Welcome to the "Proofs from the book" seminar. This seminar is part of the course requirements of students for "MATHUN3952_001_2026_1 - UNDERGRADUATE SEMINARS II". The students are expected to give talks every week on a chapter from Ziegler and Aigner's book (see References). We will be covering elegant proofs from various areas of maths with varying difficulty levels. Please email me by the Sunday before the talk with your selected topic so that I can update the website. You can sign up for a talk by putting you name down here.
We have 12 days of seminar this semester. Each seminar will have 2 talks of 50 minutes. The base grading criteria would be the following:
Organiser: Aditya Ghosh
Date: Wednesdays, Spring Semester 2026
Time: 12-2 PM
Location: Room 622, Mathematics Hall
| Serial No. | Date | Speaker | Title and Abstract | Notes |
|---|---|---|---|---|
| 1 | 28 Jan | Aditya Ghosh | Proofs of infinitude of primes. We discuss various proofs of the infinitude of prime numbers. We start with the classical proof by Euclid and move on to more advanced proofs involving topology and analysis. | Ch 1 |
| 2A | 4 Feb | Jane Chai | Representing two numbers as squares. I will discuss which numbers can be represented as a sum of two squares via Fermat's Theorem on Sums of Two Squares. We differentiate between "good" and "bad" primes and show some elegant proofs for the condition under which a prime or composite number can be written as the sum of two squares. | Ch 4 |
| 2B | 4 Feb | Yiming Song | Hilbert's third problem. In two dimensions, a polygon can be cut up and rearranged into another polygon if and only if they have the same area. In three dimensions, the question of whether volume plays the same role is known as Hilbert's third problem. The answer is no: there exist polyhedra with equal volume, but cannot be cut up and rearranged into one another. We will construct and prove a counterexample. | Ch 10 |
| 3A | 11 Feb | David Kuang | Having Fun with Inequalities. In today's talk I will be covering the importance of inequalities in analysis, beginning with the Cauchy-Schwarz inequality and moving on to interesting examples with means, polynomials, and graph theory. | Ch 20 |
| 4A | 18 Feb | Chih Huang | Pigeon-hole and double counting. We discuss Pigeon-hole principle and double counting, and show some interesting results in number theory, graph theory, and topology that can be elegantly proved through the two principles. | Ch 28 |
| 4B | 18 Feb | Joshua Fan | <Wedderburn's Little Theorem. We will discuss Ernst Witt's proof that every finite division ring is a field. | Ch 6 |
| 5A | 25 Feb | Donovan Morgan | Cayley's formula for the number of trees. We will prove that the number of labeled trees on n vertices is nⁿ⁻² two ways: a bijection via Prüfer codes and a double counting argument on rooted forests. | |
| 5B | 25 Feb | Keir Dorchen | The Borromean Rings don't exist. This talk will focus on a class of links called Brunnian Links, of which the Borromean links are a member. It will discuss the nature of these links and whether or not they can be constructed using perfect circles. | |
| 6A | 4 Mar | Jane Chai | Completing Latin Squares. We will look at Latin squares, one of the oldest combinatorial objects, and deduce when a partial Latin square can be completed to a full Latin square of the same order. | Ch 36 |
| 6B | 4 Mar | Vishal Muthuvel | Permanents and the power of entropy. We prove Brégman's theorem: an upper bound for the permanent of integral matrices with prescribed row sums. We present Radhakrishnan's method of proof, which entails a clever application of the concept of entropy from information theory. | Ch 37 |
| 7A | 11 Mar | Philip Weaver | How many times do you need to shuffle a deck of cards before it is truly random? We will formalize this question using the variation distance between probability distributions on permutations and analyze two shuffling models. For "top-in-at-random" shuffles, we show that roughly $n \log n$ shuffles suffice via a connection to the coupon collector's problem. For the more realistic riffle shuffle we prove that only about $2 \log_2 n$ shuffles are needed, with the analysis relying on a strong uniform stopping rule whose behavior is governed by the birthday paradox. For a standard 52-card deck, this means roughly 7 shuffles bring the deck close to uniformly random. Time allowing, I might try to link 'psuedo-randomness' to common parlor tricks. | Ch 31 |
| 7B | 11 Mar | Anabelle Odell | Symmetric Matrices, Orthogonal Diagonalization, and Hadamard's Bound. This lecture presents two results in linear algebra through the lens of elegance of proof. I begin with the Spectral Theorem, that every real symmetric matrix admits an orthonormal eigenbasis, proved via Herb Wilf's 1981 minimization argument. Rather than constructing an eigenbasis directly, we minimize off-diagonal energy over the compact group O(n) and show the minimizer must be diagonal. No eigenvalues, no induction; just compactness and a first-order perturbation. I then turn to Hadamard's 1893 determinant problem, bounding |det A| for matrices with entries in [−1, 1]. Writing (det A)² = det(AAᵀ) reduces the problem to bounding a product of eigenvalues, which AM-GM dispatches cleanly to give the sharp bound n^(n/2). The lecture closes with the question of when equality holds, Hadamard matrices, and the observation that their existence for all n divisible by 4 remains open after 130 years. | Ch 7 |
| 18 Mar | Spring Break! | |||
| 25 Mar | TBD |
Aditya Ghosh: ag4794 (at) columbia (dot) edu