# Book: Theorems

See: Map of Math

Discover patterns in how a mathematical theorem holds mathematical knowledge.

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

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

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

• 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.