Spring 2024

Where the Wild Forms Are

Speakers: Hahn Lheem

One can only go so far into number theory before encountering modular forms. Their definition seems innocuous, but they contain incredible arithmetic information. We will see how modular forms arise to find the number of solutions to some Diophantine equations, then map out the relation between modular forms and other important objects in number theory.

(Almost) Complex Structures on 4-Manifolds

Speakers: Dhruv Goel

Have you ever wondered why the connected sum of two complex projective planes is not a(n) (almost) complex manifold? (Hint: it's because 14 is not the sum of two squares.) The theory of characteristic classes is one of the most powerful tools from 20th-century algebraic topology that can be used to study smooth manifolds. In this talk, I will discuss the fundamentals of Chern and Pontryagin classes, and explain how they can be used to compute topological invariants (such as the Euler characteristic or signature) of manifolds. I will then explain how simple numerology can be used to prove results about the existence of almost complex structures on 4-manifolds, and end by talking about connections to the moduli space of solutions to Seiberg-Witten Equations on 4-manifolds.

Polynomial analogs of questions in number theory

Speakers: Hari Iyer

We discuss some analogs of hard, classical questions in number theory --counting prime numbers, solving equations-- which sometimes become easier when numbers are replaced by polynomials; this is known as a "function field analog," since the polynomials in question can also be viewed as functions on curves defined modulo a prime. Famous examples of such questions include the Riemann hypothesis and Fermat's Last Theorem. Time permitting, we may mention a curious application of this perspective to resolving a conjecture of Ramanujan on coefficients of modular forms.

The statistics of increasing subsequences in biased random permutations

Speakers: Jonas Iskander

How does the cycle type of a random permutation affect the expected number of increasing subsequences of a given length? Recently, there has been growing interest in how the group structure on the symmetric group interacts with the natural ordering on the letters 1 through n. In a 2022 paper, Gaetz and Pierson studied the expected value of the number of length k increasing subsequences times a symmetric group character, and they found explicit formulas for these expected values in several special cases. In this talk, I will discuss how I generalized the results of Gaetz and Pierson to obtain a much broader class of explicit formulas for these expected values. In particular, my results imply explicit formulas for the expected number of length k increasing subsequences when a permutation is randomly generated according to a distribution that favors certain cycle types.

Anosov Actions of PSL2(Z) on T^3

Speakers: Katherine Tung

Hurder (1992) showed that the natural action of SL2(Z) on the torus T^2 is not topologically rigid: there is a family of deformations that are not topologically conjugate to the standard action. By contrast, in dimensions higher than 2, the standard action of SLn(Z) on T^n is rigid, so there are no such families of deformations. There is a natural irreducible representation from PSL2(Z) to SL3(Z) inducing an action of PSL2(Z) on the 3-torus. Is this action rigid or not? We resolve this question by extending a technique of Hurder while preserving the elementary nature of his argument. This talk is based off of a Northwestern REU project I worked on with Tanner Leonard, Nathan Louie, and Paul Shin.

Graduate Student Panel

Speakers: NA

Merrick Cai - Mathematics graduate student at Harvard 

Ricky Li Economics - graduate student at MIT 

Hanna Mularczyk - Mathematics graduate student at MIT

Statistical Inference of Finite-Population Galton-Watson Branching Processes

Speaker: Ivan Specht

Abstract: The Galton-Watson process is a stochastic branching process in which the offspring distribution of each vertex is an independent and identically distributed (i.i.d.) N-valued random variable following a prespecified distribution ξ. Here, we consider a modification of the process in which the set of vertices in any realization is sampled from a fixed, finite set. In this modification, which we refer to as a finite-population Galton-Watson process, offspring distributions are no longer i.i.d., and must be re-normalized based on the remaining population size after each draw. We show that for large population size, the probability of a finite-population Galton-Watson process containing a fixed subtree admits a tractable approximation with tight error bounds for several choices of ξ. Finally, we discuss the importance of approximating this probability for modeling infectious disease transmission, a common application of the Galton-Watson process.

The Pursuit of Quasi-excellence: Using complete local rings to study the prime spectra of quasi-excellent local integral domains

Speaker: AnaMaria Perez

Abstract: Complete local rings are well understood due to a powerful theorem from Cohen.The completion of a local ring with maximal ideal $M$ can be defined using the $M$-adic metric. Characterizing the relationship between a local ring and its completion allows us to deduce results about the original ring. In particular, we characterize the relationship between quasi-excellent local domains and their completions. We prove that $T$, a complete local ring of characteristic 0 with maximal ideal $M$, is the completion of a local quasi-excellent integral domain if and only if no nonzero integer of $T$ is a zero-divisor and $T$ is reduced. We also provide necessary and sufficient conditions for countable local quasi-excellent domains. One notable application of our results is that there is no bound on how non-catenary a quasi-excellent integral domain can be, a result that was previously unknown. Finally, we use these results to motivate the study of formal fiber rings and provide a characterization for completions of local integral domains where we can control formal fibers of countably many principally generated ideals. We may extend these characterizations to completions of countable local domains, quasi-excellent local domains, and excellent local domains.