Zoom Link

Zoom Link

Zoom Link

Zoom Link

For a tensor triangulated $R$-linear category $T$, we study its fibers $\Gamma_{\mathfrak p}T$  at a prime ideal $\mathfrak p \in R$. We ask questions like “What does it mean for $T$ to be locally regular?” or “Gorenstein?” – as detected on these fiber categories $\Gamma_{\mahtfrak p} T$.

Inspiration and motivation often comes from commutative algebra and stable homotopy theory but the results will be for modular representations of a finite group.  This is joint work with D. Benson, S. Iyengar, and H. Krause.

Zoom Link

Christopher O'Neill: "Classifying numerical semigroups using polyhedral geometry"

Abstract: A numerical semigroup is a subset of the natural numbers that is closed under addition.  There is a family of polyhedral cones C_m, called Kunz cones, for which each numerical semigroup with smallest positive element m corresponds to an integer point in C_m.  It has been shown that if two numerical semigroups correspond to points in the same face of C_m, they share many important properties, such as the number of minimal generators and the Betti numbers of their defining toric ideals.  In this way, the faces of the Kunz cones naturally partition the set of all numerical semigroups into "cells" within which any two numerical semigroups have similar algebraic structure.  In this talk, we survey what is known about the face structure of Kunz cones, and how studying Kunz cones can inform the classification of numerical semigroups.

Aldo Conca: "Two bounds on Castelnuovo-Mumford regularity"

I will report on bounds on the Castelnuovo-Mumford regularity for ideals with polynomial parametrization (joint work with F.Cioffi) and for ideals associated with general subspace arrangements (joint work with M.Tsakiris).

Zoom Link

Zoom Link

We shed new light on local cohomology with support in an ideal defining a central linear subspace
arrangement.  We do so by making concrete relations between these local cohomology modules and the singular
homology of a family of simplicial complexes, which are related to the Hochster-Huneke graph of the associated
quotient ring.  This is work with Abraham Pascoe.  

Zoom Link

Zoom Link

In 2023 several groups of researchers independently proved the existence of short virtual resolutions on a smooth projective toric variety X, meaning that for each multigraded module there exists a free complex of length at most dim X which gives a resolution of the corresponding sheaf. I will discuss constructions of such resolutions using commutative algebra.

Zoom Link

We will discuss some ways to create integral kernels motivated by windows.

Zoom Link

In this talk, we delve into the intricate interplay between vanishing orders of functions along singularities and ideal containment problems in Noetherian rings. We start by focusing on the polynomial ring R = C[x1, . . . , xn] and explore the behavior of prime ideals and their symbolic powers. If P is a prime ideal then the nth symbolic power P(n) is the ideal of polynomial functions that vanish to order at least n at the generic point of V (P ). The Zariski-Nagata Theorem, established by Zariski in 1949, reflects the non- singular nature of n-space by asserting that if P ⊆ Q are prime ideals, then P (n) ⊆ Q(n) for all natural numbers n. However, when dealing with rings with singularities, it is no longer the case that P (n) ⊆ Q(n) for all prime ideal containments P ⊆ Q.

Pioneered by Huneke, Katz, and Validashti, The Uniform Chevalley Theorem introduces a linear adjust- ment, requiring functions to vanish along the generic point of V (P ) to ensure a vanishing order of at least n along the generic point of V (Q). Specifically, it establishes the containment of ideals as P(Cn) ⊆ Q(n) for all P ⊆ Q and n ∈ N, where C depends on the singular prime Q.

A notable limitation of the theorem is the dependency of the constant C on the singular prime Q, which restricts its applicability. This talk will explore methodologies to determine a constant C that remains inde- pendent of the prime Q, thereby enhancing the theorem’s utility. The investigation will also shed light on the role of the Izumi-Rees Theorem and unveil novel uniform properties exhibited by affine rings.

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link