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I am interested in identifying, studying and understanding key results in mathematics. I want to include them in my map of mathematics to show the role they play in defining concepts and establishing and relating branches of mathematics. The results may typically be understood as theorems. Some theorems are meaningful for their own sake and others for helping prove results. I want to understand how a great theorem works and what makes it so important.

Collections of theorems

• 维基百科: List of theorems
• 维基百科: List of fundamental theorems
• 维基百科: List of lemmas
• Math Stack Exchange: Overpowered Theorems
• The Hundred Greatest Theorems
• Oliver Knill. Favorite math theorems
• Oliver Knill. Some Fundamental Theorems in Mathematics. 2018.
• 播客: My Favorite Thorem

Fundamental theorems. Elementary symmetric functions are analogous to prime factorizations of numbers - monomial symmetric functions are analogous to the numbers as such ("natural" basis - "natural" numbers +1)

Fundamental theorems

• Orbit-stabilizer
• Analytic function
• Classification of closed surfaces
• Short exact sequence of chain complexes implies long exact sequence of their homology groups
• Changing the order of integration
• The existence of a primitive root in (Z/pZ)×
• Halting problem
• Existence of nonmeasurable Lebesgue sets
• Row-rank equals column-rank (f.t. of linear algebra?)
• Convergence criteria for geometric series (r<1)
• Convergence of monotone sequences: Any Lp-integrable function can be Lp-approximated by step functions.
• Fourier transform and Fourier series
• Laplace transform
• flowbox theorem - Frobenius theorem

  • ABC Conjecture
  • Atiyah-Singer Index Theorem
  • Banach-Tarski Paradox
  • Birch-Swinnerton-Dyer Conjecture
  • Carleson's Theorem
  • Central Limit Theorem
  • Classification of Finite Simple Groups
  • Dirichiet's Theorem
  • Ergodic Theorems
  • Fermat's Last Theorem
  • Fixed Point Theorems
  • Four-Color Theorem
  • Fundamental Theorem of Algebra
  • Fundamental Theorem of Arithmetic
  • Goedel's Theorem
  • Gromov's Polynomial-Growth Theorem
  • Hilbert's Nullstellensatz
  • Independence of the Continuum Hypothesis
  • Inequalities
  • Insolubility of the Halting Problem
  • Insolubility of the Quintic
  • Liouville's Theorem and Roth's Theorem
  • Mostow's Strong Rigidity Theorem
  • P versus NP Problem
  • Poincare Conjecture
  • Prime Number Theorem and the Riemann Hypothesis
  • Problems and Results in Additive Number Theory
  • From Quadratic Reciprocity to Class Field Theory
  • Rational Points on Curves and the Mordell Conjecture
  • Resolution of Singularities
  • Riemann-Roch Theorem
  • Robertson-Seymour Theorem
  • Three-Body Problem
  • Uniformization Theorem
  • Weil Conjectures

Proofs from THE BOOK

• Number Theory
  • Six proofs of the infinity of primes
  • Bertrand's postulate
  • Binomial coefficients are (almost) never powers
  • Representing numbers as sums of two squares
  • The law of quadratic reciprocity
  • Every finite division ring is a field
  • The spectral theorem and Hadamard's determinant problem
  • Some irrational numbers
  • Three times pi2/6
• Geometry
  • Hilbert's third problem: decomposing polyhedra
  • Lines in the plane and decomposition of graphs
  • The slope problem
  • Three applications of Euler's formula
  • Cauchy's rigidity theorem
  • The Borromean rings don't exist
  • Touching simplices
  • Every large point set has an obtuse angle
  • Borsuk's conjecture
• Analysis
  • Sets, functions, and the continuum hypothesis
  • In praise of inequalities
  • The fundamental theorem of algebra
  • One square and an odd number of triangles
  • A theorem of Polya on polynomials
  • On a lemma of Littlewood and Offord
  • Cotangent and the Herglotz trick
  • Buffon's needle problem
• Combinatorics
  • Pigeon-hole and double counting
  • Tiling rectangles
  • Three famous theorems on finite sets
  • Shuffling cards
  • Lattice paths and determinants
  • Cayley's formula for the number of trees
  • Identities versus bijections
  • The finite Kakeya problem
  • Completing Latin squares
• Graph Theory
  • The Dinitz problem
  • Permanents and the power of entropy
  • Five-coloring plane graphs
  • How to guard a museum
  • Turan's graph theorem
  • Communicating without errors
  • The chromatic number of Kneser graphs
  • Of friends and politicians
  • Probability makes counting (sometimes) easy

Ideas about theorems

• Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information.
• For example, in this proposition about pullbacks, the statement about the pullback is much more explicit than that for the function f because it includes f as a special case when Z = x, for then f*(idX)={$f^{-1}$}. But that special case leverages the framework to establish all the other cases.
• When you get the definitions right, the theorems are easy to prove. When the theorems are hard to prove, then the definitions are not right. (Tobias Osborne) So this shows how definitions and theorems coevolve.

Literature

Readings about famous theorems

• What Does the Spectral Theorem Say?
• In Search of God’s Perfect Proofs: Proofs from THE BOOK