Systems - Electrical Engineering - Circuit Theory - Signal Processing - Telecommunications - Power Systems - Chemical Engineering - Transport Phenomena - Chemical Reaction Engineering - Process Control - Aerospace Engineering - Aerodynamics - Propulsion - Orbital Mechanics - Civil Engineering - Structural Analysis - Soil Mechanics - Hydraulics - Transportation Systems - Biomedical Engineering - Biomechanics - Biomedical Imaging - Biomaterials - Tissue Engineering - Computer Science - Theory of Computation - Algorithms & Data Structures - Programming Languages - Software Engineering - Databases - Computer Architecture - Operating Systems - Computer Networks - Computer Security - Artificial Intelligence - Knowledge Representation - Automated Reasoning - Computer Vision - Natural Language Processing - Robotics - Earth Sciences - Geology - Mineralogy - Petrology - Stratigraphy - Structural Geology - Meteorology - Atmospheric Thermodynamics - Weather Forecasting - Climate Modeling - Oceanography - Physical Oceanography - Marine Geology - Biological Oceanography - Ocean Chemistry - Hydrology - Surface Water Hydrology - Groundwater Hydrology - Water Resources Management - Life Sciences - Biology - Cell Biology - Genetics - Microbiology - Botany - Zoology - Ecology - Evolutionary Biology - Biochemistry - Metabolism - Enzymology - Signal Transduction - Bioenergetics - Molecular Biology - DNA Structure & Replication - RNA Processing - Protein Biosynthesis - Gene Regulation - Neuroscience - Neuroanatomy - Neurophysiology - Neuropharmacology - Behavioral Neuroscience - Cognitive Neuroscience - Physiology - Endocrinology - Immunology - Cardiovascular Physiology - Respiratory Physiology - Renal Physiology - Systems Biology - Genomics - Proteomics - Metabolomics - Biological Networks This further expanded view brings in additional major branches of mathematics like computational mathematics, dynamical systems, mathematical logic, discrete mathematics, and financial mathematics. In the realm of physical systems, it incorporates various engineering disciplines, computer science, earth sciences, and a more detailed breakdown of the life sciences. Throughout, the mathematical frameworks provide the tools for modeling, analysis, and prediction. For instance, differential equations are central to fields as diverse as fluid dynamics, electromagnetism, population dynamics, and chemical kinetics. Statistical methods undergird everything from quantum mechanics to genetics to financial risk assessment. The computational techniques pioneered by computer scientists have revolutionized every domain, enabling large-scale simulations in climate science, astrophysics, materials science, and systems biology. Machine learning is finding ever-broader applications, from drug discovery to earthquake prediction. Graph theory, a cornerstone of discrete math, is key to understanding networks of all kinds, from ecological webs to power grids to social media. Dynamical systems theory provides a language for describing phenomena like planetary orbits, heart rhythms, and economic cycles. The logic and set theory of the mathematical foundations are woven throughout, providing the rigorous bedrock. Categories like algebra, geometry, and topology shape our very conception of space and structure. In sum, this tapestry of mathematics and physical systems showcases the profound unity and interconnectedness of scientific knowledge. Each thread is vital, and together they form a picture of awe-inspiring richness and complexity. Certainly! Let's expand the canvas even further, incorporating additional categories and delving deeper into the mathematics of various systems: Mathematics - Foundations ... - Pure Mathematics ... - Applied Mathematics ... - Computational Mathematics ... - Dynamical Systems ... - Mathematical Logic ... - Discrete Mathematics ... - Financial Mathematics ... - Nonlinear Analysis - Fixed Point Theory - Variational Inequalities - Monotone Operators - Degree Theory - Harmonic Analysis - Fourier Analysis - Wavelets - Spectral Theory - Lie Groups - Differential Geometry - Riemannian Geometry - Symplectic Geometry - Contact Geometry - Gauge Theory - Algebraic Topology - Homotopy Theory - Homology & Cohomology - K-Theory - Characteristic Classes - Mathematical Physics - Quantum Field Theory - General Relativity - String