Schedule and Abstracts

All talks are expected to take place in Avery Hall. (Avery Hall Floorplan)

Saturday, April 23, 2016 | |

10:00 – 10:20 | Registration |

10:20 – 10:30 | Opening Remarks |

10:30 – 11:10 | William Taylor F-singularities of the generic link |

11:15 – 11:35 | Basanti Poudyal Existence of exact zero divisors in Artinian Gorenstein rings |

11:40 – 12:20 | Jason Lutz Homological characterizations of quasi-complete intersections |

12:30 – 2:00 | Lunch |

2:00 – 2:20 | John Myers Multiplicity and the homotopy Lie algebra |

2:30 – 3:10 | Tony Se On Affine Semigroup Rings of Dimension 2 |

3:20 – 3:40 | Nathan Steele Support and Rank Varieties of Totally Acyclic Complexes |

3:40 – 4:00 | 20 Minute Break |

4:00 – 4:40 | Thomas Polstra Global Hilbert-Kunz Multiplicity |

4:50 – 5:10 | Brittney Falahola Characterizing Gorenstein Rings Using Frobenius |

Sunday, April 24, 2016 | |

9:30 – 10:10 | Alessandro De Stefani Frobenius Betti Numbers |

10:20 – 10:40 | Shalan Alkarni Three Dimensional Jacobian Derivations And Divisor Class Groups |

10:50 – 11:30 | Patricia Klein Associated Primes of Local Cohomology Modules |

11:40 – 12:00 | Brent Holmes On Simplicial Complexes with Serre Property $S_2$ |

12:00 | Closing Remarks |

**Speaker:** Shalan Alkarni

**Title:** Three Dimensional Jacobian Derivations And Divisor Class Groups

**Abstract:**
We use P. Samuel’s purely inseparable descent methods to investigate the
divisor class groups of the intersections of pairs of hypersurfaces of the form $w_1 = f$,
$w_2 = g$ in affine 5-space with $f , g \in A = k[x, y, z]$; $k$ is an algebraically closed
field of characteristic $p > 0$. This corresponds to studying the divisor class group of the
kernels of three dimensional Jacobian derivations on $A$ that are regular in codimension
one. Our computations focus primarily on pairs where $f , g$ are quadratic forms. We
find results concerning the order and the type of these groups. We show that the divisor
class group is a direct sum of up to three copies of $\mathbb{Z}_p$, is never trivial, and
is generated by those hyperplane sections whose forms are factors of linear combinations of
$f$ and $g$.

**Speaker:** Brittney Falahola

**Title:** Characterizing Gorenstein Rings Using Frobenius

**Abstract:**
We will give two characterizations of Gorenstein rings of prime characteristic
using the Frobenius functor in various ways. The first characterization, which
only applies for rings which possess a canonical module, states that a ring is
Gorenstein if and only if the Frobenius functor preserves the injective dimension
of the canonical module. As a result of this characterization, we'll see that in
rings possessing a canonical module, the canonical module serves as a test module
for when the Frobenius functor preserves injective dimension. The second
characterization depends on the existence of a certain finitely generated module
$M$ of finite injective dimension for which the Frobenius functor “plays well”
with an injective resolution of $M$.

**Speaker:** Brent Holmes

**Title:** On Simplicial Complexes with Serre Property $S_2$

**Abstract:**
Given an ideal $I$ in a polynomial ring, $S$, one can form a
graph from the minimal prime ideals of $R = S/I$, where the vertices
of the graph are the minimal prime ideals of $R$ and an edge connects
two vertices, $v_1, v_2$, if and only if $\mathrm{height}(v_1+v_2) = 1$.
This graph is known as the *Hochster-Huneke* or *dual graph*
of $R$. The $S_2$ property of $R$ implies the connectedness of this graph.
We will discuss lower bounds and upper bounds for the diameter of the
dual graph in the case that $R$ is $S_2$ and $I$ is a square free
monomial ideal.

**Speaker:** Patricia Klein

**Title:** Associated Primes of Local Cohomology Modules

**Abstract:** We will define and briefly discuss some motivation
for studying local cohomology. We will then turn to the question of which
rings $R$ have the property that $H^i_I(R)$ has finitely many associated primes
for all $i$ and $I$. We will survey some results from the past 15 years on rings
that do have this property and then share some examples due to Singh and Swanson
of hypersurface domains that do not. Lastly, we will state some vanishing theorems
for local cohomology and discuss how those results restrict the class of graded
hypersurface domains of dimension 4 that may have local cohomology modules with
infinitely many associated primes.

