# Program and Abstracts

All talks will take place in Jepson Hall Room 109.

8:15 – 9:00

Coffee, tea, and light breakfast in Jepson Hall outside Room 108.

9:00 – 9:50

Marius Junge, University of Illinois – Urbana Champaign

Entropy inequalities for matrices

Abstract: Motivated by quantum information theory, we prove so called Log-Sobolev inequalites for matrix valued functions. These inequalities are related to the famous Talagrand inequality and the work by Otto and Villani on optimal transport. We will discuss a transference principle allowing us derive results on noncommutative ergodic systems from well-known kernel estimates, due to Saloff-Coste and his coauthors, on Lie groups and finite groups.

10:00 – 10:30

Scott Atkinson, Vanderbilt University

A selective version of Lin’s Theorem

Abstract: Lin’s Theorem says that a pair of nearly commuting self-adjoint matrices is near a pair of commuting self-adjoint matrices where “near” is with respect to the operator norm, and the estimates are independent of dimension. This resolved a long-standing question posed by Halmos. Such a result is subject to many generalizations and variations—we will discuss some that fail and some that hold. To add to the list of affirmative generalizations and variations, we will show that in a finite factor von Neumann algebra, an n-tuple of self-adjoints for which pairs in a certain selection nearly commute is near an n-tuple of self-adjoints for which the pairs from the corresponding selection truly commute. In this case “near” is with respect to the Hilbert-Schmidt norm coming from the trace. In fact a slightly more general statement holds. To prove this theorem we obtain results on the tracial stability of certain graph products of abelian C*-algebras. Time permitting, we will show how such results apply further to characterize the amenable traces of certain right-angled Artin groups.

10:40 – 11:30

J. Alejandro Chavez-Dominguez

Frame potential for finite-dimensional Banach spaces

Abstract: Frames for Hilbert spaces, as overcomplete versions of bases, are quite useful in applications because they provide decompositions that are more robust. Those frames that consist of vectors of norm one and are additionally tight have even more computational advantages, e.g. they provide fast convergence for said decompositions.

Benedetto and Fickus have defined a frame potential, a numerical quantity that can be assigned to a collection of finitely many vectors in a Hilbert space, which characterizes unit norm tight frames: a sequence of k norm-one vectors in an n-dimensional Hilbert space (where $k \ge n$) has frame potential at least \$latex  k^2/n\$, with equality if and only if the sequence forms a tight frame (and there always exist frames achieving this bound).

The main result of this paper is a generalization of the aforementioned result to the context of finite-dimensional Banach spaces. We use the 2-summing norm to define a frame potential for a sequence of \$k\$ norm-one vectors in an n-dimensional Banach space (where $k \ge n$), which generalizes the Hilbert-space notion. This generalized potential is also bounded below by $k^2/n$, with equality if and only if the sequence is a tight frame. The arguments rely on the geometry of the space of 2-summing maps from a finite-dimensional Banach space to itself. For a wide class of spaces, in particular complex n-dimensional Banach spaces with a 1-unconditional basis, we show that the equality case does occur.

This is joint work with Dan Freeman and Keri Kornelson.

11:40 – 2:00

Lunch at the University of Richmond dining hall.

2:00 – 2:50

Dominique Guillot, University of Delaware

Matrix positivity preserver problems

Abstract: Determining which transformations map the cone of positive semidefinite matrices into itself is a classical problem that continues to attract attention. I will give a historical account of matrix positivity and of operations that preserve it, and will discuss recent characterizations of entrywise positivity preservers. I will also discuss several applications of positivity preservers in geometry, combinatorics, and statistics.

3:00 – 3:30

Ray Cheng, Old Dominion University

An extension of the Pythagorean Theorem and some applications

Abstract: We will begin by wondering what the Pythagorean Theorem might be for a normed linear space that does not have an inner product. There will need to be a notion of orthogonality, and we will adopt the one developed by Birkhoff and James. Our exploration will lead us to define the Weak Parallelogram Laws. We’ll see that Weak Para spaces enjoy an associated Pythagorean Theorem, which takes the form of a family of inequalities. We will then look at some applications of these inequalities to a variety of areas – bounds for the roots of polynomials, prediction theory of heavy-tailed processes, and function theory on the open unit disk.

3:40 – 4:30

Christina Brech, Universidade de Sao Paulo

Basic sequences in nonseparable Banach spaces

Abstract: Schauder bases play a fundamental role in the theory of Banach spaces. After Enflo’s example of a Banach space with no Schauder basis, the existence of basic sequences of special types gained more importance, the most famous result in the field being the solution given by T. Gowers and B. Maurey in the nineties to the “unconditional basic sequence problem”. A lot of work has been done during the last decade, moving from the separable to the nonseparable setting. We shall present some of these results, including our recent joint work with J. Lopez-Abad and S. Todorcevic on subsymmetric basic sequences.

6:30

Dinner. Location TBD.