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ABSTRACTS GPOTS 2019 — Texas A&M University ABSTRACTS GPOTS 2019 | Texas A&M University Plenary talks Kelly Bickel, Bucknell University Portraits of rational inner functions This talk focuses on the structure of two-variable rational inner functions (i.e. the two variable general- izations of finite Blaschke products) with singularities on the two torus. We analyze how singular such functions can be using information pulled from their zero sets, their unimodular level sets, and some associ- ated pictures. Time permitting, we will discuss some generalizations and complications in the three-variable setting. This is joint work with Alan Sola and James Pascoe. March Boedihardjo, University of California, Los Angeles Brown-Douglas-Fillmore theorem for lp A version of the Brown{Douglas{Fillmore theorem for lp is obtained. Joint work with Chris Phillips. Sarah Browne, The Pennsylvania State University Quantitative E-theory and the Universal Coefficient Theorem In recent years, many people have been working on classifying C*-algebras and these results assume the Universal Coefficient Theorem (UCT), which we would like to try to extend to a bigger class of C*-algebras. I will give a summary of the Theorem and then talk about ongoing joint work with Nate Brown on Quantitative E-theory. This aims to create a new approach to tackling results like the UCT for new classes of C*-algebras. Our inspiration is work by Oyono-Oyono-Yu, who used a quantitative approach of K-theory to prove the K¨unnethTheorem for new classes of C*-algebras. Jos´eCarri´on, Texas Christian University Classifying ∗-homomorphisms We report on a joint project with J. Gabe, C. Schafhauser, A. Tikuisis, and S. White that develops a new approach to the classification of C∗-algebras and the homomorphisms between them. Mahya Ghandehari, University of Delaware On non-commutative weighted Fourier algebras Beurling-Fourier algebras are analogues of Beurling algebras in the non-commutative setting. These algebras for general locally compact groups were defined as the predual of certain weighted von Neumann algebras, where a weight on the dual object is defined to be a suitable unbounded operator affiliated with the group von Neumann algebra. In this talk, we present the general definition of a Beurling-Fourier algebra, and discuss how their spectra can be identified. In particular, we determine the Gelfand spectrum of Beurling- Fourier algebras for some representative examples of Lie groups, emphasizing the connection of spectra to the complexification of underlying Lie groups. This talk is based on joint work with Lee, Ludwig, Spronk, 1 and Turowska. Elizabeth Gillaspy, University of Montana Spectral triples and wavelets for higher-rank graphs Joint work with C. Farsi, A. Julien, S. Kang, and J. Packer has revealed that the wavelets for higher-rank graphs introduced by Farsi{Gillaspy{Kang{Packer in 2016 are intimately tied to the Pearson-Bellissard spectral triples for Cantor sets. To be precise, the infinite path space of a higher-rank graph is often a Cantor set. Investigating the associated Pearson{Bellissard spectral triple reveals that the FGKP wavelets describe the eigenspaces of the Laplace-Beltrami operators of the spectral triple. We also discuss the relationship between the measure underlying the construction of the wavelets, which was first introduced by an Huef et al in 2014, and certain Dixmier traces associated to the spectral triples. This talk will strive to be a friendly introduction to the main concepts mentioned above (spectral triples, Dixmier traces, higher-rank graphs, wavelets) and to showcase the neat connections between noncommuta- tive geometry and wavelets that appear in the setting of higher-rank graphs. Hanfeng Li, SUNY at Buffalo Entropy and combinatorial independence Topological entropy is a numerical invariant for group actions on compact spaces or C∗-algebras. Whether the entropy is positive or not makes a fundamental difference for the dynamical haviour. When a group G acts on a compact space X, a subset H of G is called an independence set for a finite family W of subsets of X if for any finite subset M of H and any map f from M to W , there is a point x of X with sx in f(s) for all s in M. I will discuss how positivity of entropy can be described in terms of density of independence sets, and give a few applications including the relation between positive entropy and Li-Yorke chaos. The talk is based on various joint works with David Kerr and Zheng Rong. Weihua Liu, Indiana University Bloomington The atoms of the free additive convolution of two operator-valued distributions Given two freely independent random variables X and Y , the pioneering work of Bercovici and Voiculescu on regularity questions of free convolutions showed the relation between the atoms of X; Y and the atoms of X + Y . In this work, by Voiculescus operator valued subordination functions, we study the atoms of X + Y under the assumption that X and Y are free with amalgamation over a von Neumann subalgebra B of M with respect to a trace-preserving conditional expectation. Furthermore, via the linearization trick, we provide some applications to study the kernels of selfadjoint polynomials in free variables. Joint work with Serban Belinschi and Hari Bercovici. Zhengwei Liu, Harvard University and Tsinghua University Quantum Fourier analysis We propose a program Quantum Fourier Analysis to investigate the analytic aspects of quantum symmetries and their Fourier dualities. We introduce a topological analogue of the Brascamp{Lieb inequality. We discuss some recent results and perspectives. 