# Commutative Algebra Seminar

## Fall 2021, Tuesdays 2:00 pm, LCB 222

### Every other week, we will attend the Fellowship of the Ring virtual seminar, hosted by MSRI

Date | Speaker | Title — click for abstract |

August 31 | Fellowship of the ring Bernd Sturmfels Virtual seminar |
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September 14 | Fellowship of the ring Wenliang Zhang Virtual seminar |
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September 28 | Fellowship of the ring Steven Sam Virtual seminar |
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October 12 | Fellowship of the ring Yairon Cid-Ruiz Virtual seminar |
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October 26 | Fellowship of the ring Julia Pevtsova Virtual seminar |
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November 2 | Eamon Quinlan-Gallego University of Utah |
Bernstein-Sato theory in positive characteristic
Given a holomorphic function f, its Bernstein-Sato polynomial is a classical invariant that detects the singularities of the zero locus of f in very subtle ways; for example, its roots recover the log-canonical threshold of f and the
eigenvalues of the monodromy action on the cohomology of the Milnor fibre. In this talk I will describe a characteristic-p analogue of this invariant and some recent developments that I proved in my thesis. I will also describe some open
problems and future directions that I am working on at the moment. |

November 9 | Fellowship of the ring Patricia Klein Virtual seminar |
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November 16 | Vaibhav Pandey University of Utah |
Are natural embeddings of determinantal rings split?
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. |

November 23 | James Cameron University of Utah |
Local cohomology modules of group cohomology rings
For G a finite group and k a field of characteristic dividing |G| there is a rich history of studying the geometry and commutative algebra of the group cohomology ring of G with
coefficients in k. These rings are complicated, but some of their geometric features can be described in group theoretic terms. These geometric features can often be phrased in
terms of local cohomology. I will survey some of the results around local cohomology modules of group cohomology rings, and describe how to study these modules using a
topological perspective. I will also mention some open problems. |

November 30 | Fellowship of the ring Robin Baidya Virtual seminar |
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December 7 | Selvi Kara University of Utah |
Blow-Up algebras of strongly stable ideals
Let A be a polynomial ring and I_1,..., I_r be a collection of ideals in A. The multi-Rees algebra of I_1,..., I_r
encode many algebraic properties of these ideals, their products, and powers. Additionally, the multi-Rees
algebras arise in successive blowing up of Spec(A) at the subschemes defined by I_1,..., I_r. Due to this
connection, Rees and multi-Rees algebras are also called blow-up algebras in the literature.
In this talk, we will focus on Rees and multi-Rees algebras of strongly stable ideals. In particular, we will discuss the Koszulness of these algebras through a systematic study of these objects via three parameters: the number of ideals in the collection, the number of Borel generators of each ideal, and the degrees of Borel generators. In our study, we utilize combinatorial objects such as fiber graphs to detect GrÃ¶bner bases and Koszulness of these algebras. This talk is based on a joint work with Kuei-Nuan Lin and Gabriel Sosa. |

This web page is maintained by Srikanth Iyengar, Karl Schwede, Anurag K. Singh.