GSU-USC Commutative Algebra Seminar
Columbia, SC
November 14-16, 2008
All of the talks will take place in LeConte College on the corner of Greene Street and Pickens Street. Map, Hotels, and Parking.
Schedule
- Friday, 3:00 -- 3:30 PM, room 408 LeConte, Refreshments.
- Friday, 3:30 -- 4:20 PM, room 412 LeConte, Colloquium: "Numerical criteria for integral dependence", Bernd Ulrich, Purdue University.
- Friday, 5:00 --5:50 PM, room 310 LeConte, "Finiteness theorems in algebraic statistics", Seth Sullivant, North Carolina State University.
- Friday, 7:00 PM, Dinner, Delhi Palace Restaurant, 1029 Briargate Circle and Broad River Road, Columbia, SC 29210. Click here for a map.
Here are driving directions from LeConte to Delhi Palace. Please let Andy know if you are coming to dinner.
- Saturday, 9:00 -- 9:30 AM, room 310 LeConte, Refreshments.
- Saturday, 9:30 -- 10:20 AM, room 310 LeConte, "Commutative
algebra and group cohomology", Jon Carlson, University of Georgia.
- Saturday, 10:45 -- 11:35 AM, room 310 LeConte, "The interplay between cores, adjoints (or multiplier ideals) and
algebraic properties of the Rees ring", Claudia Polini, University of Notre Dame.
- Saturday, Lunch, Blue Cactus Cafe, 2002 Greene Street, Columbia, South Carolina 29205. The Blue Cactus is a short walk down Greene Street from LeConte.
- Saturday, 2:30 -- 3:20 PM, room 310 LeConte, "The weak Lefschetz property", Uwe Nagel, University of Kentucky.
- Saturday, 3:45 -- 4:35 PM, room 310 LeConte, "Properties of the Frobenius endomorphism that imply regularity", Yongwei Yao, Georgia State University.
- Saturday, 5:00 -- 5:50 PM, room 310 LeConte, "Free summands of cokernels and syzygies", Bart Snapp, Coastal Carolina University.
- Saturday, 7:00 PM, Dinner Al Amir Restaurant, 7001 St. Andrews Rd, Columbia, South Carolina 29212.
Click here for a map.
Here are driving directions from LeConte to Al Amir.
Please let Andy know if you are coming to dinner.
- Sunday, 9:30 -- 10:00 AM, room 310 LeConte, Refreshments
- Sunday, 10:00 -- 10:50 AM, room 310 LeConte, "The $cl$-core of an ideal", Louiza Fouli, University of Texas.
- Sunday, 11:15 -- 12:05 AM, room 310 LeConte, "Anti-nilpotent modules and primary decomposition with
respect to Frobenius", Florian Enescu, Georgia State University.
Organizers
Other participants include
- Alberto Corso, University of Kentucky
- Alina Iacob, Georgia Southern University
- Jinjia Li, Middle Tennessee State University
- Matt Miller, University of South Carolina
- Sara Malec, Georgia State University
- Sandy Spiroff, University of Mississippi
- Javid Validashti, University of Kansas
Abstracts
- Florian Enescu (Georgia State University): "Anti-nilpotent modules and primary decomposition with
respect to Frobenius".
The talk will present the notion of anti-nilpotent modules and
discuss ways of developing a primary decomposition theory for
modules with Frobenius action over a ring of prime characteristic. Part
of the work is joint with M. Hochster.
- Louiza Fouli (University of Texas): "The $cl$-core of an ideal".
We expand the notion of the core of an ideal to $cl$-core for Nakayama closures $cl$. Let $(R, \mathfrak{m})$ be a Noetherian local ring of characteristic $p>0$ and infinite residue field. In general $\core{I} \subset *$-$\core{I}$, where $*$ denotes the tight closure operation and $I$ is an $R$-ideal. We show that the $*$-$\core{I}=\core{I}$ in a local Cohen--Macaulay normal
domain with perfect infinite residue field, if the analytic spread,
$\ell$, is equal to the $*$-spread and $I$ is for example $\mathfrak{m}$-primary. We also generalize the notion of general reductions to general $*$-reductions. This is joint work with Janet Vassilev. Here is a .pdf version of this abstract.
- Uwe Nagel (University of Kentucky): "The weak Lefschetz property".
An artinian standard graded algebra over a field has the Weak
Lefschetz Property (WLP) if multiplication by a general linear form,
from any component to the next, has maximal rank. In many situations it
is expected that the algebra under consideration has the WLP. However,
there are rather few general results that establish the presence of the
WLP. We describe several open problems. We also discuss the subtlety of
the WLP by presenting various examples. In particular, the surprising
role of the characteristic of the ground field is illustrated.
- Bart Snapp (Coastal Carolina University) "Free summands of cokernels and syzygies".
In this talk we will discuss properties of rings which can be
characterized when certain syzygies and certain cokernels have free
summands.
- Seth Sullivant (North Carolina State University): "Finiteness theorems in algebraic statistics". I will describe a range of new finiteness results for
statistical models, in particular, results which say that, up to symmetry,
many models in random variables with state space "tending to infinity" have
finite algebraic descriptions. The focus will be on applications of these
ideas to Markov bases (that is, generating sets of toric ideals), but the
techniques apply to many other statistical models. The results follow
from studying polynomial rings in infinitely many indeterminates under the
action of the infinite symmetric group, which leads to a theory that is of
independent interest. I will focus on the commutative algebraic aspects
of the theory, with the statistical problems serving as motivation. This
is joint work with Chris Hillar.
- Bernd Ulrich (Purdue University): "Numerical Criteria for Integral Dependence". Given a family of singularities one would like to use numerical
invariants to distinguish between them. The corresponding algebraic
problem is to prove multiplicity based criteria for the integral
dependence of modules. This requires, among other things, generalizations
of the classical notions of multiplicity. Part of the talk will be a
report on recent joint work with Javid Validashti.
- Yongwei Yao (Georgia State University): "Properties of the Frobenius endomorphism that imply regularity". Let R be a Noetherian ring of prime characteristic p. Then, for any positive integer e, there is the Frobenius endomorphism F^e of R. Thus, for any R-module M, there is an induced R-module structure on with the scalar multiplication twisted by F^e. In this talk, we show that if there is a non-zero finitely generated R-module M such that the induced R-module is flat over R, then R is regular. This is joint work with Mel Hochster.
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