Geometry and Topology

Speaker: Tim Campion (Johns Hopkins University)

"Cubical subdivision"

Time: 15:30 - 16:30

Room: MC 107

We study the test model structure (as well as a “projective" variant) on several categories of cubical sets (cubical sets with or without connections / symmetries / reversals), study their monoidal properties, and establish Quillen equivalences among them and to simplicial sets and topological spaces.

With this groundwork in place, we then consider (on most of these cubical sites) cubical subdivision — an analog of simplicial subdivision (which is even better because it is strong monoidal!). This leads to a cubical analog of Kan’s $\mathrm{Ex}^\infty$ functor which we show to be a functorial fibrant replacement with good properties. As a corollary, we obtain cubical approximation theorems for various flavors of cubes.

Geometry and Combinatorics

Speaker: Prajwal Dhondiram Udanshive (Western)

"Katz's proof of Rota-Heron-Welsh (season finale)"

Time: 15:30 - 16:30

Room: MC 108

Mathematics Departmental Presentation 2026

Speaker: Vladimir Gorchakov, Deepak Sadanandan, Mac Martin (Western)

"TBA"

Time: 15:30 - 17:00

Room: MC 107

Geometry and Topology

Speaker: Tim Campion (Johns Hopkins University)

"The categorified Berkovich spectrum (a lowbrow approach)"

Time: 15:30 - 16:30

Room: MC 108

The set $M(A)$ of multiplicative seminorms on a commutative ring $A$ carries a natural topology. By considering completions of $A$ with respect to these seminorms, Berkovich obtained a structure sheaf of $\mathcal O_A$ of "analytic" functions on $M(A)$.

We consider (the rudiments of) a categorification of Berkovich’s theory, associating to every symmetric monoidal stable infinity category $\mathcal A$ a topological space $M(\mathcal A)$ and sheaf of categories $\mathcal O_{\mathcal A}$. We emphasize the concrete nature of these objects. For example, for $k$ a field and any $r > 0$ the function $$N_{r,k}: \mathsf{Ob}(\mathsf{Sp}^\mathrm{fin}) \to \mathbb{R}_{\geq 0}$$ $$ X \mapsto \sum_i \mathrm{dim} H_i(X; k) r^i$$ is a point in $\mathcal M(\mathsf{Sp}^\mathrm{fin})$. Moreover, the skeletal filtration of a spectrum is often a "Cauchy sequence."

We carry through the theory far enough to compute $M(\mathsf{Sp}^\mathrm{fin})$ (which the reader familiar with chromatic homotopy theory may now guess).

Department Meeting

Speaker: (Western)

"Department meeting"

Time: 15:30 - 16:30

Room: MC 108

Graduate Seminar

Speaker: Yunhai Xiang (Western)

"Trace of Frobenius"

Time: 16:30 - 17:30

Room: MC 108

Given a smooth projective variety X over F_q, a natural question is what is the size of X(F_q^n) for all n. This question leads to the famous Weil conjectures. In this talk, we will talk about the first tiny element of it called the Grothendieck-Lefschez trace formula, which finds X(F_q^n) in terms of alternating sums of traces of Frobenius maps acting on l-adic étale cohomology of X. 

Algebra Seminar

Speaker: David Zureick-Brown (Amherst College)

"l-adic Images of Galois for Elliptic Curves over Q"

Time: 14:30 - 15:30

Room: MC 108

I will discuss recent joint work with Jeremy Rouse and Drew Sutherland on Mazur’s “Program B” — the classification of the possible “images of Galois” associated to an elliptic curve (equivalently, classification of all rational points on certain modular curves XH). The main result is a provisional classification of the possible images of l-adic Galois representations associated to elliptic curves over Q and is provably complete barring the existence of unexpected rational points on modular curves associated to the normalizers of non-split Cartan subgroups and two additional genus 9 modular curves of level 49.

