Notre Dame
Algebra Seminar
Fall 2009
(12:45 – 1:45 on Wednesdays in
HAYE 127 and/or Fridays in HAYE 125)
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Date |
Time/Place |
Speaker |
Title |
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Sept 2 , Wed (colloquium talk) |
4pm HAYE 117 |
Mihnea Popa ( |
BGG correspondence and the cohomology of compact Kaehler manifolds |
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Sept 9, Wed |
12:45-1:45 HAYE 127 |
Yu Xie (Notre Dame) |
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Sept 18, Fri |
12:45-1:45 HAYE 125 |
Roi Docampo Álvarez ( |
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Sept 23, Wed |
12:45-1:45 HAYE 127 |
Michael Broshi (Notre Dame) |
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Sept 26, Sat |
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Oct 2, Fri |
12:45-1:45 HAYE 125 |
Donu Arapura ( |
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Oct 7, Wed (colloquium talk) |
4pm HAYE 117 |
Andrew Kustin (Univ of |
Singularities of plane curves which are parameterized by homogeneous forms of small degree |
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Oct 9, Fri |
12:45-1:45 HAYE 125 |
Paolo Aluffi ( |
Chern classes of singular varieties, graph
hypersurfaces, and Feynman integrals |
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Oct 9 (colloquium talk) |
4pm Room TBA |
Tom Nevins (UIUC) |
TBA |
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Oct 16-18 |
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Oct 19-23 |
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Mid-semester BREAK |
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Oct 26, Mon (colloquium talk) |
4 pm Room TBA |
Elisa Gorla ( |
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Nov 4, Wed |
12:45-1:45 HAYE 127 |
Laurenţiu Maxim (UW Madison) |
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Nov 11, Wed |
12:45-1:45 HAYE 127 |
Jeb Willenbring (UW Millwaukee) |
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Nov 13, Fri |
12:45-1:45 HAYE 125 |
Zach Teitler ( |
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Nov 18, Wed |
12:45-1:45 HAYE 127 |
Yu Xie (Notre Dame) |
Formulas for the multiplicity of graded algebras (continuation) |
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Nov 25-27 |
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Thanksgiving BREAK |
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Past seminars: Spring 2009, Fall 2008, Spring 2008.
Other
seminars: Student Algebraic Geometry /
Commutative Algebra Seminar.
Abstract of Talks:
Sept 2 (colloquium talk): Mihnea Popa (UIC) - BGG correspondence and the cohomology of compact Kaehler manifolds
Abstract: The cohomology algebra of the sheaf of holomorphic functions on a compact Kaehler manifold can be naturally viewed as a module over the exterior algebra of a vector space. A well-known result of Bernstein-Gel'fand-Gel'fand gives a correspondence between such "exterior" modules and linear complexes of modules over the symmetric algebra, i. e. the polynomial ring. I will explain how one can use a modern view on this correspondence, together with the Generic Vanishing theory developed by Green and Lazarsfeld via Hodge-theoretic methods, in order to understand subtle algebraic structures of the cohomology algebra. As a bonus, homological and commutative algebra tools can be applied on the polynomial ring side to obtain new inequalities for the holomorphic Euler characteristic and the Hodge numbers of compact Kaehler manifolds. This is joint work with R. Lazarsfeld.
Sept 9: Yu Xie (Notre Dame)- Formulas for the multiplicity of graded algebras
Abstract: For simplicity, we
will assume that $A$ is contained in $B$, a homogeneous inclusion of standard
graded Noetherian domains over a field. We want to express the multiplicity of
$A$ in terms of that of $B$ and local multiplicities along the projective
spectrum of $B$. One of the applications is to find the multiplicity of the
special fibre ring of an ideal generated by forms of the same degree in a
standard graded Noetherian algebra over a field.
Observe that the dimension of $A$ is always less than or equal to that of $B$.
They are equal if and only if their quotient fields extension is algebraic of
degree $r$. If $B$ is integral over $A$, i.e. the dimension of $B/A_1B$ is
zero, then $e(B)=re(A)$. In 2001, Simis, Ulrich and Vasconcelos gave a
formula when both rings have the same dimension and the dimension of $B/A_1B$
is equal to one. We generalize their formula to arbitrary dimensions of $B/A_1B$.
We also provide the formula for the case when the dimension of $A$ is strictly
less than the dimension of $B$. Thus we give a complete answer to the original
question. The techniques we use are $j$-multiplicities and filter-regular
sequences.
The formulas we obtain can be used to find the degree of dual varieties for any
hypersurfaces without any restrictions on its dual varieties and singularities.
In particular, it gives a generalization of Teissier's Pl\"{u}cker formula
to hypersurfaces with non-isolated singularities.
Sept 18: Roi Docampo Álvarez (
Abstract: We study the
structure of the arc space and the jet schemes of generic
determinantal varieties. Via
the use of group actions, I will show how
to compute the number of
irreducible components of all the jet
schemes, find formulas for
log canonical thresholds, and compute some
motivic volumes. If time
permits, I will explore extensions of the
results to the case of
spherical varieties.
