Symplectic structures roughly lie between smooth topology and complex geometry and exhibit both flexible and rigid phenomena. However, we still do not have a good understanding of what a general symplectic manifold should look like.

In this talk, I will begin by surveying some classical results of Gromov and McDuff from the 1980s, which use pseudoholomorphic curves to classify symplectic manifolds containing certain distinguished submanifolds.

I will then discuss a question of Donaldson in the 1990s, which relates smooth structures in dimension four to symplectic structures in dimension six.

Finally, I will describe some recent refinements of the Gromov-McDuff techniques and their applications to classification problems in the compact setting, which has many interactions with low-dimensional topology.

Spaces of complex polynomials contain a surprising amount of combinatorial structure, particularly in connections to noncrossing partitions. In this talk we will discuss how to extract this noncrossing data from a polynomial and how it determines a compactification and cell structure for a space of polynomials. This is joint work with Jon McCammond.

Zeta and L-functions play a central role in modern number theory. In this talk, I will discuss some classical contexts where these functions and their properties arise and how we can study them via a different kind of analytic object: a modular or automorphic form. The process of relating a modular form to an L-function is by forming a “period integral”. My goal is to give a sense (through examples) of how the study of periods has taken a central place in modern number theory. I will give an overview of some recent techniques and results in this area of research.

The infinite earring group, i.e. the fundamental group of a shrinking wedge of circles, is one of the first examples of a fundamental group that cannot be computed using standard methods in algebraic topology. This group is uncountable, not free, and perhaps best thought of as an infinitary analogue of a free group. In this talk, I’ll give a friendly introduction to the methods used to characterize this group and derive its properties. I’ll also discuss how it has been applied in recent decades and mention a few open questions.

Many groups arise naturally as the symmetry group of a geometric or topological object. In this talk we will focus on infinite but 0-dimensional topological spaces, specifically those that arise as end spaces. Their homeomorphism groups can be thought of as infinite analogs of symmetric groups. We will start with an overview of some basic ideas in geometric group theory, before focusing in on the homeomorphism groups of countable end spaces. These are large topological groups, which are not finitely generated, so classical tools and techniques don’t apply. Instead, we turn to descriptive set theoretic techniques of Rosendal to study their large-scale geometry and present a trichotomy classification theorem. This talk includes joint work with George Domat and Robbie Lyman.