## Fall 2022 Speakers / Abstracts

August 31: Vaibhav Pandey (Purdue University)
Title:
When are the natural embeddings of determinantal rings split?
Abstract: Over an infinite field, a generic determinantal ring is the fixed subring of an action of the general linear group on a polynomial ring; this is the natural embedding of the title. If the field has characteristic zero, the general linear group is linearly reductive, and it follows that the invariant ring is a split subring of the polynomial ring. We determine if the natural embedding is split in the case of a field of positive characteristic. Time permitting, we will address the corresponding question for Pfaffian and symmetric determinantal rings. This is ongoing work with Mel Hochster, Jack Jeffries, and Anurag Singh.

Septermber 14: Hunter Simper (Purdue University)
Title:
Ext and Local Cohomology of Thickenings of Ideals of Maximal Minors
Abstract: Let $R$ be the ring of polynomial functions in $mn$ variables with coefficents in $\mathbb{C}$, where $m>n$. Set $X$ to be the matrix in these variables and $I$ the ideal of maximal minors of this matrix. I will discuss the R-module structure of certain Ext and local cohomology modules arising from the rings $R/I^t$. In particular, for $i$ equal to the cohomological dimension of $I$, I will discuss the embedding of $Ext^i_R(R/I^t,R)$ into $H_\frak{m}^{mn}(R)$, explicitly describing this embedding when $X$ is size $n \times (n-1)$. More generally for $X$ of arbitrary size I will describe the annihilator of $Ext^i_R(R/I^t,R)$ and thereby completely determine the $R$-module structure of $H_\frak{m}^{mn-i}(R)$.

Septermber 21: Swaraj Pande (University of Michigan)
Title:
The F-signature function of the ample cone of a globally F-regular variety
Abstract: The F-signature of a strongly F-regular local ring R is an interesting invariant of its singularities. In this talk, we will discuss this invariant when R is the normalized homogeneous coordinate ring of a projective variety. In particular, we study how the F-signature varies as we vary the embedding of a fixed projective variety X into various projective spaces. For this purpose, we will introduce the F-signature function, a real valued function on the ample cone of X, and discuss its continuity properties. We will also present some analogies and comparisons to the well-known volume function, which records the Hilbert-Samuel multiplicities. This is joint work with Seungsu Lee.

October 5: Wenbo Niu (University of Arkansas)
Title:
Multiplier ideals on varieties and local properties
Abstract: In this talk, we discuss the notion of Mather-Jacobian ideals defined on an arbitrary variety. It was introduced by Ishii-Ein-Mustata and de Fernex-Docampo extending the notion of multiplier ideals on normal varieties. We also discuss local syzygies of MJ-multiplier ideals, extending the work of Lazarsfeld-Lee and Lazarsfeld-Lee-Smith. This is a joint work with Ulrich.

October 12: Rabeya Basu (Indian Institute of Science Education and Research)
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October 19: Alapan Mukhopadhyay (University of Michigan)
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November 9: Takumi Murayama (Purdue University)
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November 16: Kriti Goel (University of Utah)
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November 30: Jennifer Kenkel (University of Michigan)
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December 7: Omar Colon Reyes (University of Puerto Rico)
Zoom talk
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December 14: Pham Hung Quy (FPT University)
Title:
Tight Hilbert Polynomial and F-rational local rings
Abstract: I will define the Buchsbaum property for tight closure. After that we discuss tight Hilbert polynomial, Hilbert coefficients. The talk is based on the paper A Buchsbaum theory for tight closure with Linquan Ma, and Tight Hilbert Polynomial and F-rational local rings with Saipriya Dubey, and Jugal Verma.