List of Talks
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the abstracts below.

Some thoughts on the jacobian conjecture
by Shreeram ABHYANKAR, Purdue University
Abstract:
I shall report on some recent progress in this long standing conjecture.

The image of the Chern class map for singular varieties
by Donu ARAPURA, Purdue University
Abstract:
When X is a smooth complete variety over C, its well known that the
Chern classes of a vector bundle in rational cohomology are Hodge classes,
i. e. classes of type (p,p). When X is singular, the notion of Hodge class
can be generalized using mixed Hodge theory, and it is not difficult
to see that Chern classes of vector bundles are again of this type.
However there are now further constraints on these classes that I want to
explain. This phenomenon was first noticed by BarbieriViale and Srinivas
for line bundles about 10 years ago.

Hochschild cohomology and Gorenstein algebras
by Luchezar AVRAMOV, University of Nebraska at Lincoln
Abstract:
Let K be a noetherian ring and S a commutative algebra, which is
essentially of finite type over K and is projective as a Kmodule. The
Kalgebra S is said to be Gorenstein if the nontrivial fibers of the
structure homomorphism K → S are Gorenstein rings. This property will be
characterized in terms of the vanishing of appropriate Hochschild cohomology
groups HH^{n}(SK;S ⊗_{K}S).
This is joint work with Srikanth Iyengar.

BuchsbaumRim multiplicity and HilbertSamuel multiplicities
by C.Y. Jean CHAN, University of Arkansas
Abstract:
Let $R$ be a regular local ring of dimension $2$ with maximal ideal
$\mathfrak m$. We study the BuchsbaumRim multiplicity $e_{BR}(M)$ of
a finitely generated module $M$ of finite colength in a free module
$F$.\par
Let $\mathfrak a$ be an $\mathfrak m$primary ideal in
$R$. We first investigate the colength $\ell(R/ \mathfrak a)$ of
$\mathfrak a$ and its
HilbertSamuel multiplicity $e(\mathfrak a)$ using linkage theory. As
applications, we establish several multiplicity formulas that express
$e_{BR}(M)$ in terms of the Hilbert
multiplicities of ideals related to an arbitrary minimal reduction $U$ of
$M$. In the special case where the maximal Fitting ideal of $F/U$ is
integrally closed, $e_{BR}(M)$ is directly related to all Fitting
ideals of $F/U$. \par
There exists $\mathfrak m$primary Bourbaki ideals $I$ and $J$ of the
modules $F$ and $M$ respectively such that $F/M \cong I/J$. We also
have a formula for $e_{BR}(M)$ in terms of $e(I)$ and $e(J)$. This is
related to a graphical interpretation of the multiplicities in the
case of monomial ideals.
(This is joint work with JungChen Liu and Bernd Ulrich.)

On the CohenMacaulayness of residual intersections and extended Rees
rings
by Christine CUMMING, Purdue University
Abstract:
I will talk about when residual intersections and extended Rees rings are
CohenMacaulay when the ring is merely CohenMacaulay.

On Extension of Gabber's Work
by Sankar DUTTA, University of Illinois at UrbanaChampaign
Abstract:
We will first point out the key steps in Gabber's proof of the nonnegativity
part of Serre's conjecture on Intersection Multiplicity over regular local
rings. Next we will show that his technique cannot be extended to regular
schemes of finite type over an excellent discrete valuation ring. We would like
to derive some special cases of positivity if time permits.

Adjoint ideals
by Lawrence EIN, University of Illinois at Chicago
Abstract:
Adjoint ideal is a modification of multiplier ideal. We'll describe the
properties of adjoint ideals and give some applications of these ideals to
extension theorems and singularities of pairs.

Matroids arising from certain closure operations on ideals
by Neil EPSTEIN, University of Michigan
Abstract:
I will discuss some cases where matroids (infinite, with finite basis
condition) arise naturally when considering an ideal, a closure operation, and
the set of ideals which have the same closure as the given ideal. This opens
up a wide range of combinatorial tools for dealing with closure operations for
which this situation occurs.

