**List of Some of the Hardest and Greatest Mathematical Proofs** 1. **Fermat's Last Theorem** (Andrew Wiles, 1994) - Proved that there are no three positive integers \( a, b, c \) that satisfy \( a^n + b^n = c^n \) for any integer \( n > 2 \). 2. **Poincaré Conjecture** (Grigori Perelman, 2003) - Demonstrated that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. 3. **Four Color Theorem** (Kenneth Appel and Wolfgang Haken, 1976) - Established that any planar map can be colored with no more than four colors without adjacent regions sharing the same color. 4. **Classification of Finite Simple Groups** (Collaborative effort, completed in 1983) - A monumental project classifying all finite simple groups into several broad categories. 5. **Feit–Thompson Theorem (Odd Order Theorem)** (Walter Feit and John G. Thompson, 1963) - Proved that every finite group of odd order is solvable. 6. **Modularity Theorem** (Andrew Wiles and others, 1995) - Showed a deep connection between elliptic curves and modular forms, pivotal in proving Fermat's Last Theorem. 7. **Kepler Conjecture** (Thomas Hales, 1998) - Proved that no arrangement of equally sized spheres filling space has a greater average density than that of the face-centered cubic packing. 8. **Monstrous Moonshine Conjecture** (Richard Borcherds, 1992) - Revealed unexpected relations between the Monster group and modular functions. 9. **Fundamental Lemma** (Ngo Bao Chau, 2008) - A key component in the Langlands program, connecting number theory and representation theory. 10. **Sphere Packing Problem in Dimensions 8 and 24** (Maryna Viazovska, 2016-2017) - Solved the densest sphere packing problems in these specific high dimensions. 11. **Atiyah–Singer Index Theorem** (Michael Atiyah and Isadore Singer, 1963) - Connected analysis, topology, and geometry by relating the analytical index of an elliptic differential operator to its topological index. 12. **Calabi Conjecture** (Shing-Tung Yau, 1977) - Proved the existence of Ricci-flat metrics on compact Kähler manifolds, leading to Calabi-Yau manifolds used in string theory. 13. **Weil Conjectures** (Pierre Deligne, 1974) - Provided analogs of the Riemann Hypothesis for varieties over finite fields. 14. **Mordell Conjecture (Faltings' Theorem)** (Gerd Faltings, 1983) - Demonstrated that a curve of genus greater than one over a number field has only finitely many rational points. 15. **Positive Mass Theorem** (Richard Schoen and Shing-Tung Yau, 1979) - Showed that, in general relativity, the total mass of a spacetime satisfying the dominant energy condition is non-negative. 16. **Green–Tao Theorem** (Ben Green and Terence Tao, 2004) - Proved that the sequence of prime numbers contains arbitrarily long arithmetic progressions. 17. **Bieberbach Conjecture** (Louis de Branges, 1985) - Solved a long-standing problem in complex analysis regarding the coefficients of univalent functions. 18. **Resolution of Singularities** (Heisuke Hironaka, 1964) - Demonstrated that singularities of algebraic varieties over fields of characteristic zero can be resolved. 19. **Oppenheim Conjecture** (Gregory Margulis, 1986) - Proved results about values of indefinite quadratic forms at integral points. 20. **Carleson's Theorem** (Lennart Carleson, 1966) - Established almost everywhere convergence of Fourier series for square-integrable functions. 21. **Solving of Hilbert's Tenth Problem** (Yuri Matiyasevich, 1970) - Showed that there is no general algorithm to solve all Diophantine equations, proving the problem undecidable. 22. **Deligne's Proof of the Riemann Hypothesis over Finite Fields** (Pierre Deligne, 1974) - Confirmed the last of the Weil conjectures, a major milestone in algebraic geometry. 23. **Fundamental Theorem of Algebraic K-Theory** (Daniel Quillen, 1973) - Developed higher algebraic K-theory, linking algebraic topology and algebraic geometry. 24. **Langlands Program Contributions** (Robert Langlands and others) - A vast network of conjectures and theorems connecting number theory and representation theory. 25. **Kadison–Singer Problem** (Adam Marcus, Daniel Spielman, and Nikhil Srivastava, 2013) - Solved a fundamental problem in operator theory with implications in signal processing. 26. **Gödel's Incompleteness Theorems** (Kurt Gödel, 1931) - Showed that any sufficiently powerful axiomatic system cannot be both consistent and complete. 27. **Perelman's Proof of the Geometrization Conjecture** (Grigori Perelman, 2003) - Extended his proof of the Poincaré Conjecture to a broader classification of 3-manifolds. 28. **Donaldson's Theorems on Smooth 4-Manifolds** (Simon Donaldson, 1983) - Revealed exotic smooth structures on four-dimensional spaces using gauge theory. 29. **Seiberg–Witten Theory** (Edward Witten and Nathan Seiberg, 1994) - Provided new invariants for 4-manifolds, revolutionizing the field of differential topology. 30. **Wiles–Taylor Proof of Fermat's Last Theorem** (Andrew Wiles and Richard Taylor, 1995) - Completed the proof by addressing gaps in the initial attempt, solidifying the Modularity Theorem. 