GSU-USC Commutative Algebra Seminar
Columbia, SC
November 14-16, 2008

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

• Andy Kustin (University of South Carolina) kustin@math.sc.edu
  
  
• Adela Vraciu (University of South Carolina) vraciu@math.sc.edu
  

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.

Previous seminars

• Fall 2008, Atlanta.
• Spring 2008, Atlanta.
• Spring 2008, Columbia.
• Fall 2007, Columbia.
• Fall 2007, Atlanta.
• Spring 2007, Columbia.
• Spring 2007, Atlanta.
• Fall 2006, Columbia.
• Fall 2006, Atlanta.