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 Florida
Jon F. Carlson, University of Georgia
Shuhong Gao, Clemson University
Anton Leykin, Georgia Institute of Technology
Sandra Spiroff, University of Mississippi
Adela 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 ):
Friday
2-3pm (Colloquium held in 796 College of Education Bldg): Andy Kustin
3:30-4:30pm: Joe Brennan
Saturday
9:30-10:30am: Jon Carlson
11-12pm: Sandra Spiroff
Lunch break
1:30-2:30pm: Shuhong Gao
3-4pm: Anton Leykin
4:15-4:45pm: Brett Barwick
Sunday
9:30-10:30am: Josephine Yu
11-12pm: Adela Vraciu
Titles and abstracts:
Andy Kustin, University of South Carolina
The bi-graded
structure of Symmetric Algebras with applications to Rees
rings
Brett Barwick, University of South Carolina
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.
Joe Brennan, University of Central Florida
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.
Jon F. Carlson, University of Georgia
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.
Shuhong Gao, Clemson University
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.
Anton Leykin, Georgia Tech
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
Sandra Spiroff, University of Mississippi
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.
Adela Vraciu, University of South Carolina
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.
Josephine Yu, Georgia Tech
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.
List of Participants
Brett Barwick, University of South Carolina
Joe Brennan, University of Central Florida
Jon F. Carlson, University of Georgia
Florian Enescu, Georgia State University
Shuhong Gao, Clemson University
Earl Hampton, University of South Carolina
Andy Kustin, University of South Carolina
Doug Leonard , Auburn University
Alina Iacob, Georgia Southern University
Anton Leykin, Georgia Institute of Technology
Sara Malec , Georgia State University
Sandra Spiroff, University of Mississippi
Thomas 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.