Spring 2012

Algebra, Geometry and Cryptology Seminar

The AGC seminar will meet on Wednesdays,  2:40 p.m. - 3:30 p.m. in MG 124,
  alternating with the Topology seminar

January  25
Speaker: Zach Teitler

Waring decompositions of monomials
A Waring decomposition of a polynomial is an expression of the polynomial as a sum of powers of linear forms, where the number of summands is minimal possible. In this talk, I will describe joint work with Weronika and Jarek Buczynski which proves that any Waring decomposition of a monomial is obtained from a complete intersection ideal, and determines the dimension of the set of Waring decompositions. (All these terms will be explained in the talk.) In addition I will describe a number of open questions, some (or all?) of which could be research projects for a student.

February 8
Meeting cancelled

March 7
Speaker: Hirotachi Abo, University of Idaho

Ideals, varieties, and vector bundles
The area of mathematics that I work in is algebraic geometry. At its most basic, algebraic geometry is the study of the common zero loci of systems of polynomial equations. Such systems arise very naturally throughout mathematics, theoretical physics, mathematical biology, statistics, computer science and engineering. These objects considered geometrically are called varieties. The first goal of this talk is to discuss one of the guiding problems in algebraic geometry called Hartshorne's conjecture, which asserts that every non-singular variety $X$ in projective $n$-space $\mathbb{P}^n$ should be a complete intersection if the codimension of $X$, i.e., $n-\dim X$, is small compared with $n$. The second goal is to explore the connection between the question whether varieties of small codimension of $\mathbb{P}^n$ are complete intersections and the question whether there are so-called indecomposable vector bundles of small rank on $\mathbb{P}^n$. If time permits, I would like to discuss how this talk is related to the talk I will give at the colloquium.
 

March 19
Speaker: Zach Teitler

Decompositions of determinantal ideals
Let X be a 2-by-3 matrix whose entries are variables. Then X has three maximal (2-by-2) minors, each obtained by deleting one of the columns of X. It is well-known that the ideal generated by all three of the maximal minors is a prime ideal; its vanishing locus is the set of rank 1 matrices. The ideal generated by a single one of the minors is also a prime ideal. What about the ideal generated by two of the minors? More generally, what is the primary decomposition of an ideal generated by a subset S of the minors of a generic m-by-n matrix? Theorem: Under suitable hypotheses, this ideal is the intersection of two prime ideals, first the ideal generated by all the maximal minors of the matrix, second the ideal generated by the maximal minors of the set of columns that appear in all the minors included in S.
As motivation, the Hilbert-Burch theorem asserts that any finite set of points in the plane is defined by a system of equations given by the maximal minors of a generic k-by-(k+1) matrix whose entries are polynomials. Taking a subset of the minors corresponds to solving a subsystem of the system of defining equations of the point set. Our decomposition result yields some situations in which some of the defining equations are redundant.
If time permits I'll mention some questions that remain open (as far as I know): deal with special matrices (symmetric, skew-symmetric, etc); deal with special choices of minors (e.g., adjacent minors); work out other properties of these ideals (e.g., free resolution).

April  25
Speaker: Jason Smith, College of Western Idaho
Title: Some Interesting Infinite Sums
Abstract: pdf