Events of the Week

In this talk, we study quadratic twists of elliptic curves over totally real number fields. Using ideas from class field theory and Hecke characters, we show that under suitable hypotheses, there exist large families of quadratic twists with trivial ell-Selmer group, ell is a prime in S(n). The argument imposes local conditions on quadratic extensions and at the bad primes of the elliptic curve, and relates these conditions to Selmer groups via results of Morrow and Takai, without requiring an explicit description of the twists. This is ongoing joint work with my advisor, Professor Olivia Beckwith.

A toric vector bundle is a vector bundle over a toric variety which is equipped with a lift of the action action of the associated torus. As a source of examples, toric vector bundles and their projectivizations provide a rich class of spaces that still manage to admit a combinatorial characterization. Toric vector bundles were first classified by Kaneyama, and later by Klyachko using the data of decorated subspace arrangements. Klyachko's classification is the foundation of many interesting results on toric vector bundles and has recently led to a connection between toric vector bundles, matroids, and tropical geometry.

After explaining some of this background, I'll introduce the notion of a tropical toric vector bundle over a toric variety. These objects are discrete analogues of vector bundles which still have notions of positivity, a sheaf of sections, an Euler characteristic, and Chern classes. The combinatorics of these invariants can reveal properties of their classical analogues as well as point the way to new theorems for tropical vector bundles over a more general base. Time permitting I will discuss some new results on higher Betti numbers of a tropical vector bundle.

We will go over the proof of the Isometry Theorem.
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This learning seminar introduces the foundations of persistent homology, a key tool in topological data analysis for capturing multiscale topological features of general metric spaces. We will study persistence modules, barcodes, and distances such as the bottleneck and interleaving metrics, with a focus on their geometric meaning.

A central goal of the seminar is to understand and prove stability theorems for persistent homology, including bounds relating bottleneck distance to the Gromov–Hausdorff distance. Core examples will come from Morse theory and Vietoris–Rips filtrations of metric spaces.

The seminar is structured as a guided, collaborative reading course and is aimed at graduate students and faculty generally interested in this aspect of topology and geometry.

We will share a story that brings together geometry, dynamics, and number theory in interesting and novel ways, and also creates some very compelling imagery- the talk will have lots of pictures, and all relevant background notions will be introduced and explained. Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat 3-manifolds which we call translation prisms. Using ideas of Furstenberg, and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot-Vijayraghavan expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular n-gon for n = 7, 9, 14, 16, 18, 20, 24, 30. This is joint work with Nicolas Bedaride, Pat Hooper, and Pascal Hubert.

A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. Algebraic topology can provide a lens to understand complex data by studying its shape. One such method is a crocker plot, a 2-dimensional image that displays the topological information at all times simultaneously. We discuss how this method has shown promise in a variety of contexts: to perform exploratory data analysis, to choose between two models of collective motion, and to investigate parameter recovery via machine learning. We also empirically quantify the stability of crocker plots with respect to noise, random particle deletion, and finite-size effects in various self-propelled particle models that exhibit collective behavior and phase separation.

Theta series play a central role in many areas of mathematics, especially number theory. In this talk, we begin with a brief overview of two applications of theta series: Counting the number of solutions of the congruent quadratic forms, and the evaluation of special values of L-functions via Ramanujan’s theory of elliptic functions to alternative bases for modular forms. Along the way, we state our main results in each setting. In the second part of the talk, we return to these applications to outline the key ideas and techniques involved in the proofs, as time permits.

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Developing a suitable notion of vector bundles in tropical geometry has recently attracted considerable interest. In a recent work, Khan and Maclagan introduced tropical vector bundles using matroids, inspired by Klyachko’s classification of toric vector bundles, and studied their stability properties. In this talk, we reinterpret their notion of stability through the framework of the categorical approach to stability proposed by André. This perspective clarifies the structure underlying their results and places them in a broader conceptual setting. This is joint work with Alex Sistko and Cameron Wright.