The core of ideals
by Louiza FOULI, Purdue University
Abstract:
Let R be a Noetherian local ring with infinite residue field k
and I an Rideal. We say that J ⊂ I is a reduction of
I if I and J have the same integral closure. A reduction is
called minimal if it is minimal with respect to inclusion. In
general minimal reductions are not unique. To remedy this lack of
uniqueness one considers the intersection of all (minimal)
reductions, namely the core of I, core(I). The core
encodes information about all possible reductions of the ideal. On
the other hand reductions are connected with the study of blowup
algebras. The core is a mysterious object that appears naturally in
the context of BriançonSkoda kind of theorems. Hyry and Smith
have shown that Kawamata's conjecture on the existence of sections
of certain line bundles is equivalent to a statement about the core
of particular ideals in section rings. Under some technical
conditions (which are automatically satisfied in case I is
equimultiple) Polini and Ulrich have shown that for a Gorenstein
local ring, core(I)=J^{n+1}:I^{n}, for n ≫ 0.
We present some recent work on the core of ideals.

Comparison of local and global liaison questions
by Robin HARTSHORNE, University of California at Berkeley
Abstract:
We study the analogy between the global theory of liaison of varieties in
projective space and the local theory of liaison of ideals in a local ring. To
what extent do the two theories give parallel results? Can one deduce global
from local results or vice versa? We give some examples to show this is not
always possible.

Embedding modules of finite projective dimension
by Melvin HOCHSTER, University of Michigan
Abstract:
The main result is that over a Noetherian ring finitely generated modules
of finite projective dimension can be embedded in
in a very special modules with this property. Applications
of this result will also be described. (Th is joint work with Yongwei Yao)

Integral Closures of Modules and Hyperplane Sections
by Jooyoun HONG, Purdue University
Abstract:
The theory of the normalization of ideals has been extended to
modules and, as in the case of ideals, the Rees algebras of modules serve as a
useful tool to detect normality of modules. Let $R$ be a Noetherian ring and
$E$ a finitely generated torsionfree $R$module having a rank. To enable the
extension, we consider two technical devices to attach an ideal $I$ of a ring
$S$ to the $R$module $E$ so that the comparison can be made between the Rees
algebra of $E$ and the Rees algebra of $I$. One of the ideals that can be used
is the ideal generated by the module in the polynomial ring which contains the
Rees algebra of the module. The other is known as the generic Bourbaki ideal of
$E$. In this joint work with B. Ulrich, we
show that integrally closedness of any ideals of height at least $2$ is
compatible with a specialization of generic elements using a vanishing theorem
of local cohomology of certain degrees. By turning a module into an ideal, we
relate the integral closure of a module to that of the ideal.
(This is joint work with Bernd Ulrich)

Coniveau and the Grothendieck group of varieties
by SuJeong KANG, Purdue University
Abstract:
We prove that if the generalized Hodge conjecture, or some weaker form of it,
holds for a CalabiYau variety then it holds for any CalabiYau varieties
birationally equivalent to it. This is joint work with Donu Arapura.

The resolution of the Frobenius module
by Andrew KUSTIN, University of South Carolina
Abstract:
Let R be a ring containing the field of positive characteristic p and
let ^{f}R to represent the ring R endowed with an Rmodule structure given
by the Frobenius automorphism f:R → R with f(r)=r^{q}, where q=p^{e}
for some fixed exponent e.
We study the resolution of ^{f}R by free Rmodules. (This is joint work with
C. Miller and A. Vraciu.)

