# Difference between revisions of "Algebraic Geometry Seminar Spring 2016"

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First I will discuss Beilinson's resolution of the diagonal and some of the applications of that construction including the notion of a Beilinson. Then I will discuss new work, joint with Steven Sam, where we use supernatural bundles to build GL-equivariant resolutions supported on the diagonal of P^n x P^n, in a way that extends Beilinson's resolution of the diagonal. I will discuss some applications of these new constructions. | First I will discuss Beilinson's resolution of the diagonal and some of the applications of that construction including the notion of a Beilinson. Then I will discuss new work, joint with Steven Sam, where we use supernatural bundles to build GL-equivariant resolutions supported on the diagonal of P^n x P^n, in a way that extends Beilinson's resolution of the diagonal. I will discuss some applications of these new constructions. | ||

+ | |||

+ | ===Eric Ramos=== | ||

+ | The Local Cohomology of FI-modules | ||

+ | |||

+ | Much of the work in homological invariants of FI-modules has been concerned with properties of certain right exact functors. One reason for this is that the category of finitely generated FI-modules over a Noetherian ring very rarely has sufficiently many injectives. In this talk we consider the (left exact) torsion functor on the category of finitely generated FI-modules, and show that its derived functors exist. Properties of these derived functors, which we call the local cohomology functors, can be used in reproving well known theorems relating to the depth, regularity, and stable range of a module. We will also see that various facts from the local cohomology of modules over a polynomial ring have analogs in our context. This is joint work with Liping Li. |

## Revision as of 22:30, 7 February 2016

The seminar meets on Fridays at 2:25 pm in Van Vleck B113.

The schedule for the previous semester is here.

## Contents

## Algebraic Geometry Mailing List

- Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## Spring 2016 Schedule

date | speaker | title | host(s) |
---|---|---|---|

January 22 | Tim Ryan (UIC) | Moduli Spaces of Sheaves on \PP^1 \times \PP^1 | Daniel |

January 29 | Local | ||

February 5 | Botong Wang (Wisconsin) | Topological Methods in Algebraic Statistics | Local |

February 12 | Jay Yang (Wisconsin) | Random Toric Surfaces | Local |

February 19 | Daniel Erman (Wisconsin) | Supernatural Analogues of Beilinson Monads | |

February 26 | TBD | ||

March 4 | Claudiu Raicu (Notre Dame) | TBA | Steven |

March 11 | Eric Ramos (Wisconsin) | Local Cohomology of FI-modules | Local |

March 18 | Spring break | ||

March 25 | TBD | ||

April 1 | TBD | ||

April 8 | TBD | ||

April 15 | TBD | ||

April 22 | TBD | ||

April 29 | David Anderson (Ohio State) | TBA | Steven |

May 6 | TBD |

## Abstracts

### Tim Ryan

Moduli Spaces of Sheaves on \PP^1 \times \PP^1

In this talk, after reviewing the basic properties of moduli spaces of sheaves on P^1 x P^1, I will show that they are $\mathbb{Q}$-factorial Mori Dream Spaces and explain a method for computing their effective cones. My method is based on the generalized Beilinson spectral sequence, Bridgeland stability and moduli spaces of Kronecker modules.

### Botong Wang

Topological Methods in Algebraic Statistics

In this talk, I will give a survey on the relation between maximum likelihood degree of an algebraic variety and it Euler characteristics. Maximam likelihood degree is an important constant in algebraic statistics, which measures the complexity of maximum likelihood estimation. For a smooth very affine variety, June Huh showed that, up to a sign, its maximum likelihood degree is equal to its Euler characteristics. I will present a generalization of Huh's result to singular varieties, using Kashiwara's index theorem. I will also talk about how to compute the maximum likelihood degree of rank 2 matrices as an application.

### Daniel Erman

Supernatural analogues of Beilinson monads

First I will discuss Beilinson's resolution of the diagonal and some of the applications of that construction including the notion of a Beilinson. Then I will discuss new work, joint with Steven Sam, where we use supernatural bundles to build GL-equivariant resolutions supported on the diagonal of P^n x P^n, in a way that extends Beilinson's resolution of the diagonal. I will discuss some applications of these new constructions.

### Eric Ramos

The Local Cohomology of FI-modules

Much of the work in homological invariants of FI-modules has been concerned with properties of certain right exact functors. One reason for this is that the category of finitely generated FI-modules over a Noetherian ring very rarely has sufficiently many injectives. In this talk we consider the (left exact) torsion functor on the category of finitely generated FI-modules, and show that its derived functors exist. Properties of these derived functors, which we call the local cohomology functors, can be used in reproving well known theorems relating to the depth, regularity, and stable range of a module. We will also see that various facts from the local cohomology of modules over a polynomial ring have analogs in our context. This is joint work with Liping Li.