Spring 2024

On the Power of Proof: Characterizing the Gap Between Provability and Truth

Speaker: Ava Zinman

Abstract: This talk seeks to distinguish between mathematical terms that provide a strong characterization of correctness; namely, proof and truth. We first formalize our motivating question, the Henkin Problem, and subsequently answer it by proof of Löb’s Theorem. Gödel’s 2nd Incompleteness Theorem, a corollary to Löb’s result, inspires the use of forcing to model ZFC-provability. We ultimately conclude that mathematical uncertainty stems from the inaccessibility of “natural” truth, which must be resolved by use of intuition.

Class Groups, Forms, and Arithmetic Statistics

Speaker: Eliot Hodges

Abstract: The ideal class group of a number field F measures the extent to which the integral elements of F fail to factor uniquely into primes. Of central importance in number theory, ideal class groups were first studied by Gauss under the guise of binary quadratic forms. From this perspective, multiplication of ideals corresponds to Gauss's celebrated composition law. In this talk, we develop Gauss's theory of binary quadratic forms and discuss its arithmetic reinterpretation. We then present a modern reinterpretation of Gauss composition à la Bhargava and give an overview of many astounding arithmetic-statistical consequences of this theory.

Random Mosaics in Machine Learning

Speaker: Calvin Osborne

Abstract: Stochastic geometry, at the intersection of convex geometry and probability, is the study of spatial point processes such as random mosaics. Separately, random feature methods are a technique in machine learning used to decrease the computational cost of kernel machines in large-scale problems. In this talk, I will describe the surprising application of random mosaics to random feature methods from research I conducted at the California Institute of Technology; fist, we will investigate the abstract definition and a few examples of random mosaics; next, we will discuss a number of important constructions relating to random mosaics, such as the typical cell and associated zonoid; and finally, I will discuss the particular application of random mosaics that I studied as related to the uniformly rotated Mondrian kernel.

Word Problems Are Hard

Speaker: William Hu

Abstract: Elementary schoolers think word problems are hard. In 1955, Pyotr Novikov showed that computers also think that (certain types of) word problems are hard. Namely, there exists a group whose word problem is uncomputable: it is impossible to write a computer program that takes as input any word inside that group and determines whether it is equal to the identity element. We will talk about why this result is true, its connections to the analogous problem on semigroups, and other related properties. No familiarity with the notion of computability (or even computers) is needed.

Graduate Student Panel

Panelists: 

• Enrico Colon (Harvard, Math)

• Thomas Kaminsky (Harvard, Computer Science)

• Dora Woodruff (MIT, Math)

\zeta(3) and Friends

Speaker: Daniel Hu

Abstract: By now, values of the Riemann zeta-function at odd integers, such as \zeta(3), have averted our understanding for hundreds of years. Numbers like these are responsible for an entire industry known as special values of zeta-functions. I will share (what I know about) certain numerical, combinatorial, and algebro-geometric aspects of this problem.

Ricci Curvature of Directed Graphs

Speaker: Ellie Wiesler

Abstract: This talk will explore the notion of curvature on undirected and directed graphs using concepts from discrete differential geometry and introduce the exciting applications of graph curvature in geometric machine learning and network science. We will focus on a local notion of curvature called Ricci curvature and how it can be generalized to metric spaces endowed with probability measures or random walks as done with Ollivier Ricci curvature. We will then discuss recent research in defining Ollivier Ricci curvature on undirected and directed graphs using optimal transport and fundamental differential geometry, and finally the mathematical challenges involved with capturing discrete local curvature when directionality constraints are present.