Events of the Week

"So," Ioana said quietly, "you’ve seen the patterns. Catalan numbers. Non-crossing pairings. The moments… they’re no longer mysterious."
"But we still don’t know the distribution," the student replied. "We only know its shadows."
Alan nodded. "Then it’s time to reconstruct the object casting them."
"From the moments?" the student asked.
"Exactly. If you know how to sum them."
In this lecture, we take the sequence of moments arising from the Gaussian ensembles and encode them into a generating function. From this perspective, the limiting eigenvalue distribution emerges explicitly and is identified as the Wigner semicircle law.
"So the randomness disappears?" the student asked.
"Not disappears," Ioana said. "It organizes itself."
We then show that the semicircle law is not merely a consequence of combinatorics, but the solution to a variational problem: it is the equilibrium measure minimizing an energy functional with logarithmic interaction.
Alan leaned forward. "This is the part most people miss."
"Which part?"
Ioana smiled. "The matrix was never the point. The equilibrium was."
The student paused. "So all those moments…"
"…were leading you here," Alan said. "They had no choice."

Motivated by the dichotomy between acute and chronic phases of Chagas disease, we analyze two within-host immune response models. Both models couple identical parasite dynamics with distinct mechanisms for CD8⁺ T-cell proliferation. While both exhibit bistability and hysteresis, the model incorporating homeostatic immune maintenance additionally permits backward bifurcation, enabling endemic persistence even when the basic reproduction number R₀ < 1

In the first part of this talk, we will provide a friendly introduction to different aspects of hypergeometric motives. In the second part, we will explain recent work by the author that relates these aspects to modular forms.

The Persistent Homology Learning Seminar resumes after the break. We will begin with a brief recap of the material covered so far, followed by preparation for 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.

The Andrews-Curtis conjecture is a long-standing open problem about presentations of the trivial group, related to some of the central problems in low-dimensional topology. I will discuss a generalization of the Andrews-Curtis conjecture and recent approaches using 4-manifolds and quantum topology.

Retrieval-Augmented Generation (RAG) has emerged as an important paradigm for improving large language model systems by grounding generation in external knowledge, and it is now widely used in modern AI applications. In practice, the quality of a RAG system often depends critically on its matching layer, which determines what evidence is retrieved and made available to the generator. This talk focuses on the matching layer through the lens of relevance, ranking, and retrieval quality. We begin with a brief overview of the RAG pipeline used in industry and then focus on how query–document relevance is modeled and computed. In particular, we trace the progression from discrete lexical retrieval methods such as BM25 to continuous semantic representations, and then to more advanced interaction and fusion methods. Finally, we discuss system-level constraints and practical challenges in deploying RAG systems.

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 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.

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