SCIENTIFIC PROGRAMS AND ACTIVITIES

Nate Brown (Penn State University)
On the Toms-Winter Conjecture

I'll discuss the history and current state of this conjecture, emphasizing analogies with Connes's work on injective factors. There have been tremendous advances in the last year, this conjecture could fall soon. I'll discuss what remains to be done.

Marius Dadarlat (Purdue University)
K-theoretic quasidiagonality and almost flat bundles

We will discuss connections between quasidiagonality of group C*-algebras C*(G) and almost flat bundles on the classifying space of G.
The talk is based on the paper "Group quasi-representations and almost flat bundles" and on recent joint work with Jose Carrion.

Caleb Eckhardt (Miami University)
Nuclear dimension of nilpotent group C*-algebras

We discuss recent work with Paul McKenney showing that C*(G) has finite nuclear dimension when G is a finitely generated nilpotent group. We'll explain what all the terms of the last sentence mean as well as implications of the result for Elliott's classification program and representation theory of nilpotent groups.

Isaac Goldbring (University of Illinois at Chicago)
Existentially closed C* algebras and operator systems

A C* algebra B is said to be existentially closed if, roughly speaking, every set of equations involving the norms of noncommutative *polynomial with parameters in B that has a solution in B(H) already has approximate solutions in B. There is an analogous notion of existentially closed operator system, where now one looks at equations involving matrix norms of linear *polynomials. In this talk, we investigate the relationship between existential closedness and other well known properties in operator algebras, e.g. nuclearity, WEP, and LLP. This is joint work with Thomas Sinclair.

Hanfeng Li (University ofBuffalo)
Ergodicity of principal algebraic actions

For a countable group G and an element f of the integral group ring ZG of G, one may consider the G-action on the Pontryagin dual of ZG/ZGf. I will discuss when such a principal algebraic action is ergodic with respect to the Haar measure. This is joint work with Jesse Peterson and Klaus Schmidt.

Leonel Robert (University of Louisiana, Lafayette)
Commutators and regularity of C*-algebras

An element of a C*-algebra belongs to the kernel of all bounded traces if and only if it is a limit of sums of commutators (a consequence of Hahn-Banach's theorem). For many C*-algebras, more can be said: the number of terms in the sums of commutators can be kept fixed; even better: no approximation is needed. This phenomenon has been explored by several authors (Fack, Thomsen, Pop, Marcoux, Ng, et. al.). On the other hand, examples exist of simple nuclear C*-algebras where no approximation by a fixed number of commutators is possible. In fact, regularity properties such as nuclear dimension, Z-stability, and pureness, play a significat role in this question. I will discuss these and other results, some obtained in recent work with Ping Wong Ng.

Frederic Latremoliere (University of Denver)
The Gromov-Hausdorff propinquity

We survey the notion of a quantum metric space, including the locally compact notion we recently introduced, and then a few particular generalizations of the Gromov-Hausdorff distance to the noncommutative context, which we introduced under the name of the Gromov-Hausdorff propinquity. These new metrics on the class of quantum compact metric spaces are devised to provide a natural framework for the study of noncommutative metric geometry: they are defined within the category of C*-algebras, make *-isomorphisms a necessary condition for distance zero, and some of them are complete. These metrics were in particular devised to address some issues raised by the recent research in noncommutative metric geometry regarding the continuity of certain C*-algebraic structures. We also will address how these new forms of convergences for quantum compact metric spaces can be adapted to our notion of quantum locally compact metric spaces.

Farzad Fathizadeh (University of Western Ontario)
Scalar curvature and Einstein-Hilbert action for noncommutative tori

Noncommutative geometry has many overlaps with operator algebras and spectral geometry. In this talk, I will present a review of some recent developments on local differential geometry of noncommutative tori equipped with curved metrics, which were stimulated by a seminal paper of A. Connes and P. Tretkoff on the Gauss-Bonnet theorem for the noncommutative two-torus. In joint works with M. Khalkhali, we extend this result to general translation-invariant conformal structures on the noncommutative two-torus and compute the scalar curvature, where our final formula matches with the independent computation by A. Connes and H. Moscovici. For the noncommutative four-torus, after computing the scalar curvature, we consider the analog of the Einstein-Hilbert action and show that flat metrics are extremums of this action within a conformal class. We also prove the analog of Weyl's law and Connes' trace theorem for these spaces.

Serban Belinschi (Queen's University)
The linearization trick

The idea of what we call here "the linearization trick" is the following: given a non-linear object (polynomial or rational function - not necessarily commutative) with "simple" (for ex. complex) coefficients, find a linear polynomial with matrix coefficients such that the relevant properties of the initial non-linear object can be easily deduced from properties of the linear polynomial. This approach is by no means new, and it can be found under one guise or another in several subfields of mathematics. In our talk, we will be concerned primarily with versions of the linearization trick used in random matrix theory and free probability.

