# Difference between revisions of "Algebraic Geometry Seminar Fall 2014"

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|Local | |Local | ||

|- | |- | ||

− | |November | + | |November 14 |

+ | |No seminar (room will be used for a specialty exam) | ||

+ | | | ||

+ | |- | ||

+ | |December 5 | ||

|[http://www.math.umass.edu/~markman/ Eyal Markman] (UMass Amherst) | |[http://www.math.umass.edu/~markman/ Eyal Markman] (UMass Amherst) | ||

− | | | + | |Integral transforms from a K3 surface to a moduli space of stable sheaves on it |

|Andrei | |Andrei | ||

|- | |- | ||

− | |December | + | |December 12 |

− | |DJ Bruce (UW) | + | |[http://www.math.wisc.edu/~djbruce/ DJ Bruce] (UW) |

− | | | + | |Betti Tables of Graph Curves |

|local | |local | ||

+ | |- | ||

+ | |March 13 | ||

+ | |[http://www3.nd.edu/~jrodri18/ Jose Rodriguez] (Notre Dame) | ||

+ | |TBD | ||

+ | |Daniel | ||

|- | |- | ||

|} | |} | ||

Line 69: | Line 78: | ||

===Eyal Markman=== | ===Eyal Markman=== | ||

− | + | Let S be a K3 surface, v an indivisible Mukai vector, | |

+ | and M(v) the moduli space of stable sheaves on S with Mukai vector v. | ||

+ | The universal sheaf gives rise to an integral functor F from the derived category of | ||

+ | coherent sheaves on S to that on M(v). We show that the functor F is faithful (but not full). | ||

+ | The bounded derived category of M(v) is rather mysterious at the moment. | ||

+ | As a first step, we provide a simple conjectural description of its full subcategory | ||

+ | whose of objects are images of objects on S via the functor F. | ||

+ | We verify that description whenever M(v) is the Hilbert scheme of points on S. | ||

+ | This work is joint with Sukhendu Mehrotra. | ||

===Lars W Christensen=== | ===Lars W Christensen=== | ||

Line 80: | Line 97: | ||

report on recent progress in this direction. The talk is based on | report on recent progress in this direction. The talk is based on | ||

joint work with Olgur Celikbas, Li Liang, and Grep Piepmeyer. | joint work with Olgur Celikbas, Li Liang, and Grep Piepmeyer. | ||

+ | |||

+ | ===Anatoly Libgober=== | ||

+ | I will discuss elliptic genus of singular varieties and | ||

+ | its extension to Witten's phases of N=2 theories. In particular | ||

+ | McKay correspondence for elliptic genus will be described. | ||

+ | As one of applications I will show how to derive relations | ||

+ | between elliptic genera of Calabi Yau manifolds and related Witten phases | ||

+ | using equivariant McKay correspondence for elliptic genus. | ||

===Vlad Matei=== | ===Vlad Matei=== | ||

In 2012, J.P Keating and Z. Rudnick published a paper where they resolved a function field version of the Montgomery-Goldston pair correlation conjecture. Their proof relies on a recent equidistribution result of N. Katz. In joint work with Daniel Hast, we reprove their result by counting points on a certain variety using a twisted Grothendieck-Lefschetz formula and obtain also information about higher moments. | In 2012, J.P Keating and Z. Rudnick published a paper where they resolved a function field version of the Montgomery-Goldston pair correlation conjecture. Their proof relies on a recent equidistribution result of N. Katz. In joint work with Daniel Hast, we reprove their result by counting points on a certain variety using a twisted Grothendieck-Lefschetz formula and obtain also information about higher moments. | ||

Moreover our method allows us to also give a proof of the autocorrelation of the Mobius function on average in the function field setting, also known as the Chowla conjecture. | Moreover our method allows us to also give a proof of the autocorrelation of the Mobius function on average in the function field setting, also known as the Chowla conjecture. | ||

+ | |||

+ | ===DJ Bruce=== | ||

+ | Given a graph one may obtain a reducible algebraic curve by associating a P^1 to each vertex with two P^1’s intersecting if there is an edge between the associated vertices. Such curve are called graph curves, or line arrangements, and were introduced by Bayer and Eisenbud in studying Green’s conjecture. I will discuss how the combinatorics of the graph affect the Betti table of its associated curve. In particular, I will present formulas for the Betti table for all graph curves of genus zero and one. Additionally, I will give formulas for the graded Betti numbers for a class of curves of higher genus. This talk is based on joint work with Pete Vermeire, Evan Nash, Ben Perez, and Pin-Hung Kao. |

## Latest revision as of 13:26, 19 December 2014

The seminar meets on Fridays at 2:25 pm in Van Vleck B131.

