Events of the Week

The numerical invariant $F$-volume was first introduced by Wágner Badilla-Céspedes et al. in 2022 as a generalization of the $F$-threshold for a pair of ideals $I$ and $J$. We extend this invariant to the case of a sequence of filtrations of ideals $\mathcal{I}(1),\ldots, \mathcal{I}(t)$ and a $p$-family $J_{\bullet}$ of ideals in a Noetherian ring. Under the assumption that the filtrations are $J_{\bullet}$-admissible, we show that the $F$-volume
\[
\mathrm{Vol}_F^{J_{\bullet}}(\underline{\mathcal{I}})=\lim\limits_{e\to \infty}\dfrac{\left|V^{J_{\bullet}}_{\underline{\mathcal{I}}}(p^e)\right|}{p^{et}}
\]
exists, where $V^{J_{\bullet}}_{\underline{\mathcal{I}}}(p^e):=\{(a_1,\ldots,a_t)\in \mathbb{N}^t \mid I(1)_{a_1}\cdots I(t)_{a_t}\nsubseteq J_{p^e} \}$. This is joint work with Thai Thanh Nguyen and Souvik Dey.

Consider an invariant that behaves nicely for smooth varieties, such as Euler number, Betti numbers, or Hodge numbers. Suppose we want a version of this invariant for singular varieties that sees interesting information about the singularities. I will discuss how this naturally leads to the notion of crepant resolutions of singularities. However, crepant resolutions (by varieties) are rare in practice. I will discuss joint work with M. Satriano in which we show that crepant resolutions actually exist in broad generality, as long as one is willing to consider algebraic stacks. Specifically, any variety with log-terminal singularities admits a crepant resolution by a smooth algebraic stack. As one consequence, in joint work with J. Huang and M. Satriano, we obtained a cohomological interpretation for Batyrev's stringy Hodge numbers. This talk will not assume familiarity with stacks.

We will move into the realm of the bottleneck distance and 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.

In this joint work with Markus Neuhauser, we investigate D'Arcais polynomials, which extend k-colored partitions and encompass all powers of the Dedekind eta function. We present new results on their zero distributions, examine their coefficients, and apply both analytic and algebraic number theoretic techniques, particularly excluding nontrivial roots of unity as zeros. I will also report on a joint work with Kathrin Bringmann and Olivia Beckwith on k-regular partitions.

In a polynomial ring, the d-th Veronese subring is generated as a k-algebra by all monomials whose degree is a multiple of d. In a polynomial ring with standard grading (degree of each variable is 1), the Veronese subrings have many nice properties: they are normal, Cohen-Macaulay, and Koszul. Furthermore, their defining ideals are quadratic, binomial, and determinantal, generated by 2x2 minors of suitable matrices that also form a Groebner basis for the ideal. In this talk, we will discuss Veronese subrings of a non-standard graded polynomial ring. We will see that many of the nice properties are satisfied in two-variable case, but no longer hold in general in more variables. This is based on joint work with Bek Chase, Luca Fiorindo, Thiago de Holleben, Emanuela Marangone, Alexandra Seceleanu, and Srishti Singh.

The square-root cancellation hypothesis, in its original form, concerns cancellation in certain GL(1) sums with applications to the distribution of zeros of L-functions associated with GL(2) cusp forms. Building on Ye’s work on a varying GL(2) cusp form and my work (jointly with Ye and Gillespie) on the Rankin Selberg convolution of two GL(2) cusp forms, both allowed to move, I will discuss a variant in which only one form is permitted to vary. This leads naturally to shifted convolution sums and new analytic challenges. I will outline my methods and preliminary results in this setup and discuss how these fit into the broader concept of the square root cancellation hypothesis.

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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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