I will also discuss the framework and various applications (for example: a very fast algorithm to rigorously compute the l-adic image of Galois of an elliptic curve over Q), and then highlight several new ideas from the joint work, including techniques for computing models of modular curves and novel arguments to determine their rational points, a computational approach that works directly with moduli and bypasses defining equations, and (with John Voight) a generalization of Kolyvagin’s theorem to the modular curves we study.

Geometry and Combinatorics

Speaker: Tyler Dunaisky (Purdue University)

"Cosmological Correlators and Triangulating the Dual Cosmological Polytope"

Time: 15:30 - 16:30

Room: MC 108

A cosmological correlator is an Euler integral, associated to a graph G, which encodes information about the state of the early universe. Evaluation of these integrals is extremely challenging, even in simple cases. However, it turns out the integrand can be identified with the so-called canonical form of the cosmological polytope, revealing a rich combinatorial structure and allowing the application of techniques from commutative algebra. I'll sketch my contribution to this story and advertise the fledgling field of positive geometry, which seeks to generalize the notion of canonical forms to geometric objects more exotic than polytopes.

Ph.D. Public Lecture

Speaker: Jeremy Gamble (Western)

"Random matrices, noncommutative geometry, and Dirac ensembles"

Time: 14:00 - 15:00

Room: MC 107

Dirac ensembles lie at the intersection of random matrix theory and noncommutative geometry. In particular, we look at the (0,1) fuzzy Dirac ensemble with fermion in the case where the fermion mass is positive. We give an introduction to random matrix models, both analytic and formal, including the relationship between matrix integrals and generating functions of maps in the formal case. We also motivate and introduce the concept of a spectral triple from noncommutative geometry. One of these applications is Dirac ensembles, consisting of a certain finite spectral triple which describes a toy model of quantum gravity. We describe Dirac ensembles and how they lead to hermitian matrix models, giving special attention to the (0,1) fuzzy Dirac ensemble with a fermion. Loop equations for this model can be found through complex analytic techniques, and one allows for a perturbative solution of the moments. In the case where the potential is Gaussian, an exact solution for moments can be obtained using complex analytic techniques and elliptic integrals.

Ph.D. Public Lecture

Speaker: Yucen Jin (Western)

"Delay-induced bifurcations and chaos dynamics in a host competitor parasite model"

Time: 14:00 - 15:00

Room: MC 108

: Latency is a feature of nearly all real-world biological systems, yet many models simplify analysis by assuming such delays to be negligible. Here we examine a three-dimensional host–competitor–parasite model in which a maturation delay is introduced in the parasite’s reproduction term. We analyze how this latency alters the system’s dynamics, particularly its ability to generate more complex bifurcations. In the delayed model, we identify delay-induced Hopf bifurcations, a generalized Hopf (Bautin) point obtained through a fifth-order normal form, and a codimension-two Hopf–zero bifurcation arising when a boundary transcritical condition intersects the Hopf curve.

Beyond local bifurcations, the corresponding non-delayed ODE exhibits a classical period-doubling cascade to chaos. In contrast, the delayed system produces only narrow “chaos windows,” indicating that parasite maturation latency plays a limited role in generating chaos. We also examine bistability between coexistence dynamics, such as periodic or chaotic attractors, and a boundary equilibrium, and we discuss how the introduction of delay influences the extent of this bistability.

Geometry and Combinatorics

Speaker: Prajwal Udanshive (Western)

"arrangement properties of simple Schubert matroids"

Time: 15:30 - 16:30

Room: MC 107

Homotopy Theory

Speaker: Daniel Carranza (Johns Hopkins University)

"The Discrete Homotopy Hypothesis for n-types"

Time: 14:00 - 15:00

Room: MC 107

Homotopy Theory

Speaker: Mark Behrens (University of Notre Dame)

"The degree spectral sequence"

Time: 15:30 - 16:30

Room: MC 107

Homotopy Theory

Speaker: Nathan Kershaw (Western)

"Computations of discrete homology"

Time: 10:00 - 11:00

Room: MC 107

Homotopy Theory

Speaker: Jacob Ender (Western)

"Towards the basic shapes conjecture"

Time: 11:30 - 12:30

Room: MC 107