Events for 04/19/2024 from all calendars

Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Matthew Faust, TAMU

Title: Irreducibility of the Bloch Variety

Abstract: Given a ZZ^d-periodic graph G, a discrete periodic operator, a periodic potential together with a weighted graph laplacian, acts on functions on the vertices of G. Floquet theory allows us to study the spectrum through a finite matrix with Laurent polynomials entries. The zero set of the corresponding characteristic polynomial is called the Bloch variety. We will focus our attention on the irreducibility of this variety, which provides insight into quantum ergodicity. In particular we study how irreducibility of the Bloch variety is affected as one varies the period of the potential.

Noncommutative Geometry Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 306

Speaker: Henri Moscovici, Ohio State University

Title: Prolate wave operators beyond the archimedean place

Abstract: As we have seen, the negative spectrum of the prolate spheroidal wave operator tries to simulate the nontrivial zeros of Zeta. To improve the degree of approximation one needs to go beyond the archimedean place and involve the primes, i.e. the finite places of the adeles over Q. This talk, based on joint work with A. Connes and C. Consani, discusses a process for obtaining such an extension in the semilocal adelic framework, which involves finitely many primes at a time.

Geometry Seminar

Time: 4:00PM - 5:00PM

Location: BLOC 302

Speaker: Alex Cohen, MIT

Title: An optimal inverse theorem for the rank of tensors

Abstract: A polynomial f(x_1, … x_n) over a finite field has a large bias if its output distribution is far from uniform. It has rank `r' if we can write `f' as a function of polynomials g_1, …, g_r that each have smaller degree. Bias measures the amount of randomness, and rank measures the amount of structure. It is known that if `f' has small rank, it must have large bias. Green and Tao proved an inverse theorem stating that if `f' is significantly biased, its rank is bounded. Their bound was qualitative, however, and several authors gave quantitative improvements. We prove an optimal inverse theorem: the rank and the log of the bias are equivalent up to linear factors (over large enough fields). Our techniques are very different from the usual methods in this area, we rely on algebraic geometry rather than additive combinatorics. This is joint work with Guy Moshkovitz.