Events for 12/05/2016 from all calendars

Probability Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Steven Heilman, UCLA

Title: Noncommutative Majorization Principles and Grothendieck's Inequality

Abstract: The seminal invariance principle of Mossel-O'Donnell-Oleszkiewicz implies the following. Suppose we have a multilinear polynomial Q, all of whose partial derivatives are small. Then the distribution of Q on i.i.d. uniform {-1,1} inputs is close to the distribution of Q on i.i.d. standard Gaussian inputs. The case that Q is a linear function recovers the Berry-Esseen Central Limit Theorem. In this way, the invariance principle is a nonlinear version of the Central Limit Theorem. We prove the following version of one of the two inequalities of the invariance principle, which we call a majorization principle. Suppose we have a multilinear polynomial Q with matrix coefficients, all of whose partial derivatives are small. Then, for any even K>1, the Kth moment of Q on i.i.d. uniform {-1,1} inputs is larger than the Kth moment of Q on (carefully chosen) random matrix inputs, minus a small number. The exact statement must be phrased carefully in order to avoid being false. Time permitting, we discuss applications of this result to anti-concentration, and to computational hardness for the noncommutative Grothendieck inequality. (joint with Thomas Vidick) https://arxiv.org/abs/1603.05620

Geometry Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 220

Speaker: Roberto Barrera, TAMU

Title: A finiteness result for local cohomology modules of Stanley-Reisner rings

Abstract: While local cohomology modules of a ring may not be finitely generated, they still may possess other finiteness properties. In 1990, Craig Huneke asked if the number of associated prime ideals of a local cohomology module is finite. Huneke's question has since been answered in the affirmative for various families of rings by using different methods in characteristic 0 and in positive characteristic. In 2010, Gennady Lyubeznik gave a characteristic free proof that the local cohomology modules of the polynomial ring have finitely many associated prime ideals. In this talk, I will give the necessary background from D-module theory and local cohomology and then answer Huneke's question for local cohomology modules of Stanley-Reisner rings using techniques inspired by Lyubeznik. This is joint work with Jeffrey Madsen and Ashley Wheeler.