Events for 03/11/2022 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 302

Speaker: Matthew Faust, TAMU

Title: The number of Critical Points of Discrete Periodic Operators

Abstract: The spectral gap conjecture is a well known and widely believed conjecture in mathematical physics concerning the structure of the Bloch variety (dispersion relation) of periodic operators. The Bloch variety of a discrete operator is algebraic, inviting methods from algebraic geometry to their study. Motivated by this conjecture, this talk will introduce a bound on the number of critical points of the dispersion relation for discrete periodic operators, and provide a general criterion for when this bound is achieved. We also present a class of periodic graphs for when this criteria is satisfied for Laplace-Beltrami operators. This is joint work with Frank Sottile.

Noncommutative Geometry Seminar

Time: 2:00PM - 3:00PM

Location: ZOOM

Speaker: Rufus Willett, University of Hawai’i

Title: Decomposable C*-algebras and the UCT

Abstract: A C*-algebra satisfies the UCT if it is K-theoretically the same as a commutative C*-algebra, in some sense. Whether or not all (separable) nuclear -algebras satisfy the UCT is an important open problem; in particular, it is the last remaining ingredient needed to prove the ‘best possible’ classification result for simple nuclear C*-algebras in the sense of the Elliott classification program. We introduce a notion of a ‘decomposition’ of a C*-algebra over a class of C*-algebras. Roughly, this means that there are almost central elements of the C*-algebra that cut it into two pieces from the class, with well-behaved intersection. Our main result shows that the class of nuclear C*-algebras that satisfy the UCT is closed under decomposability. Decomposability introduces a natural ‘complexity hierarchy’ on the class of -algebras: one starts with finite-dimensional C*-algebras, and the ‘complexity rank’ of a C*-algebra is roughly the number of decompositions one needs to get to down to the finite-dimensional level. There are interesting examples: we show that all UCT Kirchberg (i.e. purely infinite, separable, simple, unital, nuclear) C*-algebras have complexity rank one or two, and characterize when each of these cases occur. The UCT for all nuclear -algebras thus becomes equivalent to the statement that all Kirchberg algebras have finite complexity rank. This is based on joint work with Arturo Jaime, and with Guoliang Yu.

URL: Event link

Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Harshit Yadav, Rice University

Title: Filtered Frobenius algebras in monoidal categories

Abstract: We develop filtered-graded techniques for algebras in monoidal categories with the goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. We then produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form. These two results of independent interest are used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric finite tensor category is represented by a Frobenius algebra in it. This is joint work with Dr. Chelsea Walton (Rice University).

Geometry Seminar

Time: 4:00PM - 4:50PM

Location: BLOC 302

Speaker: Thomas Brazelton, University of Pennsylvannia

Title: An enriched degree of the Wronski

Abstract: Given $mp$ different $m$-planes in general position in $(m+p)$-dimensional space, a classical problem is to ask how many $p$-planes intersect all of them. For example when $m=p=2$, this is precisely the question of "lines meeting four lines in 3-space" after projectivizing. The Shapiro conjecture asserts that all solutions to a real Schubert problem of this type will be real in the setting where the planes are chosen to osculate a rational normal curve. In this setting, the Brouwer degree of the Wronski map provides an answer to this question, first computed by Schubert over the complex numbers and Eremenko and Gabrielov over the reals. We provide an enriched degree of the Wronski for all $m$ and $p$ even, valued in the Grothendieck--Witt ring of an arbitrary field. We further demonstrate in all parities that the local contribution of a $p$-plane is a determinantal relationship between certain Plücker coordinates of the $m$-planes it intersects.