Events of the Week

This talk provides an introduction to combinatorial design theory, focusing on balanced incomplete block designs (BIBDs). These structures arise when organizing elements of a finite set into subsets in such a way that pairs of elements appear together in a controlled and uniform manner.
We begin with classical motivating examples, such as the Kirkman schoolgirl problem, and introduce the basic definition and parameters of BIBDs. Fundamental relationships between these parameters are derived, along with key results such as Fisher’s inequality. We then explore the algebraic perspective through incidence matrices, highlighting the connection between combinatorics and linear algebra.

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.

We will go over the stability result involving the Gromov–Hausdorff distance and the Rips complex construction.

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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 or geometry.

The Gromov-Hausdorff distance is a notion of dissimilarity between two datasets or between two metric spaces. It is an important tool in geometry, but notoriously difficult to compute. I will show how to provide new lower bounds on the Gromov-Hausdorff distance between unit spheres of different dimensions by combining Borsuk-Ulam theorems with Vietoris-Rips complexes. This joint work with 15 coauthors is available at https://arxiv.org/abs/2301.00246. Many questions remain open!

TBA

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.