Notre Dame
Algebraic Geometry / Commutative
Algebra Seminar
Fall 2008
(11-12 on Tuesdays in DBTL 138 or
Thursdays in H 258)
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Date |
Time/Place |
Speaker |
Title |
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Tu, Sept 2 |
11am DBTL 138 |
Alan Stapledon (U. |
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Tu, Sept 16 |
11 am DBTL 138 |
Andy Kustin (U. |
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Tu, Sept 23 |
11am DBTL 138 |
Nero Budur (Notre Dame) |
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Th, Oct 9 |
11am H 258 |
Bonnie
Smith (Notre Dame) |
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Tu, Oct 14 |
11 am DBTL 138 |
Izzet Coskun (U. Illinois -Chicago) |
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Th, Oct 30 |
11am H 258 |
Kyungyong Lee (Purdue) |
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Th, Nov 6 |
11am H 258 |
Yihuang
Shen (Purdue) |
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Wed, Nov 19 CANCELED |
3pm HH 125 |
Giulio Caviglia (Purdue) |
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Mo, Dec 1 |
3:05pm HAYE 125 |
Adam Boocher ( |
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Tu, Dec 2 |
11am DBTL 138 |
Jeff Mermin (U. Kansas) |
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Th, Dec 11 |
11am H 258 |
Luca Scala (U. |
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Next
semester: Spring 2009. 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.
Nov 6.
Yiuang Shen (Purdue):
Abstract: We will introduce the
Nov 19. Giulio Caviglia (Purdue): Properties deducible from the generic initial ideal.
Dec 1.
Adam Boocher (
Dec 2. Jeff Mermin (U. Kansas): On the lex-plus-powers conjecture.
Abstract: Let $S=k[x_1,\dots,x_n]$ be a polynomial ring and
$F=(f_1,\dots,f_r)$ be a regular sequence with deg (f_i)=e_i, 2\leq e_1\leq
\dots \leq e_r. Set $P=(x_{1}^{e_{1}},\dots,x_{n}^
Dec 11. Luca Scala (U. Chicago): Strange duality on the projective plane and cohomology of tautological bundles on Hilbert schemes of points on a surface.
Abstract: We study tensor powers of tautological bundles on Hilbert schemes of points over a surface with particular interest in their cohomology. By making use of the derived McKay correspondence of Bridgeland-King-Reid and of some results by Haiman, we prove general formulas for the cohomology of the double tensor power and of general exterior powers of tautological bundles on the Hilbert scheme. We will explain how similar computations can be used to approach the strange duality conjecture on the projective plane, and why this problem is connected with some properties of Barth morphism.
For problems with this page, email Nero Budur.