Undergraduate Seminar II

Schedule

Organiser: Aditya Ghosh

Date: Wednesdays, Spring Semester 2026

Time: 12-2 PM

Location: Room 622, Mathematics Hall

Topics Covered

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!
8A	25 Mar	David Kuang	Politicians and Friends Todays talk will cover the "friendship theorem," a graph theory problem that states that in a finite graph where any two vertices have precisely one common neighbor, there is a vertex which is adjacent to all other vertices.
8B	25 Mar	-	[Cancelled]	-
9A	1 Apr	Philip Weaver	Coloring the Map: Why Five Colors Always Suffice How many colors do you need to color a map so that no two neighboring regions share a color? We will formalize this question by passing to the dual graph, reducing map-coloring to vertex-coloring of plane graphs. After a warm-up showing that Euler's formula forces every plane graph to be 6-colorable, we turn to the sharper result: every plane graph can be properly 5-list colored. The proof proceeds by induction on a stronger statement about near-triangulated graphs, splitting into two cases depending on whether the outer boundary cycle has a chord. When it does, the graph separates cleanly into two smaller near-triangulated pieces, each handled by induction. When it does not, a clever removal of one vertex and a two-color reservation argument in its neighborhood allows the inductive coloring to extend. Together, these cases yield a self-contained proof that the χℓ(G) ≤5 for all planar graphs G - one that, surprisingly, never once invokes Euler's formula.	-
9B	1 Apr	Joshua Fan	Dissecting Squares Oddly Can a square be partitioned into an odd number of triangles of equal area? While the even case is trivial, the odd case remained a mystery until 1970, when Paul Monsky provided a negative answer.	Ch 22
10A	8 Apr	Donovan Morgan	How to guard a museum. We prove the Art Gallery Theorem: any simple polygon with n walls can be guarded by at most ⌊n/3⌋ guards, using Fisk’s proof via triangulation, 3-coloring, and the pigeonhole principle.
10B	8 Apr	Leopold Brandl	Lines in the Plane We begin our discussion with the Sylvester–Gallai theorem, showing that any non-collinear set of points must determine an ordinary line containing exactly two points. Building on this, we prove that such configurations generate at least as many distinct lines as points, using elegant arguments like minimal counterexamples and induction. We then use our proof concepts to follow Aigner's proof on the bipartite decomposition of graphs.
11A	15 Apr	Odysseas Konstantakopoulos		Ch 11
11B	15 Apr	Yiming Song	Euler's formula and applications We introduce Euler's formula and briefly explain how it generalizes to higher dimensional shapes via the Euler characteristic. Then we give some fun applications to combinatorics.	Ch 13
12A	22 Apr	Vishal Muthuvel	Cotangent and the Herglotz trick We prove that $\pi \cot (\pi x) = \sum_{n \in \mathbb{Z}} \frac{1}{x+n}$ using the Herglotz trick. We also present an application of this series expansion of the cotangent function to the calculation of the Riemann zeta function at even integers.
12B	22 Apr	Keir Dorchen	The Chromatic Number of Kneser Graphs This talk discusses Kneser's conjecture, a statement about the colorability of Kneser Graphs. These graphs are an important link between combinatorics and topology, as illustrated by the conjecture's proof using the Borsuk-Ulam theorem.
12C	22 Apr	Chih Huang	Partition of natural numbers In this talk, we introduce the partitioning of a natural number, along with the interesting phenomenon that the number of odd partitions equals the number of unique partitions. We present two proofs showing why this is - the first proof is rooted in identity, while the second is based on bijections.	Ch 34
13A	27 Apr	Leopold Brandl	Cauchy's Rigidity Principle	Ch 14
13B	27 Apr	Anabelle Odell	This lecture presents Chapter 16 of Proofs from THE BOOK, exploring the question of how many d-dimensional simplices can be arranged in ℝᵈ such that every pair shares a complete (d−1)-dimensional face. Defining f(d) as the maximum number of such simplices, the lecture develops the answer through small cases (f(1) = 2, f(2) = 4, f(3) = 8) and motivates Bagemihl's 1956 conjecture that f(d) = 2ᵈ, which remains open for d ≥ 4. The lecture then establishes the bounds 2ᵈ ≤ f(d) < 2ᵈ⁺¹, stating the lower bound (Zaks, 1981) without proof and focusing on the upper bound (Perles). The upper bound is established through a combinatorial counting argument: each simplex is encoded as a row in a sign matrix recording its position relative to its supporting hyperplanes, and expanding this matrix by filling in all possible sign combinations yields a count that is constrained by the total number of possible sign vectors. Crucially, any point exterior to all simplices produces a sign vector necessarily absent from the matrix, making the inequality strict and yielding f(d) < 2ᵈ⁺¹.	Ch 16
13C	27 Apr	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.	Ch 41

UG Seminar II: Proofs from the Book

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