KUMUNU 2017 Commutative Algebra
Greg Blekherman, Georgia Institute of Technology
Do Sums of Squares Dream of Free Resolutions?
A real polynomial is called nonnegative if it takes only nonnegative values. Sums of squares of polynomials (and rational functions) are obviously nonnegative. I will briefly review the rich history of this area of real algebraic geometry. I will explain how this topic is inextricably linked to classical topics in algebraic geometry and commutative algebra, such as free resolutions. I will discuss a specific example of square-free monomial ideals, which connects this area to positive semidefinite matrix completion problems.
Alessandro De Stefani, University of Nebraska
Symbolic Powers in Mixed Characteristic
The Zariski-Nagata theorem in one of its classical versions states that, if P is a prime ideal in a polynomial ring over the complex numbers, then the n-th symbolic power of P consists of all the polynomial functions that vanish of order at least n at all points of the variety defined by P. We prove analogous results in mixed characteristic, combining the properties of differential operators and p-derivations. This talk is based on joint work with Eloísa Grifo and Jack Jeffries.
Daniel Hernández, University of Kansas
Frobenius Powers of Ideals in Regular Rings
In this talk, we introduce the notion of the generalized Frobenius powers of an ideal in a regular ring of prime characteristic. As an application of this theory, we establish a prime characteristic analog of Howald's result relating the log canonical threshold of a polynomial with that of its term ideal. Time permitting, we will also discuss applications of this theory to that of Bernstein-Sato polynomials, and other invariants of singularities. This is joint work with Emily Witt and Pedro Teixeira.
Sam Payne, Yale University
A Tropical Motivic Fubini Theorem with Applications to Donaldson-Thomas Theory
I will present a new tool for the calculation of Denef and Loeser’s motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan’s theory of motivic volumes of semi-algebraic sets. As time permits, I will discuss applications of this method, which include the solution to a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Lê Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson-Thomas theory.
Ilya Smirnov, University of Michigan
Lech's Inequality and its Improvements
In 1960 Lech found a simple inequality that relates the colength and the multiplicity of a primary ideal in a local ring. Unfortunately, his proof also shows that the inequality is never sharp if dimension is at least two. The goal of this talk is to present a stronger form of Lech's inequality and an even stronger conjecture that will make the inequality sharp. I will also discuss a Lech-type inequality on the multiplicity and the minimal number of generators of an integrally closed ideal.
Mark Walker, University of Nebraska
On Complexes of Free Modules
Let R be local ring of dimension d and F a minimal complex of free R-modules. We consider the Conjecture: If F is bounded and has non-zero finite length homology, then the total rank of F must be at least 2d. When F is the minimal resolution of a module of finite length and finite projective dimension, this conjecture is a weak form of the well-known Buchsbaum-Eisenbud-Horrocks Conjecture. A variant of this conjecture is well-known in topology as the Toral Rank Conjecture.
In this talk I will discuss recent progress, both positive and negative, toward settling this conjecture. In particular, I will discuss examples, constructed in joint work with Srikanth Iyengar, showing that the conjecture fails in general. Finally, I will describe how these counter-examples lead also to counter-examples of some related conjectures.
Dana Weston, University of Missouri
Descent and Ascent of Perinormality in Some Ring Extension Settings
An integral domain A with quotient field K is defined as perinormal if the only local rings, lying between A and K, to satisfy the going-down theorem over A are of the type Ap for p ∈ Spec(A). Let B stand for A[X] or Aˆ (if A is local). We investigate conditions under which perinormality of one of A or B implies the perinormality of the other. (joint work with Andrew McCrady)