Schedule and Abstracts

All talks took place in Avery Hall 106. Refreshments were found in Avery Hall 110.
Avery Hall Floorplan

The tentative schedule is:
 Saturday, April 8, 2017 10:30 – 10:50 Registration 10:50 – 11:00 Opening Remarks 11:00 – 11:20 Alessandra Costantini Cohen-Macaulay property of Rees algebras of modules *11:30 – 12:10 Zachary Flores The Weak Lefschetz Property for a Graded Module 12:10 – 2:00 Lunch *2:00 – 2:40 Andrew Bydlon Restriction of Test ideals to Hypersurfaces 2:50 – 3:10 William Taylor Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicities 3:10 – 3:30 20 minute break 3:30 – 3:50 Jenny Kenkel Local Cohomology of Thickenings *4:00 – 4:40 Ashley Wheeler Local cohomology modules over Stanley-Reisner rings
 Sunday, April 9, 2017 *9:30 – 10:10 Andrew McCrady On perinormality in polynomial rings and completions *10:20 – 11:00 Jonathan Montaño The Core of Monomial Ideals *11:10 – 11:50 Andrew Windle Cohomological Operators on Quotients by Exact Zero Divisors 12:00 Closing Remarks
* 40 minute talk

## Abstracts

*Speaker: Andrew Bydlon
Title: Restriction of Test ideals to Hypersurfaces
Abstract: In positive characteristic, the test ideal has served as a measure of singularities. In contrast with the characteristic 0 picture, in this talk I show that there are varieties for which every general hypersurface has strictly worse singularities, and use it to answer a question stemming from work of Hochster and Huneke

Speaker: Alessandra Costantini
Title: Cohen-Macaulay property of Rees algebras of modules
Abstract: In this talk I will introduce Rees algebras of modules and discuss their Cohen-Macaulay property. This includes the case of Rees algebras of ideals $R[It]$, whose Cohen-Macaulayness is well-understood in connection with the Cohen-Macaulay property of the associated graded ring $\mathrm{gr}_I(R)$. Unfortunately, there is no module analogue for the associated graded ring, hence most of the techniques used for Rees algebras of ideals do not apply to the case of Rees algebras of modules. I will give a brief overview of some known techniques for the study of Cohen-Macaulayness of Rees algebras of modules and, if time allows, present possible alternative strategies to attack the problem.

*Speaker: Zachary Flores
Title: The Weak Lefschetz Property for a Graded Module
Abstract: If $k$ has characteristic zero and $R = k[x,y, z]$, it is well known that complete intersections in $R$ have the Weak Lefschetz Property. This result follows from a beautiful blend of commutative algebra and algebraic geometry. We discuss a modest generalization of this result by extending these techniques to investigate when the cokernel of an $R$-linear map $\varphi: \bigoplus_{j=1}^m R(-b_j) \rightarrow \bigoplus_{i=1}^n R(-a_i)$ has the Weak Lefschetz Property.

Speaker: Amy Huang
Title: Equations of Kalman Varieties
Abstract: Given a subspace $L$ of a vector space $V$, the Kalman variety consists of all matrices of $V$ that have a nonzero eigenvector in $L$. We will discuss how to apply Kempf Vanishing technique with some more explicit constructions to get a long exact sequence involving coordinate ring of Kalman variety, its normalization and some other related varieties in characteristic zero. We might also discuss how to extract more information from the long exact sequence including the minimal defining equations for Kalman varieties.

Speaker: Robert Laudone
Title: The spin-Brauer diagram algebra
Abstract: Schur-Weyl duality is an important result in representation theory that connects the representations of the general linear group and symmetric group. In this talk, we will discuss the Spin-Brauer diagram algebra that arises from studying an analogous form of Schur-Weyl duality for the action of pin group on ${\bf V}^{\otimes n} \otimes \Delta$. Here ${\bf V}$ is the standard $N$-dimensional complex representation of ${\rm Pin}(N)$ and $\Delta$ is the spin representation. We will give a combinatorial construction for this algebra and discuss its connection to the centralizer of this action of the pin group.

Speaker: Jenny Kenkel
Title: Local Cohomology of Thickenings
Abstract: Let $R$ be a standard graded polynomial ring that is finitely generated over a field, let $\mathfrak{m}$ be the homogeneous maximal ideal of $R$, and let $I$ be a homogeneous prime ideal of R. Bhatt, Blickle, Lyubeznik, Singh and Zhang examined the local cohomology of $R/I^{t}$, as t goes to infinity, in characteristic 0. I will discuss their results and give a concrete example.

Title: On perinormality in polynomial rings and completions
Abstract: In 2015 Neil Epstein and Jay Shapiro introduced perinormal domains. A domain $R$ is perinormal if each birational extension $R\subset(S,\mathfrak n)$ that satisfies the going-down property is actually flat. Epstein and Shapiro suggested studying the transference of perinormality between $R$ and $R[X]$, and between $(R,\mathfrak m)$ and $(\widehat{R},\mathfrak m\widehat{R}).$ We show that if $R$ is universally catenary of dimension at most 2, then perinormality descends from $R[X]$ to $R$. We also show that if $R[X]$ is globally perinormal, then $R$ is weakly normal. Finally, we give an example to show that perinormality need not ascend from $(R,\mathfrak m)$ to $(\widehat{R},\mathfrak m\widehat{R}).$

*Speaker: Jonathan Montaño
Title: The Core of Monomial Ideals
Abstract: Let $R$ be a commutative ring and $I$ an $R$-ideal. Rees and Sally defined the core of $I$ as the intersection of all of its reductions. In general, it is very hard to explicitly compute the core of an ideal as it may require the intersection of infinitely many ideals. In this project we find a formula for the core of monomial ideals satisfying certain residual assumptions. This is work in progress jointly with Louiza Fouli, Claudia Polini, and Bernd Ulrich.

Speaker: William Taylor
Title: Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicities
Abstract: We define and study a function that interpolates continuously between Hilbert-Samuel and Hilbert-Kunz multiplicity of ideals. We see several existing theorems as special cases of more general theorems and relate the new function to F-thresholds. In addition, we define an infinite family of closures that have a strong relationship to the multiplicities we consider.

*Speaker: Ashley Wheeler
Title: Local cohomology modules over Stanley-Reisner rings
Abstract: Local cohomology modules, even over a Noetherian ring $R$, are typically unwieldy. As such, it is of interest whether or not they have finitely many associated primes. We prove the affirmative in the case where $R$ is a Stanley-Reisner ring over a field, whose associated simplicial complex is a $T$-space. This work is joint with R. Barrera and J. Madsen.

*Speaker: Andrew Windle
Title: Cohomological Operators on Quotients by Exact Zero Divisors
Abstract: Let $S$ be a commutative, local, Noetherian ring. If $I \subset S$ is an ideal generated by a regular sequence, Eisenbud provides a construction for cohomological operators over $R = S/I$ that play an important role in studying the homological properties of modules over complete intersections. In this talk, we provide a similar construction of cohomological operators in the case that $I$ is generated by an exact zero divisor instead of a regular sequence.