Atlanta, GA

February 10-12, 2012

The Commutative Algebra Conference at Georgia State in February 2012 continues the tradition of commutative algebra meetings
started by the "Commutative Algebra in the Southeast" series. It aims to bring together experts and students in
commutative algebra and related areas to facilitate the exchange of exciting contributions and the developing of new avenues of
collaboration, with some emphasis on research from neighboring schools.
(For most recent meeting held at Georgia State see the 2010 Atlanta National Meeting ; a link
with past meetings can be found here.)

Florian
Enescu (Georgia State University) `fenescu@gsu.edu`

Yongwei Yao (Georgia State University) `yyao@gsu.edu`

Confirmed
conference speakers are:

Brett Barwick, University of South Carolina Joe Brennan, University of Central FloridaJon F. Carlson, University of GeorgiaShuhong Gao, Clemson University |
Anton Leykin, Georgia Institute of Technology Sandra Spiroff, University of MississippiAdela Vraciu, University of South Carolina Josephine Yu, Georgia Institute of Technology |

Schedule (All talks, other than the colloquium, will be held in 124 PETIT SCIENCE CENTER ):

Titles and abstracts:

The bi-graded structure of Symmetric Algebras with applications to Rees rings

Generic Hilbert-Burch Matrices for Ideals Generated by Triples of Homogeneous Forms in k[x,y]

Abstract: We consider the space of triples g = (g_1 , g_2 , g_3) of homogeneous forms in B = k[x,y] of degrees d_1, d_2, and d_3. We may identify this space with a (d_1+d_2+d_3+3)-dimensional affine space over k by identifying the triple of polynomials with a point which lists the coefficients of the polynomials. If we restrict to the space of triples which generate height 2 ideals I in B, then the minimal graded free resolution of B/I is described by the Hilbert-Burch Theorem. We will generalize some recent work of Cox-Kustin-Polini-Ulrich which describes how to construct an open cover of this space so that on each open set the coefficients of the entries in a Hilbert-Burch matrix for g may be explicitly recovered as polynomials in the coefficients of the generators.

Resolutions of almost bipartite graphs

Abstract: A graph is a degree two monomial mapping of projective spaces. This talk will consider the resolutions of the homogeneous coordinate rings of the almost bipartite graphs with particular attention to (generalized) ladders and subdivides rectangles.

Thick subcategories of the bounded derived category

Abstract: This is joint work with Srikanth Iyengar. It is all about using methods from commutative algebra to study group representations. A new proof of the classification for tensor ideal thick subcategories of the bounded derived category, and the stable category, of modular representations of a finite group is obtained. The arguments apply more generally to yield a classification of thick subcategories of the bounded derived category of an artinian complete intersection ring. One of the salient features of this work is that it takes no recourse to infinite constructions, unlike the previous proofs of these results.

A new algorithm for computing Groebner bases.

Abstract: Polynomial systems are ubiquitous in Mathematics, Sciences and Engineerings, and Gröbner basis theory is one of the most powerful tools for solving them. Buchberger introduced in 1965 the first algorithm for computing Gröbner bases and it has been implemented in most computer algebra systems (e.g. Maple, Mathematica, Magma, etc). Faugere presented two new algorithms: F4 (1999) and F5 (2002), with the latter being the fastest algorithm known in the last decade. In this talk, I shall first give a brief overview on Gr\"{o}bner bases then present a new algorithm that matches Buchberger's algorithm in simplicity yet is several times faster than F5.

Real log canonical threshold

Abstract: The log canonical threshold (lct) is a birational invariant of a complex algebraic variety that can be computed either using the resolution of singularities or through $D$-modules algorithms for Bernstein-Sato polynomials. However, the latter can't produce its real analogue, the real lct (rlct), directly. We describe several approximate numerical approaches that determine rlct as well as a possible application of rlct in statistics

Some Invariants on Complete Intersections

Abstract: We discuss some invariants for a pair of modules over a complete intersection, with special focus on the graded case. In particular, we introduce a new invariant when the ring has only isolated singularity at the irrelevant maximal ideal and show that it shares many of the same properties as Hochster's original theta invariant, defined for hypersurfaces.

On the degrees of relations on $x_1^{d_1}, \dots, x_n^{d_n}, (x_1+ \dots +x_n)^c$.

Abstract: We discuss the smallest possible degree of a relation on the elements $x_1^{d_1}, \dots, x_n^{d_n}, (x_1+ \dots +x_n)^c$ in a polynomial ring $k[x_1, \dots, x_n]$, both in characteristic zero and in positive characteristic. In positive characteristic, this is related to the question of whether the weak Lefschetz property holds for monomial complete intersections, and also to calculations of Hilbert-Kunz multiplicities.

Computing Tropical Resultants

Abstract: We fix the supports A=(A_1,...,A_k) of a list of tropical polynomials and define the tropical resultant TR(A) to be the set of choices of coefficients such that the tropical polynomials have a common solution. We prove that TR(A) is the tropicalization of the algebraic variety of solvable systems and that its dimension can be computed in polynomial time. We use tropical methods to compute the Newton polytope of the sparse resultant polynomial in the case when TR(A) is of codimension 1. We also consider the more general setting in which some of the coefficients of the polynomials are specialized to some constants. This is based on joint work with Anders Jensen.

Brett Barwick, University of South Carolina Joe Brennan, University of Central FloridaJon F. Carlson, University of GeorgiaFlorian Enescu, Georgia State University Shuhong Gao, Clemson UniversityEarl Hampton, University of South Carolina Andy Kustin, University of South CarolinaDoug Leonard , Auburn University Alina Iacob, Georgia Southern University |
Anton Leykin, Georgia Institute of Technology Sara Malec , Georgia State University Sandra Spiroff, University of MississippiThomas Polstra, Georgia State University Anton Preslicka, Georgia State University Adela Vraciu, University of South Carolina Yongwei Yao, Georgia State University Josephine Yu, Georgia Institute of Technology |

There is a conference dinner planned at 6pm on Friday at Chateau de Saigon. Please email the organizers by Thursday at noon if you would like to come.

The meeting is partially supported by the Department of Mathematics and Statistics at Georgia State University.