*Fall 2021*

This seminar is constructed to promote an environment for graduate students working on algebraic geometry and neighboring fields to give expository talks on research topics, an opportunity to share ideas, and collaborate within one another. For security reasons, meeting location will be shared via the group's Zulip server. If you want the meeting information for a particular talk, contact Pat Lank.

Week of |
Speaker |
Title |

9/8/21 | University of South Carolina |
Problem session |

9/15/21 | University of South Carolina |
Problem session |

9/22/21 | Bailey HeathUniversity of South Carolina |
Introduction to Affine Group Schemes |

10/11/22 | Jonathan SmithUniversity of South Carolina |
Birational geometry on the projective plane |

10/17/22 | Pat LankUniversity of South Carolina |
Techniques in prime characteristic |

11/1/22 | Jonathan SmithUniversity of South Carolina |
Introduction to Galois Cohomology |

11/14/21 | Pat LankUniversity of South Carolina |
Singularity categories in birational geometry |

Abstracts

**Problem session - **Within these problem sessions we worked thru various problems from Vakils notes (Chp 1 and 2) and Hartshornes book (Chp 1,2 and parts of 3). This led to fruitful discussions as each student has various backgrounds, knowledge, and exposure to topics.

**Bailey Heath - Introduction to Affine Group Schemes - **There are many ways to construct a group from a ring $R$, such as the familiar group of multiplicative units $R^{\times}$ and the general linear group $\operatorname{GL}_n\left(R\right)$. These are both examples of affine group schemes, which we will introduce in this talk. After giving the key definitions, we will state Yoneda's Lemma, which provides an important correspondence between affine group schemes over $R$ and finitely-generated $R$-algebras. Finally, we will use this correspondence to introduce Hopf algebras, which are another key algebraic tool for studying affine group schemes.

**Jonathan Smith** - **TBA**

**Pat Lank - Techniques in prime characteristic - **Within this talk, I will introduce a body of work called techniques in prime characteristic. In essence, one studies the homological properties of the Frobenius morphism to understand the singularity type. I start by reminding us the definition for regularity of a Noetherian ring, move to Kunz's theorem, and discuss key properties on a singularity type called Frobenius splitting.

**Jonathan Smith** - **TBA**

**Pat Lank - Singularity categories in birational geometry - **This talks will introduce a modification to the bounded derived category of coherent sheaves on a smooth projective variety, namely its singularity category. We start by reminding ourselves of the Verdier quotient and perfect complexes, then pass to the singularity category and compute some examples. If time permits, then I will discuss the utility of studying the category.