Polar syzygies and unirational algebras
by Aron SIMIS, Universidade Federal de Pernambuco, Brazil
Abstract:
Given a finitely generated subalgebra of a polynomial ring, one considers the
syzygies of the transposed Jacobian module of its generators. A good chunk of
them is given by the polar syzygies (appeared in disguise in GordanNoether).
One deals with some special classes of algebras to understand when the polar
syzygies generate all syzygies. (Joint work with I. Bermejo and P. Gimenez)

Annihilation of elements of local cohomology modules
by Anurag SINGH, University of Utah
Abstract:
We will discuss some results and questions on the existence of big
CohenMacaulay algebras, and related big algebras. These can be viewed as statements
about the annihilation of elements of local cohomology modules. There are powerful
theorems along these lines due to HochsterHuneke, and to HunekeLyubeznik.

Asymptotic behavior of multigraded regularity
by Brent STRUNK, Purdue University
Abstract:
In this talk we discuss asymptotic behavior of multigraded regularity.

Indecomposable mixed modules of large rank
by Roger WIEGAND, University of Nebraska
Abstract:
Let (R,m,k) be a onedimensional local ring. Our goal
is to build indecomposable finitely generated modules with large
rank. In most cases, even if R has finite representation type, one
can build indecomposable mixed modules whose torsionfree
parts have large rank. Exceptions include discrete valuation rings
and Dedekindlike rings (essentially, (A_{1})singularities,
e.g., k[[x,y]]/(xy) and R[[x,y]]/(x^{2}+y^{2})).
We show that, if R is not a homomorphic image of a Dedekindlike ring,
then, for each positive integer n, there is an indecomposable
finitely generated module M such that M_{P} ≅
R_{P}^{(n)} for every minimal prime ideal P.
If char(k) ≠ 2, the existence of such modules is equivalent to wildness
of the category of finitelength modules. A key idea in our construction is to
produce an indecomposable finitelength module V and a torsionfree module
N such that Ext^{1}_{R}(N,V) has big socle dimension.
One then uses this fact to produce extensions 0 → V → M
→ N^{(t)} → 0, with t large and M indecomposable.
This is joint work with Wolfgang Hassler, Ryan Karr and Lee Klingler.

Ring extensions with trivial generic fiber
by Sylvia WIEGAND, University of Nebraska
Abstract:
We study examples of domain extensions R ⊆ S such that every nonzero
prime ideal of S intersects R in a nonzero prime ideal. In certain
circumstances the dimension of S must be less than or equal to 2.
This is joint work with Wlliam Heinzer and Christel Rotthaus.

Poincaré duality algebras as rings of coinvariants
by Clarence WILKERSON, Purdue University
Abstract:
If $V$ is a finite dimensional vector space over a field $k$, and
$W$ is a finite subgroup of $Aut(V)$, then the symmetric algebra
$S(V^\#)$ can be thought of as the algebra of polynomial functions
on $V$, and it inherits an action of the group $W$. The algebra of
invariants $S(V^\#)^W$ is of particular interest. If it is itself
a polynomial algebra, then the quotient algebra $S(V^\#)/I$, where
$I$ is the ideal generated by the positive degree elements of $S(V^\#)^W$ form
the coinvariants and is Poincare' duality algebra
under the induced multiplication.
However, if the characteristic of $k$ is positive, the coinvariants
can be a Poincare duality algebra without $S(V^\#)^W$ being polynomial. The
author and W. G. Dwyer give a generic example of
this behavior. Moreover, we give a new proof, independent of the previous
work of Steinberg, Kane, and T.C. Lin of the following theorem.\par
With notation as above, if char($k$) = 0 or $p$ with $p$ relatively
prime to the order of $W$, then $S(V^\#)/I$ is a Poincare' duality
algebra if and only if $S(V^\#)^W$ is a polynomial algebra.
This is joint work with W. G. Dwyer, Notre Dame.

Resolution of singularities in characteristic zero
by Jaroslaw WLODARCZYK, Purdue University
Abstract:
We discuss basic ideas of simplified Hironaka desingularization algorithm in
characteristic zero.