**Speaker:** Jason Lutz

**Title:** Homological characterizations of quasi-complete
intersections

**Abstract:** Let $(R,\mathfrak m, \Bbbk)$ be a
commutative, local, Noetherian ring. Let $I = (\boldsymbol f)$ be a
proper, non-zero ideal of $R$ and let $S$ denote $R/I$. Consider the
Koszul complex $E = K(\boldsymbol f; R)$; its homology $H(E)$ has
the structure of an $S$-algebra. When $H_1(E)$ is free as an
$S$-module and $H(E)$ is the exterior algebra $\wedge^SH_1(E)$, we say
that $I$ is a * quasi-complete intersection (q.c.i) ideal *. We'll explore
the ideal-theoretic properties q.c.i.$\!$ ideals, develop homological
characterizations using Tate's "adjunction of variables", and
discuss the relationship between the
q.c.i.$\!$ property of ideals and the complete intersection property of
rings.

**Speaker:** John Myers

**Title:** Multiplicity and the homotopy Lie algebra

**Abstract:**
If $R$ is a commutative noetherian local ring with residue field $k$, then
the graded vector space
$\operatorname{Ext}_R{(k,k)} = \bigoplus_{n\geq 0} \operatorname{Ext}^n_R{(k,k)}$
has a product making it a graded connected $k$-algebra. It is known that this
algebra is the universal enveloping algebra of a (graded) Lie algebra $\pi(R)$
called the homotopy Lie algebra of $R$. In this talk we will explain how rings
with minimal Hilbert-Samuel multiplicity can be detected by examining the
structure of $\pi(R)$ and its Lie subalgebras.

**Speaker:** Thomas Polstra

**Title:** Global Hilbert-Kunz Multiplicity

**Abstract:**
This talk will be a report on some results from a joint work in
progress with Alessandro De Stefani and Yongwei Yao. Hilbert-Kunz
multiplicity is a classical numerical invariant of study attached
to local rings of prime characteristic. In this talk we will
discuss how to naturally extend this numerical invariant to rings
which are not local and F-finite in a meaningful way.

**Speaker:** Tony Se

**Title:** On Affine Semigroup Rings of Dimension 2

**Abstract:** Let $k$ be a field. We consider a subring
$R = k[x^a, x^{p_1} y^{q_1}, \dots, x^{p_t} y^{q_t}, y^b]$ of the polynomial
ring $k[x,y]$. We will give simple numerical criteria for the ring $R$ to be
Cohen-Macaulay when $t=2$. We will also give an answer to related questions about $R$,
such as finding the Hilbert polynomial for the ideal $(x^a, y^b)$ in $R$ and
finding a simple algorithm that generates the monomial $k$-basis of $R/(x^a, y^b)$.
This is joint work with Grant Serio.

**Speaker:** Nathan Steele

**Title:** Support and Rank Varieties of Totally Acyclic Complexes

**Abstract:**
Support and rank varieties of modules over a group algebra
of an elementary abelian p-group have been well studied. In
particular, Avrunin and Scott showed that in this setting, the rank
and support varieties are equivalent. Avramov and Buchweitz proved an
analogous result for pairs of modules over arbitrary commutative local
complete intersection rings. We study support and rank varieties in
the triangulated category of totally acyclic chain complexes over a
complete intersection ring and show that these varieties are also
equivalent. We then show that any homogeneous affine variety is
realizable as the support of some totally acyclic complex.

**Speaker:** William Taylor

**Title:** F-singularities of the generic link

**Abstract:**
In a recent paper of Niu, a strong connection was shown
between the multiplier ideal of a pair $(R,I^c)$, where
$R$ is a ring of characteristic 0, $I$ an ideal in $R$,
$c$ is the codimension of $I$, and the singularities of
a generic link $J$ of $I$. In the same paper, a comparison
is given between the log canonical thresholds of $I$ and
$J$. In this talk we will give a brief introduction to
linkage theory and Frobenius singularities. Then we will
show analogous positive characteristic statements for
the test ideal and the F-pure threshold. Of initial
interest will be a proof of the positive characteristic
version of (one direction of) Ein's lemma [Niu 2014],
with a partial converse.