Spring 2021

Friends Prize Winners: Kenz Kallal and Lux Zhao 

Lux is speaking on The history and mathematics of the axiom of choice.

and 

Kenz is speaking on The Arthur–Selberg trace formula and some applications to arithmetic statistics.

Sperner's Lemma

Speaker: Alec Sun

Abstract:  Sperner's Lemma is a combinatorial analog of Brouwer's Fixed Point Theorem. Given a triangle ABC with triangulation T, suppose that the set of vertices of T is colored in 3 colors such that A, B, C are colored 1, 2, and 3, and each vertex on an edge of ABC is colored only with one of the 2 colors of the ends of the edge. Sperner's Lemma states that given such a 3-coloring, then there exists a triangle from T whose vertices are colored with 1, 2, and 3. After sketching a proof, we will see how Sperner's Lemma can be applied to resolve fair division problems as well as establish the Brouwer's Fixed Point Theorem.

The Congruent Number Problem and the BSD Conjecture

Speaker: CJ Dowd

Abstract:  When can a given rational number be expressed as the area of a right triangle with rational side lengths? This question has not received a complete answer despite being around for centuries. We will discuss two other equivalent forms of this problem, one involving squares in arithmetic progression and the other involving elliptic curves. We conclude by presenting Tunnell’s theorem, which provides an answer to the Congruent Number Problem but relies on the unproven Birch and Swinnerton-Dyer Conjecture, one of the million-dollar Millennium problems!

Math Pool Table

Speaker: Forrest Flesher

Abstract:  In this talk, we’ll discuss mathematical billiards, a fun but also useful topic, which combines ideas from various branches of mathematics and has many applications. In particular, we’ll investigate billiards on arbitrary smooth convex tables, and see how to transform these into dynamical systems using twist maps. Using this, we’ll see how it is possible to prove results which initially seemed intractable, including the existence of infinitely many periodic orbits.

Jobs Panel:

 Interested in learning about careers you could pursue which would use the quantitative skills you've developed studying mathematics? Come to our jobs panel to ask questions and hear from the following Harvard alums about their chosen career paths.

Meena Boppana - High School Math Teacher

Lily Ge - Quantitative Trader

Luran He - Machine Learning Engineer

A Trip to the Tropics

Speaker: Amal Mattoo

Abstract:  In this talk we will introduce tropical geometry, a young field of mathematics at the interface of algebraic geometry and combinatorics. We will explore analogues of fundamental classical results, like Bézout's Theorem, as well as surprising connections between classical and tropical objects, such as the Fundamental Theorem. We hope to show that tropical geometry is both an elegant theory and a useful tool – with applications from curve counting to auction design.

Summations and Approximations and Recurrence Relations, Oh My!

Speaker: Kyle Fridberg

Abstract:  Investigating the convergence of sequences and series lies at the heart of real and complex analysis, with diverse applications in areas such as physics, statistics, and finance. The harmonic series—a special case of the simplest type of Dirichlet series—is a classic example of a series whose terms approach zero but the full series diverges. In this talk, we will investigate how to use a recurrence relation, the AM-GM inequality, and Stirling’s approximation to split divergent Dirichlet series into sequences of convergent subseries. Then, we will use this result to numerically approximate very large sums to a high accuracy and discuss some further questions that arise. 

Class numbers, prime geodesics, and automorphic forms (after Sarnak)

Speaker: Kenz Kallal

Abstract:  Given fixed integers a, b, c, which primes can be written as aX + bXY +cY for integers X, Y? This simple question in number theory has generated vast amounts of mathematics over the past 400 years. Of central importance to the answer in the general case are more abstract quantities called class numbers. These are individually very mysterious, but on average seem to be well-behaved. This talk is about an asymptotic law for truncated averages of class numbers, originally from Sarnak's 1980 Stanford thesis. The proof will take us far away from the elementary terms in which the original question was stated: first to the Riemannian geometry of modular curves, and then (via the Selberg trace formula) to the analytic theory of automorphic forms.