Let X be an algebraic variety -- that is, the solution set to a system of polynomial equations.  Then the fundamental group of X has several incarnations, reflecting the geometry, topology, and arithmetic of X.  This talk will discuss some of these incarnations and the subtle relationships between them, and will describe an ongoing program which aims to apply the study of the fundamental group to classical problems in algebraic geometry and number theory.

Hilbert's 19th problem asks if minimizers of "natural" variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDEs. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, as well as outstanding open problems. Parts of this are joint works with A. Figalli and O. Savin.

This talk will illustrate some topological properties of the space F_k(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of the configuration spaces F_k(M) to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which these configuration spaces stabilize. In this talk I will explain these stability patterns, and describe a higher-order “secondary representation stability” phenomenon among the unstable homology classes. These results may be viewed as a representation-theoretic analogue of current work of Galatius–Kupers–Randal-Williams. The project is joint with Jeremy Miller.

Many real-world networks -- social, technological, biological -- have wonderful structures. Some structures may be apparent (such as trees) while others may be hidden (such as communities). How can we discover hidden structures? Known approaches to "structure mining" in networks come from a variety of disciplines, including probability,  statistics, combinatorics, physics, optimization, theoretical computer science, signal processing and information theory. We will focus on new probabilistic approaches to structure mining. They bring together insights from random matrices, random graphs and semidefinite programming.

This is a joint applied math and probability seminar.

I will highlight some of the methods I've previously employed in undergraduate mathematics education. These methods were drawn from the research of others and some of my own inquires. I will give insight into how the methods are brought together so that they form a cohesive whole, while fitting the needs of the both the students and the department.

Central to the discussion is a particular enlightening experience I had with one of my students.

The experience has inspired my current methods. I will discuss this my vision for further developing curriculum and more so fostering a lively dynamic atmosphere around mathematics courses, both within and without the classroom walls.

Time permitting, I'll discuss some research that is underway with regards to these approaches.