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Commutative Algebra Seminar at Purdue

Time: Wednesday, 12:30 -- 1:30 pm

Location: REC 309

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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.