Notre Dame Algebraic Geometry Seminar

Fall 2008

(11-12 on Tuesdays in DBTL 138 or Thursdays in H 258)

Date

Time/Place

Speaker

Title

 Tu, Sept 2

11am

DBTL 138

Alan Stapledon

(U. Michigan)

Weighted Ehrhart theory and orbifold cohomology

Tu, Sept 16

11 am

DBTL 138

Andy Kustin

(U. South Carolina)

Reduction numbers of planar ideals

Tu, Sept 23

11am

DBTL 138

Nero Budur

(Notre Dame)

Singularities and Hodge filtrations

Th, Oct 9

11am

H 258

Bonnie Smith

(Notre Dame)

The Core of a Monomial Ideal

Tu, Oct 14

11 am

DBTL 138

Izzet Coskun

(U. Illinois -Chicago)

The geometry of homogeneous spaces

Th, Oct 30

11am

H 258

Kyungyong Lee

(Purdue)

Hilbert schemes of points

Th, Nov 6

11am

H 258

Yihuang Shen

(Purdue)

 

Wed, Nov 19 !!!

3pm

HH 125

Giulio Caviglia

(Purdue)

 


Past seminars: Spring 2008.

Abstract of Talks:

Sept 2. Alan Stapledon (U. Michigan): Weighted Ehrhart theory and orbifold cohomology 

 

Abstract:  If P is a lattice polytope, then one can define a polynomial $\delta_{P}(t)$, which encodes the number of lattice points in any fixed dilation of P. The polynomial $\delta_{P}(t)$ is a classical combinatorial invariant, called the Ehrhart $\delta$-polynomial of $P$. We will present a new geometric interpretation of the coefficients of $\delta_{P}(t)$. That is, they are sums of dimensions of orbifold cohomology groups of a toric stack. As a combinatorial application, we will prove a weighted version of Ehrhart Reciprocity. 

 

Sept 16. Andy Kustin (U. South Carolina): Reduction numbers of planar ideals.

 

Abstract: We use the Socle Lemma of Huneke-Ulrich, a characteristic zero result, to calculate reduction numbers. Our proof works in all characteristics.

 

Sept 23. Nero Budur (Notre Dame): Singularities and Hodge filtrations.

 

Abstract: This talk consists of two parts. In the first part we review various points of view on singularities: motivation, definitions, relations, and computability. In the second part we discuss recent work on the Hodge filtration for local systems. Applications include: computability of invariants of singularities in the hyperplane arrangement case, and polynomial periodicity of Hodge numbers for congruence covers.

 

Oct 9. Bonnie Smith (Notre Dame): The Core of a Monomial Ideal.

Abstract:  The core of an ideal I is the intersection of all reductions of I--that is, the intersection of all sub-ideals of I which share certain properties of I.  The core arises naturally in the context of the Briançon-Skoda Theorem, and there is also a connection (which was shown by Hyry and Smith) between the core and a conjecture of Kawamata  about the existence of non-vanishing global sections of ample line bundles.  For the case of a monomial ideal I in a polynomial ring over a field k, we would like to have a combinatorial description of the core of I which would reflect the combinatorial properties of I itself.  I will describe some results of this type.

 

Oct 14. Izzet Coskun (U. Illinois- Chicago): The geometry of homogeneous spaces.

 

Abstract: Homogeneous spaces play a central role in geometry, representation theory and combinatorics. In this talk, I will explain how to solve enumerative problems involving homogeneous spaces such as Grassmannians, flag varieties and orthogonal flag varieties. I will describe a positive rule for intersecting Schubert varieties in flag varieties and explain some consequences of this rule. I will describe how to extend these rules to the orthogonal setting.

 

Oct 30. Kyungyong Lee (Purdue): Hilbert schemes of points.

 

Abstract : The famous n! conjecture can be stated in an elementary language. In fact it asserts that the dimension of the vector space spanned by all derivatives of a certain bivariate analogue of the n × n Vandermonde determinant is equal to n!. This seemingly elementary conjecture was solved by M. Haiman, with a highly nontrivial machinery. The proof is closely related to the algebraic and geometric properties of isospectral Hilbert schemes of points on the plane. I'll discuss how some of the results in the plane case can or cannot be generalized to the higher dimensional case.


For problems with this page, email Nero Budur.