Sept 23: Michael Broshi (Notre Dame)- Algebraic groups over a Dedekind domain
Abstract: In this talk, we extend some standard results on algebraic groups over a field to those defined over a Dedekind domain whose fraction field has characteristic zero. After reviewing basic definitions, we will state the main result, and discuss various applications.
Oct 2: Donu Arapura (Purdue)- Why the Hodge conjecture fails with integer coefficients
Abstract: Thanks to an old example by Atiyah-Hirzebruch,
the Hodge conjecture is now formulated with rational
coefficients. I want to review this along with other
examples. Depending on various factors (time,
interest of the audience, and my own ability to work
things out by the deadline) I may say something about
the status of the the "integral" Tate conjecture as well.
Oct 7: Andrew Kustin (
Consider an ideal $I$ of height two generated by three homogeneous forms of degree six in $R=k[x,y]$. On the one hand, we describe the degrees of the minimal generators of the defining ideal ${\mathcal A}$ of the Rees algebra $R[It]$. There are only a handful of possibilities. On the other hand, the ideal $I$ gives rise to a parameterization of a curve ${\mathcal C}$ in the projective plane. The curve ${\mathcal C}$ has singularities and the multiplicities of these singularities are determined by the generator degrees of ${\mathcal A}$. This work has been carried out with David Cox, Claudia Polini, and Bernd Ulrich.
Oct 9: Paolo Aluffi (
Abstract: Numerical evidence indicates that the value of individual
contributions of graphs to Feynman integrals are linear combinations
of multiple zeta values. This fact can be interpreted as a statement
concerning certain hypersurfaces of projective space determined by
graphs. I will report on work aimed at understanding this phenomenon.
Characteristic classes of singular varieties play an important role
in this study, by quantifying the singularity of graph hypersurfaces,
and by giving an algebro-geometric construction of invariants
satisfying `Feynman rules'. This is joint work with Matilde Marcolli.
Oct 26 (colloquium
talk): Elisa Gorla (
Abstract: The theory of liaison or linkage formally started in the
seventies, although it had been used before in an hoc manner. Roughly
speaking, liaison aims at understanding the class of projective schemes,
by partitioning it into families of schemes (the liaison classes) that
can all be ultimately ``linked'' to the same scheme. A linkage step
consists of taking the union of the scheme that we study with another
one, so that the union belongs to a well-studied family of schemes
(complete intersections or arithmetically Gorenstein schemes). In an
ideal situation, the scheme that we study is linked to one that we
understand better, and their union is simpler than each of the two parts.
We will start the talk by introducing the objects of our study, i.e.
schemes in projective space, and their algebraic counterparts, i.e.
ideals in a polynomial ring. We will then introduce the concept of
liaison and discuss its relevance. Schemes defined by minors of a matrix
of polyomials are classically studied in algebraic geometry. They will
be a recurrent example, and we will focus on what we can say about them
from the point of view of liaison theory.
Nov 4: Laurenţiu Maxim (UW Madison) -Hirzebruch invariants of symmetric products
Abstract: I will discuss how a very simple relationship between symmetric group actions on exterior products and the theory of lambda rings can be used to obtain generating series formulae for very general Hodge-theoretic invariants of symmetric products. This is joint work with J. Schuermann.
Nov 11: Jeb Willenbring (UW Millwaukee) -Stable Hilbert series and the Kronecker coefficients
Abstract: This talk will describe a problem in invariant theory
motivated by the concept of quantum entanglement. Specifically, we compute a
stable formula for
the Hilbert series of the invariant algebra of polynomial functions on a tensor
product of defining representations of unitary groups. The example may be interpreted
physically as the quantum analog of a classical system consisting of several
particles in which each has a finite number of classical states. The
stable formula involves
elementary combinatorics. The derivation involves the representation theory of
the symmetric group. In particular, the Kronecker coefficients play an
important role.
We also show how the value of the Hilbert function counts certain directed
graphs with colored edges. These graphs encode an explicit expression for
the
invariants. We describe how combinatorial properties of the graphs are
reflected in the structure of the ring of invariants. The results
presented are joint with M. Hero and L. Williams.
Nov 13: Zach Teitler (
Abstract: In 2001 Ein--Lazarsfeld--Smith showed the following: Let Z be a
set of
points in the plane and m an integer. If a polynomial F vanishes at
each point of Z to order at least 2m, then F is in the m'th power of
the ideal of Z. This statement is appealingly classical-sounding, yet
the only known proofs involve modern techniques such as multiplier
ideals. I will introduce multiplier ideals and explain how they give
the Ein--Lazarsfeld--Smith result. I will discuss some of the many
open questions in this area, both about multiplier ideals and about
efforts to improve the Ein--Lazarsfeld--Smith result.
For problems with this page, email Nero Budur.