Events of the Week

We would like to gauge interest and work out logistics before committing to a regular schedule.

Please join us to discuss:
- Current status and topics covered so far
- Interest in continuing as a biweekly seminar
- Scheduling preferences

**Course Registration:** This seminar is offered as MATH-7760-02 (Persistent Homology) for 3 graduate credits it will meet on Thursdays 12:30 PM – 1:30 PM in Norman Mayer MA-104. The course just appeared in the roster and registration remains open through Friday, August 28, 2026. Students interested in registering for credit should do so before the deadline.

Recently, enumerative geometry has received a lot of attention. In my talk, I will give an overview of this topic and discuss my current work on counting maps from a genus g curve to projective space \mathbb{P}^n.

When pouring some milk into hot tea, the milk forms increasingly filamented patterns before eventually becoming a uniform blend of milky tea. Mathematically, this process is modelled by the advection-diffusion equation. It is conjectured that the filementations of a passive scalar (the milk) get smaller and smaller until they reach a specific lenghscale, the Batchelor scale, at which point the cascade halts. In this talk, passive scalar advection and the Batchelor scale are introduced and we provide a simple stochastic fluid model for which the Batchelor scale conjecture can be verified.

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We develop computational approximation theory on higher-genus domains, finite unions of disjoint subintervals of the real axis that arise naturally in orthogonal polynomial theory, iterative numerical linear algebra, and the study of integrable nonlinear wave equations. In each case, exploiting this underlying structure to choose appropriate basis functions (orthogonal polynomials) is essential to obtain a fast and accurate numerical method. A unifying theme is the numerical solution of Riemann--Hilbert problems on these domains.

We connect generalizations of the classical Hurwitz class numbers coming from two different frameworks: one introduced by Pei and Wang, arising from the generalized Cohen--Eisenstein series, and another by Li, Skoruppa, and Zhou, arising from Eichler orders of quaternion algebras. As applications, we obtain new basis for Eisenstein space $E_{3/2}^{+}(4N,\mathrm{id})$, a generalization of recent results of Beckwith and Mono, and a generalization of Gauss' formula. This is joint work with Olivia Beckwith.

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