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 |
| 3B | 11 Feb | Shilpa Kesavan | Three Famous Theorems on Finite Sets. We will discuss and prove three major theorems about families of subsets of finite sets: Sperner's theorem, the Erdős–Ko–Rado theorem, and Hall's marriage theorem. |
Aditya Ghosh: ag4794 (at) columbia (dot) edu