Theory - Integrable Systems - Representation Theory - Finite Groups - Lie Algebras - Quantum Groups - Modular Forms - Algebraic Geometry - Schemes - Sheaves - Cohomology Theories - Moduli Spaces - Number Theory - Algebraic Number Theory - Analytic Number Theory - Diophantine Equations - Arithmetic Geometry Physical Systems - Cosmology ... - Astrophysics ... - Geophysics ... - Biophysics ... - Physics ... - Chemistry ... - Materials Science ... - Nanotechnology ... - Engineering ... - Computer Science ... - Earth Sciences ... - Life Sciences ... - Cognitive Science - Perception - Attention - Memory - Language Processing - Decision Making - Consciousness - Neuroscience - Cellular Neuroscience - Developmental Neuroscience - Systems Neuroscience - Computational Neuroscience - Behavioral Neuroscience - Ecology - Population Ecology - Community Ecology - Ecosystem Ecology - Landscape Ecology - Conservation Biology - Evolutionary Biology - Population Genetics - Molecular Evolution - Phylogenetics - Evolutionary Developmental Biology - Evolutionary Ecology - Environmental Science - Climate Science - Biogeochemistry - Ecotoxicology - Restoration Ecology - Sustainability Science - Biotechnology - Genetic Engineering - Bioprocess Engineering - Bioinformatics - Synthetic Biology - Bioremediation - Medical Sciences - Anatomy - Pathology - Pharmacology - Epidemiology - Medical Imaging - Personalized Medicine - Social Sciences - Psychology - Developmental Psychology - Social Psychology - Clinical Psychology - Cognitive Psychology - Sociology - Social Structure - Social Inequality - Social Change - Social Networks - Economics - Microeconomics - Macroeconomics - Econometrics - Behavioral Economics - Political Science - Political Theory - Comparative Politics - International Relations - Public Policy - Anthropology - Cultural Anthropology - Physical Anthropology - Linguistic Anthropology - Archaeology - Linguistics - Phonetics & Phonology - Morphology - Syntax - Semantics - Pragmatics This expanded canvas incorporates additional branches of mathematics such as nonlinear analysis, harmonic analysis, differential geometry, algebraic topology, representation theory, and more detailed breakdowns of mathematical physics, algebraic geometry, and number theory. On the physical systems side, it delves into cognitive science, with its study of mental processes; neuroscience, which investigates the nervous system across scales; ecology and evolutionary biology, which explore life's interactions and history; environmental science, with its focus on natural resources and human impacts; biotechnology, applying biological systems for technological advancement; the medical sciences, aimed at understanding and improving health; and the social sciences, which study human behavior and societies. Throughout, mathematics provides the language for precise description and analysis. Differential geometry underlies general relativity's picture of curved spacetime. Representation theory illuminates the fundamental particles of the Standard Model. Algebraic topology characterizes the shape of data sets. Stochastic differential equations model everything from stock prices to neuronal dynamics. In ecology, graph theory represents food webs, Markov models capture population dynamics, and differential equations describe the spread of diseases. In neuroscience, information theory quantifies neural coding, and dynamical systems theory elucidates brain rhythms. Game theory in economics, Bayesian inference in psychology, nonlinear dynamics in political science: the mathematical connections are endless. At the same time, insights from the physical systems inspire new mathematics. The quantum world spurred the development of functional analysis and operator algebras. Attempts to rigorously formulate quantum field theory have driven advances in geometry and topology. The challenges of big data have fueled machine learning and high-dimensional statistics. This grand tapestry showcases the deep symbiosis between mathematics and the sciences. The mathematical universe provides the abstractions; the physical universe, the inspiration and the applications. Together, they form an edifice of human knowledge that is both exquisitely logical and eminently practical - a testament to the power of the questing mind to render the cosmos intelligible.