31. **Jones Polynomial in Knot Theory** (Vaughan Jones, 1984) - Discovered a new knot invariant, leading to advances in low-dimensional topology. 32. **Szemerédi's Theorem on Arithmetic Progressions** (Endre Szemerédi, 1975) - Proved that any subset of integers with positive density contains arbitrarily long arithmetic progressions. 33. **Taniyama–Shimura–Weil Conjecture** (Now the Modularity Theorem) - Connected elliptic curves over the rationals with modular forms. 34. **Novikov's Proof of the Topological Invariance of the Pontryagin Classes** (Sergei Novikov, 1965) - Important in the topology of manifolds and characteristic classes. 35. **Smale's Proof of the Poincaré Conjecture in Higher Dimensions** (Stephen Smale, 1961) - Solved the Poincaré Conjecture for dimensions greater than four. 36. **Margulis Superrigidity Theorem** (Gregory Margulis, 1975) - Provided deep results in the theory of lattices in Lie groups. 37. **Vinogradov's Theorem on the Goldbach Conjecture** (Ivan Vinogradov, 1937) - Showed that any sufficiently large odd integer can be expressed as the sum of three primes. 38. **Harish-Chandra's Work on Representation Theory** (Harish-Chandra, mid-20th century) - Made significant contributions to the representation theory of semisimple Lie groups. 39. **Kolmogorov–Arnold–Moser (KAM) Theorem** (1954-1963) - Studied the stability of integrable systems under small perturbations. 40. **Borel–Weil–Bott Theorem** (Armand Borel, André Weil, Raoul Bott) - Connected representation theory and complex geometry. 41. **Grothendieck's Galois Theory** (Alexander Grothendieck, 1960s) - Extended Galois theory to schemes and étale cohomology. 42. **Witten's Work on Morse Theory** (Edward Witten, 1982) - Applied supersymmetry to provide analytic proofs of the Morse inequalities. 43. **Voevodsky's Proof of the Milnor Conjecture** (Vladimir Voevodsky, 1996) - Solved conjectures relating algebraic K-theory and Galois cohomology. 44. **Erdős–Rényi Law of Large Numbers** (Paul Erdős and Alfréd Rényi, 1970) - Fundamental results in probability theory and random graphs. 45. **Connes' Work on Noncommutative Geometry** (Alain Connes, 1990s) - Developed a framework extending geometry to noncommutative algebras. 46. **Faltings' Proof of the Shafarevich Conjecture** (Gerd Faltings, 1983) - Demonstrated finiteness results for abelian varieties over number fields. 47. **Hardy–Ramanujan Asymptotic Formula for Partition Function** (G. H. Hardy and Srinivasa Ramanujan, 1918) - Provided an exact formula for the partition function, which counts the number of ways an integer can be expressed as the sum of positive integers. 48. **Cartan's Classification of Simple Lie Algebras** (Élie Cartan, early 20th century) - Classified all simple finite-dimensional Lie algebras over the complex numbers. 49. **Lawson's Minimal Surface Conjecture** (Blaine Lawson, 1970s) - Advanced the theory of minimal surfaces in differential geometry. 50. **Riemann–Roch Theorem for Algebraic Curves** (Bernhard Riemann and Gustav Roch, 1850s) - Fundamental in algebraic geometry, relating the number of linearly independent sections of a line bundle to its degree and the genus of the curve. 51. **Noether's Theorem in Physics** (Emmy Noether, 1915) - Established a deep connection between symmetries and conservation laws in physics. 52. **Kervaire Invariant Problem** (Michael Atiyah, Daniel Quillen, and others) - Related to the existence of certain framed manifolds, solved in dimensions not equal to 126. 53. **Schmidt's Subspace Theorem** (Wolfgang Schmidt, 1972) - A key result in Diophantine approximation and transcendental number theory. 54. **Erdős–Kac Theorem** (Paul Erdős and Mark Kac, 1940) - Demonstrated the normal distribution of the number of prime factors of integers. 55. **Deligne–Lusztig Theory** (Pierre Deligne and George Lusztig, 1976) - Constructed representations of finite groups of Lie type. 56. **Coleman–Mandula Theorem** (Sidney Coleman and Jeffrey Mandula, 1967) - Restricts possible symmetries in quantum field theory. 57. **Cartan–Kähler Theorem** (Élie Cartan and Erich Kähler) - Fundamental in differential geometry and the theory of exterior differential systems. 58. **Turan's Theorem in Graph Theory** (Pál Turán, 1941) - Provides an extremal result for graphs without complete subgraphs of a given size. 59. **Hardy–Littlewood Circle Method** (G. H. Hardy and J. E. Littlewood) - Analytical technique in additive number theory. 60. **Bombieri's Theorem on Small Exponential Sums** (Enrico Bombieri) - Significant in analytic number theory and the distribution of prime numbers. This list includes many of the most challenging and impactful mathematical proofs across various fields such as number theory, topology, geometry, and mathematical physics. Each of these proofs has contributed significantly to the advancement of mathematics, often opening new areas of research or solving long-standing open problems.