The schedule for the previous semester is here.

## Contents

## Algebraic Geometry Mailing List

- Please join the Algebraic Geometry 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).

## Fall 2014 Schedule

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

September 12 | Andrei Caldararu (UW) | Geometric and algebraic significance of the Bridgeland differential | (local) |

September 19 | Greg G. Smith (Queen's University) | Toric vector bundles | (Daniel) |

October 3 | Daniel Erman (UW) | Tate resolutions for products of projective spaces | (local) |

October 10 | Lars Winther Christensen (Texas Tech University) | Beyond Tate (co)homology | Daniel |

October 17 | Claudiu Raicu (Notre Dame University) | TBA | Daniel |

October 31 | Anatoly Libgober (UIC) | Landau-Ginzburg/Calabi-Yau and McKay correspondences for elliptic genus | Max |

November 7 | Vlad Matei (UW) | Moments of arithmetic functions in short intervals | Local |

November 14 | No seminar (room will be used for a specialty exam) | ||

December 5 | Eyal Markman (UMass Amherst) | Integral transforms from a K3 surface to a moduli space of stable sheaves on it | Andrei |

December 12 | DJ Bruce (UW) | Betti Tables of Graph Curves | local |

March 13 | Jose Rodriguez (Notre Dame) | TBD | Daniel |

## Abstracts

### Andrei Caldararu

Several years ago Tom Bridgeland suggested that there should exist interesting chain maps C_*(M_{g,n}) -> C_{*+2}(M_{g,n+1}) and he conjectured some applications of these maps to mirror symmetry. I shall present a precise definition of these maps using techniques from the theory of ribbon graphs, and discuss a recent result (joint with Dima Arinkin) about the homology of the total complex associated to the bicomplex obtained from these maps. Then I shall speculate (wildly) about applications to mirror symmetry.

### Eyal Markman

Let S be a K3 surface, v an indivisible Mukai vector, and M(v) the moduli space of stable sheaves on S with Mukai vector v. The universal sheaf gives rise to an integral functor F from the derived category of coherent sheaves on S to that on M(v). We show that the functor F is faithful (but not full). The bounded derived category of M(v) is rather mysterious at the moment. As a first step, we provide a simple conjectural description of its full subcategory whose of objects are images of objects on S via the functor F. We verify that description whenever M(v) is the Hilbert scheme of points on S. This work is joint with Sukhendu Mehrotra.

### Lars W Christensen

Tate (co)homology was originally defined for modules over group algebras. The cohomological theory has a very satisfactory generalization---Tate--Vogel cohomology or stable cohomology---to the setting of associative rings. The properties of the corresponding generalization of the homological theory are, perhaps, less straightforward and have, in any event, been poorly understood. I will report on recent progress in this direction. The talk is based on joint work with Olgur Celikbas, Li Liang, and Grep Piepmeyer.

### Anatoly Libgober

I will discuss elliptic genus of singular varieties and its extension to Witten's phases of N=2 theories. In particular McKay correspondence for elliptic genus will be described. As one of applications I will show how to derive relations between elliptic genera of Calabi Yau manifolds and related Witten phases using equivariant McKay correspondence for elliptic genus.

### Vlad Matei

In 2012, J.P Keating and Z. Rudnick published a paper where they resolved a function field version of the Montgomery-Goldston pair correlation conjecture. Their proof relies on a recent equidistribution result of N. Katz. In joint work with Daniel Hast, we reprove their result by counting points on a certain variety using a twisted Grothendieck-Lefschetz formula and obtain also information about higher moments. Moreover our method allows us to also give a proof of the autocorrelation of the Mobius function on average in the function field setting, also known as the Chowla conjecture.

### DJ Bruce

Given a graph one may obtain a reducible algebraic curve by associating a P^1 to each vertex with two P^1’s intersecting if there is an edge between the associated vertices. Such curve are called graph curves, or line arrangements, and were introduced by Bayer and Eisenbud in studying Green’s conjecture. I will discuss how the combinatorics of the graph affect the Betti table of its associated curve. In particular, I will present formulas for the Betti table for all graph curves of genus zero and one. Additionally, I will give formulas for the graded Betti numbers for a class of curves of higher genus. This talk is based on joint work with Pete Vermeire, Evan Nash, Ben Perez, and Pin-Hung Kao.