Before starting my PhD I would often try to prove a theorem myself before reading its proof. On this page I have collected some of the proofs that I discovered in the process. I do not claim that these proofs are new to the mathematical community. It is possible that these proofs have already been discovered before.
The Sylow Counting Theorem Proof
Here I provide an alternate proof of the Sylow Counting theorem which talks about the ramainder left when the number of Sylow p-subgroups of a finite group G is divided by a prime p. The proof is a variant of the technique used in Wielandt's proof of the Sylow Existence theorem.
The Primary Decomposition Theorem Proof
We prove the primary decomposition theorem. The proof we give seems a little bit easier than the one given in the excellent book Linear Algebra by Hoffman and Kunze.
The Constant Rank Level-Set Theorem Proof
This document contains a proof of the Constant Rank Level-Set theorem without using the Constant Rank Theorem. We have proved the Submersion Level-Set Theorem first (the proof of which is taken from Milnors' Topology from the Differentiable Viewpoint). Then we develop some lemmas to finally give a short and neat proof of the Constant Rank Level-Set Theorem.
The Transition Map Property Proof
Here I present a proof of the transition map property which states that any two coordinate patches on a manifold in R^n are 'smoothly compatible'.
Dilworth's Theorem Proof
At least 4 different proofs of Dilworth's Theorem are already known. My proof is essentially the same as the one found by Perlez.
Sperner's Theorem Proof
Sperner's Theorem is a popular theorem in Combinatorics. A very elegant proof exists due to Lubell and can be found on Wikipedia. My proof is quite different from the proof due to Lubell.
Punctured Combinatorial Nullstellensatz Proof
The Punctured Combinatorial Nullstellensatz (due to Ball and Serra) is an important generalization of Alon's Combinatorial Nullstellensatz. In the original proof, the authors give a short proof by using cylindrical reduction and then applying Alon's Nullstellensatz. In the proof given here, we first develop the description of polynomials vanishing on punctured grids separately and from there give a rather more straightforward approach to prove PCN. The 6th problem posed at IMO 2007, widely regarded as one of the most difficult problems ever posed at the IMO, has a very short solution using PCN.
De Bruijn-Erdos Theorem Proof
A delightful theorem on finite set systems.
Folkman's Theorem Proof
Given a positive itneger k and given any fintie coloring of the naturals, one can find a multiset of size k such that all the fintie sums that can be formed using this multiset have the same color.
The 'Edge Cover - Independent Set' Theorem Proof
A very artistic proof can be found in West's 'Introduction to Graph Theory'. I was happy to find an alternate proof.
Richardson's Theorem Proof
Richardson's Theorem states that every simple digraph with no odd cycles has a kernel. The proof given here uses an interesting equivalence relation to tackle the problem.
A Generalization of Hall's Marriage Theorem Proof
Hall's theorem is covered in any introductory course in graph theory. I along with my friend Siddharth tried to generalize it and we were successful in doing so. We later found that this generalization already had a name -- 'The Harem Problem'.
Berge's Matching Condition Proof
Matching is an important concept in graph theory and Berge has provided an if and only if condition for a matching to be a maximum matching in a graph in terms of augmenting paths. The Berge's Matching Condition states that a matching M in a graph G is a maximum matching if and only if G has no M-augmenting path. A very elegant proof of this theorem goes by exploiting the simple fact that the edges of the symmetric difference of two matchings M and M' in a graph G form components which are either paths or even cycles. We provide here a different proof based on extremal arguments.
Manifolds are Paracompact Proof
We give a short proof of the fact that topological manifolds are paracompact.
Beatty's Theorem Proof
I first found out about Beatty's Theorem in a Math-Olympiad book which had a short and slick solution. I successfully proved it myself although my proof was considerably longer and very different from the elegant proof given in the book.
Wolstenholme's Theorem Proof