Schedule, Notes/Handouts & Videos

In this talk we will explore the improvisational and musical nature of mathematics and physics.

In this talk, I’ll chat about wonderful algebraic structures that were discovered in the early 1900’s: Frobenius algebras. I will survey the 100+ year history of the development and uses of these structures, ending with very recent research results from joint work with Harshit Yadav.

Knot concordance can be regarded as the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Specifically, two knots K_0 and K_1 are said to be smoothly concordant if there is a smooth embedding of the annulus S^1 × [0, 1] into the “cylinder” S^3 × [0, 1] that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set C of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. The algebraic structure of C, the concordance class of the unknot, and the set of knots that are topologically slice but not smoothly slice are much studied objects in low-dimensional topology. Gauge theoretical results on the nonexistence of certain definite smooth 4-manifolds can be used to better understand these objects. In particular, the study of anti-self dual connections on 4-manifolds can be used to shown that (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to Z^\infty, and (2) satellite operations that are similar to cables are not homomorphisms on C.

"Modular forms" on exceptional groups are a special class of automorphic functions on these groups, whose study was initiated by Gan, Gross, Savin, and Wallach.  I will define these objects, and give an overview of what is known about them.

The mathematics and statistics community has had to take a hard look at itself and assess whether the tenets that we hold most dear in our community are truly aligned with our goals of excellence in the mathematical and statistical sciences. The central dilemma is that mathematical and statistical scientists are an incredibly heterogenous group and do not agree on what this "excellence" looks like. We will look to examples within STEM from all levels of education -- from the "New Math" to the recent revisions to the California Mathematics Framework, from W. E. B. DuBois' data visualizations of institutionalized racism presented at the 1900 Paris World's Fair to the trailblazing work of Drs. David Blackwell and Richard Tapia -- to show how what was once considered strange is now familiar, and how our field has been, and always will be, changing for the better.

Link to New Math Animation video: https://www.youtube.com/watch?v=UIKGV2cTgqA

We will give an overview of some of Tatiana Toro's outstanding contributions to geometric measure theory, potential theory and free boundary theory.

In this talk we will look at domains in Euclidean space through different lenses using tools from geometric measure theory, potential theory and partial differential equations. Our goal is to illustrate how what we see through one lens predicts what we can see through another, and to establish equivalences through these observations.