2 Brent Nelson, Vanderbilt University Free products of finite-dimensional von Neumann algebras and free Araki{Woods factors In the early 90s, Dykema showed any free product of finite-dimensional von Neumann algebras equipped with tracial states is isomorphic to an interpolated free group factor (possibly direct sum a finite-dimensional algebra). In joint work with Michael Hartglass, we show that this result holds in the non-tracial case when the interpolated free groups factors are replaced by Shlyakhtenko's free Araki{Woods factors (equipped with their free quasi-free states). In particular, free products of finite-dimensional von Neumann algebras are now completely classified up to state-preserving isomorphisms. Our methods use our other recent joint work, where we show free Araki{Woods factors arise as the diffuse component of certain von Neumann algebras associated to finite, connected, directed graphs equipped with edge-weightings. In this talk I will discuss these results and show how the graph methodology is used in a simple but essential case. Sorin Popa, UCLA The ubiquitous hyperfinite II1 factor The hyperfinite II1 factor R has played a central role in operator algebras ever since Murray and von Neumann introduced it, some 75 years ago. It is the unique amenable II1 factor (Connes 1976), and the smallest, as it can be embedded in multiple ways in any other II1 factor M. Many problems in operator algebras could be solved by constructing \ergodic" such embeddings R,! M. I will revisit such results and applications, and present some new results along these lines: (1) Any separable II1 factor M admits coarse embeddings of R; (2) Any separable II1 factor admits ergodic embeddings of R. Leonel Robert, University of Louisiana at Lafayette Extended Choquet simplices By a theorem of Lazar and Lindenstrauss, a metrizable Choquet simplex is a projective limit of finite dimensional simplices. This theorem has implications for C*-algebras: Any metrizable Choquet simplex arises as the space of tracial states of a simple separable unital AF C*-algebra. In the investigations on the structure of a C*-algebra, another space of traces is also of interest; namely, the lower semicontinuous traces with values in [0; 1] (the extended nonnegative real numbers). These traces form a non-cancellative topological cone. The simplest example of one such cone is [0; 1]. Problem: Characterize through axiomatic properties those cones that are expressible as projective limits of finite powers of [0; 1]. These are also the cones of lower semicontinuous [0; 1]-valued traces on AF C*-algebras. I will discuss the solution to this problem. This is joint work with Mark Moodie. Christopher Schafhauser, York University On the classification of simple nuclear C∗-algebras A conjecture of George Elliott dating back to the early 1990's asks if separable, simple, nuclear C∗-algebras are determined up to isomorphism by their K-theoretic and tracial data. Restricting to purely infinite algebras, this is the famous Kirchberg-Phillips Theorem. The stably finite setting proved to be much more subtle and has been a driving force in research in C∗-algebras over the last 30 years. A series of breakthroughs were made in 2015 through the classification results of Elliott, Gong, Lin, and Niu and the quasidiagonality theorem of Tikuisis, White, and Winter. Today, the classification conjecture is now a theorem under two additional regularity assumptions: Z-stability and the UCT. I will discuss recent joint work with Jos´eCarri´on,Jamie Gabe, Aaron Tikuisis, and Stuart White which provides a much shorter and 3 more conceptual proof of the classification theorem in the stably finite setting. Dimitri Shlyakhtenko, UCLA Representing interpolated free group factors as group factors In a joint work with Sorin Popa, we construct a one parameter family of ICC groups G(t), with the property that the group factor of G(t) is isomorphic to the interpolated free group factor L(F (t)). Moreover, the groups G(t) have fixed cost t, are strongly treeable and freely generate any treeable ergodic equivalence relation of same cost. Paul Skoufranis, York University Non-microstate bi-free entropy The notions of free entropy developed by Voiculescu have had a profound impact on free probability and operator algebras. In this talk, joint work with Ian Charlesworth will be discuss that generalizes the notion of non-microstate free entropy to the bi-free setting.
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