Schedule and Abstracts

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

Avery Hall Floorplan

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 |

***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.

***Speaker:** Andrew McCrady

**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.