We will start our talk by presenting a very brief and selective history of applications of the linearization trick in free probability. We will then show how to use linearization for the study of the distributions of symmetric polynomials in two free selfadjoint random variables. We will conclude by presenting a more precise (although not yet fully satisfactory) characterization of atomic parts of distributions of such polynomials in terms of the atoms of the two variables involved. Most of this is based on both completed and ongoing work with Mai, Sniady and Speicher.

Dima Shlyakhtenko (University of California, Los Angeles)
Polynomials in Free Variables

We discuss the following question. Let X,Y be two non-commutative random variables. Assume some amount of freeness between X and Y. For p a non-commutative self-adjoint polynomial, what can be said about the law of p(X,Y)? We discuss our previous joint work with P. Skoufranis which proceeds under the assumption that X and Y are free and mirrors the proof of the Atiyah Conjecture for free groups, a result of Mai, Speicher and Weber which requires finiteness of free Fisher information and finally our improvement of their result which assumes vanishing of a certain homology group.

José R. Carrión (Penn State University)
Local embeddability of groups and quasidiagonality

We discuss recent developments concerning the quasidiagonality of group $C^*$-algebras and in particular how it relates to the group-theoretic notion of local embeddability.

Zhengwei Liu (Vanderbilt University)
Noncommutative uncertainty principles

In this talk, I will introduce the uncertainty principle for subfactors and discuss some recent work joint with Chunlan Jiang and Jinsong Wu. http://arxiv.org/abs/1408.1165

The uncertainty principle is a fundamental phenomenon related to a pair of objects dual to each other, such as the Heisenberg uncertainty principle for position and momentum of a particle. Subfactor theory naturally provides such a dual pair of von-Neumann algebras, such as the Pontryagin duality for locally compact abelian groups. Subfactor planar algebras were introduced by Jones as a Categorification of finite index subfactors. New concepts and tools were discovered, in particular a categorification of the Fourier transform. The classical uncertainty principles deal with functions on abelian groups. We discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the Hausdorff-Young inequality, Young's inequality, the Hirschman-Beckner uncertainty principle, the Donoho-Stark uncertainty principle. We characterize the minimizers of the uncertainty principles. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's ?-lattices, modular tensor categories etc. Once we have a connection in the sense of Ocneanu, the uncertainty principles hold.

Martino Lupini (York University)
The classification problem for separable operator systems

I will present an overview on the complexity of classifying separable operator systems up to complete order isomorphism. While it is hopeless to classify arbitrary separable operator systems, finitely generated operator systems admit a satisfactory smooth classification. For example operator systems generated by a single unitary operator are classified by the spectrum of such operator up to a rigid motion of the circle. This is joint work with Martin Argerami, Samuel Coskey, Matthew Kennedy, Mehrdad Kalantar, and Marcin Sabok.

Andre Kornell (UC Berkeley)
V*-algebras

Working in a model of set theory in which every set of real numbers is Lebesgue measurable, I will define the category of V*-algebras. Every separable unital C*-algebra has an enveloping V*-algebra, which may be identified with the space of strongly affine functions on the state space. If the C*-algebra is type I, then its enveloping V*-algebra is a direct sum of type I factors, with one summand for each irreducible representation.

Maria Grazia Viola (Lakehead University)
Coauthors: N. Cristopher Phillips
Classification of L^p AF algebras

We first define the notion of spatial L^p algebras and scaled ordered K_0 group for L^p AF algebras. Our main result is a complete classification of spatial L^p AF algebras. We show that two spatial L^p AF algebras are isomorphic if and only if their scaled ordered K_0 groups are isomorphic. Moreover, we show that any countable Riesz group can be realized as the scaled ordered K_0 group of a spatial L^p AF algebras. Therefore, the classification given by G. Elliott for AF C*-algebras also holds for spatial L^p AF algebras. Lastly, we discuss incompressibility and p-incompressibility for spatial L^p AF algebras.

Scott Atkinson (University of Virginia)
Convex Sets Associated to C*-Algebras

Given a separable C*-algebra A, we can associate to A an invariant given by a family of convex separable metric spaces. This family is closely related to the trace space of A, and we expect this invariant to be finer than the trace space invariant. This is an ongoing project based off of a 2011 paper by Nate Brown.

Yanli Song (University of Toronto)
Localization of K-Homology Fundamental Class

Let M be a Riemannian manifold with a compact Lie group action. Using the de Rham differential operator, Kasparov defined a distinguished K-homology fundamental class. We extend the fundamental class to the K-homology of crossed product of C*-algebras. Moreover, by introducing a perturbed fundamental class, we obtain a localization formula. This provides a K-homological approach to transversally elliptic operators.