Thu, 04 Aug 2022
15:00
S2.27

K-theoretic classification of inductive limit actions of fusion categories on AF-algebras

Roberto Hernandez Palomares
(Texas A&M University)
Abstract

I will introduce a K-theoretic complete invariant of inductive limits of finite dimensional actions of fusion categories on unital AF-algebras. This framework encompasses all such actions by finite groups on AF-algebras. Our classification result essentially follows from applying Elliott's Intertwining Argument adapted to this equivariant context, combined with tensor categorical techniques.

Our invariant roughly consists of a finite list of pre-ordered abelian groups and positive homomorphisms, which can be computed in principle. Under certain conditions this can be done in full detail. For example, using our classification theorem, we can show torsion-free fusion categories admit a unique AF-action on certain AF-algebras.

Connecting with subfactors, inspired by Popa’s classification of finite-depth hyperfinite subfactors by their standard invariant, we study unital inclusions of AF-algebras with trivial centers, as natural analogues of hyperfinite II_1 subfactors. We introduce the notion of strongly AF-inclusions and an Extended Standard Invariant, which characterizes them up to equivalence.

Fri, 28 Jan 2022

16:00 - 17:00
Virtual

Applications of subfactor and categorical techniques to C*-algebras

Roberto Hernandez Palomares
(Texas A&M University)
Abstract

Q-systems were introduced by Longo to describe the canonical endomorphism of a finite Jones-index inclusion of infinite von Neumann factors. From our viewpoint, a Q-system is a unitary  version of a Frobenius algebra object in a tensor category or a C* 2-category. Following work of Douglass-Reutter, a Q-system is also a unitary version of a higher idempotent, and we will describe a higher unitary idempotent completion for C* 2-categories called Q-system completion. 


We will focus on the C* 2-category C*Alg with objects unital C*-algebras, 1-morphisms right Hilbert C*-correspondences, and 2-morphisms adjointable intertwiners. By adapting a subfactor reconstruction technique called realization, and using the graphical calculus available for C* 2-categories, we will show that C*Alg is Q-system complete.

This result allows for the straightforward adaptation of subfactor results to C*-algebras, characterizing finite Watatani-index extensions of unital C*-algebras equipped with a faithful conditional expectation in terms of the Q-systems in C*Alg. Q-system completion can also be used to induce new symmetries of C*-algebras from old. 

 

This is joint work with Quan Chen, Corey Jones and Dave Penneys (arXiv: 2105.12010).

Fri, 27 Apr 2018
12:00
L4

Is dispersion a stabilizing or destabilizing mechanism? Landau-damping induced by fast background flows

Edriss Titi
(Texas A&M University)
Abstract

In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will  also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some new results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit ``Landau-damping" mechanism due to large spatial average in the initial data.

Fri, 18 May 2018
12:00
N3.12

Which neural codes are convex?

Anne Shiu
(Texas A&M University)
Abstract

This talk focuses on algebraic and combinatorial-topological problems motivated by neuroscience. Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus,where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets?

This talk describes how to use tools from combinatorics and commutative algebra to uncover a variety of signatures of convex and non-convex codes.

This talk is based on joint works with Aaron Chen and Florian Frick, and with Carina Curto, Elizabeth Gross, Jack Jeffries, Katie Morrison, Mohamed Omar, Zvi Rosen, and Nora Youngs.

Thu, 22 Mar 2018

14:00 - 15:00
C1

The Usefulness of a Modified Restricted Isometry Property

Simon Foucart
(Texas A&M University)
Abstract

The restricted isometry property is arguably the most prominent tool in the theory of compressive sensing. In its classical version, it features l_2 norms as inner and outer norms. The modified version considered in this talk features the l_1 norm as the inner norm, while the outer norm depends a priori on the distribution of the random entries populating the measurement matrix.  The modified version holds for a wider class of random matrices and still accounts for the success of sparse recovery via basis pursuit and via iterative hard thresholding. In the special case of Gaussian matrices, the outer norm actually reduces to an l_2 norm. This fact allows one to retrieve results from the theory of one-bit compressive sensing in a very simple way. Extensions to one-bit matrix recovery are then straightforward.
 

Thu, 14 May 2015

16:00 - 17:00
L6

Equidistribution of Eisenstein series

Matthew Young
(Texas A&M University)
Abstract

I will discuss some recent results on the distribution of the real-analytic Eisenstein series on thin sets, such as a geodesic segment. These investigations are related to mean values of the Riemann zeta function, and have connections to quantum chaos.

Thu, 17 Jun 2010

14:00 - 15:00
3WS SR

Towards Effective Computation with Kernels on Manifolds

Prof Joseph Ward
(Texas A&M University)
Abstract
This talk will focus on highly localized basis functions which exist for certain kernels and spaces associated with these kernels. Such kernels include certain radial basis functions (RBFs), their restrictions to spheres (SBFs), and their restrictions to more general manifolds embeddable in Rd. The first part of the talk will be of an introductory nature. It will discuss radial basis functions and their restriction to manifolds which give rise to various kernels on these manifolds. The talk will then focus on the development (for certain kernels) of highly localized Lagrange functions which serve as effective bases: i.e., bases which are stable and local. Scaled versions of these bases will then be used to establish the stability of the L2 minimization operator in Lp, 1 ≤ p ≤ ∞, thus obtaining a multivariate analogue of a result of de Boor. Since these bases are scalable with the data, they have potential uses beyond approximation including meshless methods and, more generally, computations of a multiresolution nature. The talk is primarily based on joint work with T. Hangelbroek, F. J. Narcowich and X. Sun.
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