Commutative Algebra in the Southeast Meeting
Columbia, SC
March 18 --20 , 2011.

The Schedule

• Friday, 3:00 PM, Wyman Williams Room, next to room 412 LeConte, Refreshments.
• Friday, 3:30 PM, room 412 LeConte, David Cox, Amherst College: "A Case Study in Bigraded Commutative Algebra". David Cox's talk is the first talk of the meeting and also a Math Department Colloquium. Slides from David Cox's talk.
  
• Friday, 5:00 PM, room 310 LeConte, Paul S. Aspinwall, Duke University: "D-Branes, Toric Geometry and Matrix Factorizations".
  
  
• Friday 6:45 PM, Dinner Bombay Grill. Please let us know if you plan to attend the dinner.
  
  
• Saturday, 8:30 AM, room 310 LeConte, refreshments.
  
  
• Saturday, 9:00--9:55 AM, room 310 LeConte, Ezra Miller, Duke University: "How primary decomposition in commutative monoids is wrong".
• Saturday, 10:10--11:05 AM, room 310 LeConte, Claudia Polini, University of Notre Dame: "j-multiplicity and depth of associated graded rings".
• Saturday, 11:20--12:15, room 310 LeConte, Jinjia Li, University of Louisville: "On socle of R modulo Frobenius powers".
  
  
• Lunch
  
  
• Saturday, 2:45-- 3:40 PM, room 310 LeConte, Sonja Mapes, Duke University: "Minimal free resolutions of rigid monomial ideals".
• Saturday, 3:55-- 4:50 PM, room 310 LeConte, Liana Sega, University of Missouri at Kansas City: "Quasi-complete intersection ideals".
• Saturday, 5:05--6:00 PM, room 310 LeConte, Janet Striuli, Fairfield University: "Uniform bounds of Artin-Rees type for resolutions over Generalized Cohen-Macaulay rings".
  
  
• Party
  
  No formal sessions are scheduled for Sunday.

Organizers

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

Other participants include:

• Brett Barwick (University of South Carolina)
• Florian Enescu (Georgia State University)
• Earl Hampton, III (University of South Carolina)
• Sabine El Khoury (American University of Beirut)
• Rigoberto Florez (University of South Carolina -- Sumter)
• Alina Iacob (Georgia Southern University)
• Sara Malec (Georgia State University)
• Anton Preslicka (Georgia State University)
• Bernd Ulrich (Purdue University)
• Yongwei Yao (Georgia State University)

Abstracts

• Paul S. Aspinwall, Duke University: "D-Branes, Toric Geometry and Matrix Factorizations". Abstract: I will review the basic picture of how string theory on a Calabi-Yau threefold is associated to the derived category of coherent sheaves. For complete intersections in toric varieties this is manifested in terms of categories associated to matrix factorizations. This reduces some string theory problems to computations in commutative algebra.
  
  
• David Cox, Amherst College: "A Case Study in Bigraded Commutative Algebra". Abstract: This lecture will illustrate some of the challenges of bigraded commutative algebra by describing the free resolution of the ideal generated by three bihomogeneous polynomials of bidegree (2,1) in variables x,y,z,w. After making a mild but natural geometric assumption on the three polynomials, we will see that there are two possible shapes for the free resolution, each with its own Hilbert function. A variety of tools will be used, including sheaf cohomology, Segre embeddings and mapping cones.
  
  
• Jinjia Li, University of Louisville: "On socle of R modulo Frobenius powers". Abstract: Let (R,m,k) be a d-dimensional local ring in positive characteristic p and I an m-primary ideal. It was proved by C. A. Yackel that there exists a constant C such that the length of socle of R modulo the qth Frobenius power of I, is bounded above by Cq^{d-2} when d>1. We examine some special cases where such a length can be computed concretely. We also investigate relations between this length function with some other interesting length functions. Finally, we raise some questions related to this function base on some Macaulay 2 experiments.
  
  
• Ezra Miller, Duke University: "How primary decomposition in commutative monoids is wrong". Abstract: Copying definitions from commutative rings to commutative monoids results in a seemingly natural notion of primary decomposition for monoid congruences ("kernels" of monoid morphisms), but uncomfortable phenomena ensue. For example, prime congruences need not be irreducible. Furthermore, although primary decomposition for monoid congruences ought to be very similar to the theory for binomial ideals in monoid algebras, the deficiencies on the monoid side of the story make the existing theories irritatingly different. I will talk about how insights from the binomial side provide instructions for fixing the notion of primary decomposition in monoids, and why this subsequently sheds light back on binomial primary decomposition. This is joint work with Thomas Kahle.
  
  
• Sonja Mapes, Duke University: "Minimal free resolutions of rigid monomial ideals". Abstract: It is known that for a generic monomial ideal the Scarf complex as constructed by Bayer, Peeva, and Sturmfels is a minimal free resolution. In this talk I will define what it means for a monomial ideal to be rigid and discuss in what sense this generalizes the notion of being generic. Additionally I will give a description of the minimal free resolutions of a subclass of rigid monomial ideals. The results in this talk are joint work with Tim Clark.
  
  
• Claudia Polini, University of Notre Dame: "j-multiplicity and depth of associated graded rings". Abstract: We define the minimal j-multiplicity and almost minimal j-multiplicity of an arbitrary R-ideal. For any ideal with minimal j-multiplicity or almost minimal j-multiplicity in a Cohen-Macaulay ring R, we prove that under some residual assumptions, the associated graded ring is Cohen-Macaulay or almost Cohen-Macaulay, respectively. Our work generalizes the results for minimal multiplicity and almost minimal multiplicity obtained by Sally, Rossi, Valla, Wang, Huckaba, Elias, Corso, Polini, and Vaz Pinto.
  
  
• Liana Sega, University of Missouri at Kansas City: "Quasi-complete intersection ideals". Abstract: I will introduce and discuss a class of ideals which strictly contains the class of complete intersection ideals, while sharing many of their properties. This is joint work with L. Avramov and I. Henriques.
  
  
• Janet Striuli, Fairfield University: "Uniform bounds of Artin-Rees type for resolutions over Generalized Cohen-Macaulay rings". Abstract. Let $(F,\partial)$ be a free resolution of a finitely generated R-module $M$, where $R$ is a local noetherian ring. Eisenbud and Huneke ask whether there is an integer $h$ such that I^nF_i \cap Im(\partial_i) \subseteq I^{n-h}Im(\partial_i)$ for all $n\geq h$ and for all $i\geq 0$. The answer has been known to be positive for Cohen-Macaulay rings, rings of dimension at most $2$, and for module that are free in the punctured spectrum over any local noetherian ring. In this talk I will show that the question has a positive answer over rings that are generalized Cohen-Macaulay.

Previous meetings

• September 2010, Atlanta.
• March 2010, Atlanta.
• January, 2010, Orlando.
• Spring 2009, Columbia.
• Spring 2009, Atlanta.
• Fall 2008, Columbia.
• 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.