Horizons in Mathematics

Horizons in Mathematics brings outstanding mathematicians from traditionally under-represented backgrounds to Rice University, where they speak in colloquium and participate in a small lunch with members of the department.

Spring 2020 Speakers

Some congruences and consequences in number theory and beyond, February 6.

Ellen Eischen, Associate Professor of Mathematics at the University of Oregon.

Abstract: In the mid-1800s, Kummer observed some striking congruences between certain values of the Riemann zeta function, which have important consequences in algebraic number theory, in particular for unique factorization in certain rings. In spite of its potential, this topic lay mostly dormant for nearly a century until it was revived by Iwasawa in the mid-1950s. Since then, advances in arithmetic geometry and number theory (in particular, for modular forms, certain analytic functions that play a central role in number theory) have enabled substantial extension to congruences in the context of other arithmetically significant data, and this has remained an active area of research. In this talk, I will survey old and new tools for studying such congruences. I will conclude by introducing some unexpected challenges that arise when one tries to take what would seem like immediate next steps beyond the current state of the art.

Relations and Ramification Loci in the Moduli Space of Curves, February 20.

Emily Clader, Assistant Professor of Mathematics at San Francisco State University.

Abstract: Enumerative geometry studies questions like: “given five points in the plane, how many conics pass through all five?” The modern approach to such questions is to translate them into intersection theory problems on a moduli space, and one of the most important of these is the Deligne-Mumford moduli space of curves. I will discuss what this space is, give a tour of some of its intersection theory, and describe some of my own work studying subspaces of the moduli space of curves that generalize the classical hyperelliptic locus.

Our next Horizons speaker will be Bianca Viray, Associate Professor of Mathematics at the University of Washington, on April 16. *****This visit has been canceled******

If you are interested in attending our lunch, please email one of the organizers (listed below).

Past Speakers:

April 2019: Susan Tolman, Professor of Mathematics at UIUC.

Title: On toric degenerations and cohomological rigidity for symplectic toric manifolds

Abstract: Given an integral polytope there is a naturally associated toric variety, that is, an n-dimensional algebraic variety with an n-dimensional torus action. We call it a toric manifold if the variety is smooth. In this case, it inherits a symplectic form. A conjecture, called “cohomological rigidity”, posits that any toric manifolds with isomorphic cohomology rings are diffeomorphic. This has been proved in various special cases. There’s a natural symplectic analog of this conjecture: If there is an isomorphism of cohomology rings which preserves the symplectic cohomology class, then the manifolds are symplectomorphic. Unfortunately, this has been difficult to prove, because it is hard to construct symplectomorphisms. Adapting ideas from Harada and Kaveh, we use toric degenerations to find symplectomorphisms between certain toric manifolds. This generalizes the well-known isomorphisms between different Hirzebruch surfaces. We then use this to prove the symplectic analog of cohomological rigidity under certain assumptions. For example, it holds when the cohomology ring is isomorphic to the cohomology ring of the product of n two-spheres. This talk is based on work-in-progress, which is joint with Milena Pabiniak.

February 2019: Jen Hom, Associate Professor of Mathematics at Georgia Tech.

Homology cobordism and Heegaard Floer homology

Abstract: One generally considers 3-manifolds up to homeomorphism. We consider 3-manifolds up to a weaker notion of equivalence, called homology cobordism. Under this equivalence relation, the set of 3-manifolds forms a group, with the operation induced by connected sum. We use Heegaard Floer homology to give new results about the structure of this group. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

November 2018: Amanda Folsom, Associate Professor of Mathematics at Amherst College.

Mock modular and quantum modular forms

Abstract: Mock modular forms were first formally defined in the literature by Zagier in 2007, though their roots trace back to the mock theta functions, curious functions described by Ramanujan in his last letter to Hardy in 1920. As the overarching theory of harmonic Maass forms has progressed over the last 15 years, we have seen applications of mock modular forms in many areas, including number theory, combinatorics, representation theory, and more. Zagier also defined quantum modular forms in 2010, which like mock modular forms feign modularity in some way, but unlike mock modular forms are defined on sets of rational numbers. In this talk, we will give an introduction to these subjects, and will also discuss an application in joint work with Ono (Emory) and Rhoades (CCR Princeton), in which we revisit Ramanujan's last letter and prove one of his remaining claims as a special case of a more general result.

We are indebted to Brown University's Horizons Seminar and UT Austin's Distinguished Women in Mathematics Series for many ideas and some of our language.