Spring 2015

- Date:
**1/23/15****Corey Jones, Vanderbilt University**- Title:
**Representations and Universal Norm for the Tube Algebra of Rigid C*-tensor categories** - Abstract: In the finite depth case, the tube algebra of a rigid C*-tensor category is finite dimensional, and its representation category is known to be equivalent to the Drinfeld center of the category. In the infinite depth case, the tube algebra is infinite dimensional, but the category of Hilbert space representations of the tube algebra retains the structure of a braided weakly rigid monoidal category. In analogy with discrete groups, we show that one can define a universal norm on the tube algebra and that Hilbert space representations of the tube algebra are given by representations of the C* closure of the tube algebra with respect to this norm. Recently, Popa and Vaes defined a universal norm on the fusion algebra of a rigid C*-tensor category, allowing them to define various approximation properties (Amenability, Property T, Haageruup) for rigid C*-tensor categories which they show agree with previously existing definitions from the subfactor case. The fusion algebra is naturally identified as a sub-algebra of the tube algebra, and we will show that the restriction of our norm to this subalgebra agrees with the universal norm defined by Popa and Vaes. This is joint work with Shamindra Ghosh.

- Date:
**1/30/15****Corey Jones, Vanderbilt University**- Title:
**Representations and Universal Norm for the Tube Algebra of Rigid C*-tensor categories, II** - Abstract: In the finite depth case, the tube algebra of a rigid C*-tensor category is finite dimensional, and its representation category is known to be equivalent to the Drinfeld center of the category. In the infinite depth case, the tube algebra is infinite dimensional, but the category of Hilbert space representations of the tube algebra retains the structure of a braided weakly rigid monoidal category. In analogy with discrete groups, we show that one can define a universal norm on the tube algebra and that Hilbert space representations of the tube algebra are given by representations of the C* closure of the tube algebra with respect to this norm. Recently, Popa and Vaes defined a universal norm on the fusion algebra of a rigid C*-tensor category, allowing them to define various approximation properties (Amenability, Property T, Haageruup) for rigid C*-tensor categories which they show agree with previously existing definitions from the subfactor case. The fusion algebra is naturally identified as a sub-algebra of the tube algebra, and we will show that the restriction of our norm to this subalgebra agrees with the universal norm defined by Popa and Vaes. This is joint work with Shamindra Ghosh.

- Date:
**2/13/15****Ben Hayes, Vanderbilt University**- Title:
**Polish Models and Sofic Entropy** - Abstract: Suppose G is a countable discrete group with a pmp action on a standard probability space (X,m). A topological model for the action is an action of G on a metrizable, separable topological space X' preserving a Borel probability measure m' and such that pmp action of G on (X',m') is isomorphic to the action of G on (X,m). We call the model compact (or Polish) if X' is compact (or Polish). In 2009, Lewis Bowen extended entropy for pmp actions of an amenable group to the much larger class of sofic groups, provided the action has a finite generating partition. David Kerr and Hanfeng Li removed the assumption of a finite generating partition shortly after, and moreover described how one can compute the entropy in terms of a compact model. It turns out that compact models always exist so this may be considered a satisfactory result. Nevertheless, there are examples where it easy to describe a Polish model and not so easy to describe a compact model, for example for Gaussian actions. Because of this, we describe how one computes the entropy in terms of a given Polish model. Polish models also turn out to be a natural way to handle generating families of functions which are unbounded. Using our Polish model formalism we are able to deduce properties of the Koopman representation of an action from positive entropy assumptions. For example, we are able to show that compact actions must have entropy at most zero. We can further show that entropy decreases under compact extensions. We will discuss the techniques which are mostly "linear" and rely on representations of C*-algebras. Time permitting, we will explain how our techniques can be used to recover a Theorem due to Voiculescu stating that if a von Neumann algebra has microstates free entropy dimension >1 with respect to some set of generators, then this algegra has no Cartan subalgebra.

- Date:
**2/20/15****Arnaud Brothier, Vanderbilt University**- Title:
**Approximation properties for subfactors**

- Date:
**3/6/15****No Meeting, Spring Break.**

- Date:
**3/13/15****Thomas Sinclair, Indiana University**- Title:
**On a result of Junge and Pisier: a model theoretic perspective** - Abstract: A fundamental result of Junge and Pisier states that, for n at least 3, the set of n-dimensional operator spaces is not separable in the strong topology. In particular, there cannot be a universal separable operator space, refuting Kirchberg's Conjecture (A) that there is a unique C* norm on the algebraic tensor product of B(H) with itself. In this talk, I will discuss some model-theoretic and descriptive-set theoretic issues concerning quantitive, finitary characterizations of various properties of operator spaces and show how they connect to the aforementioned result of Junge and Pisier and the foundational work of Kirchberg. No knowledge of model theory will be assumed. This is based on joint work with Isaac Goldbring.

- Date:
**Thursday, 3/19/15**, Departmental Colloquium, 4:10-5:00 in SC 5211.**David Kerr, Texas A&M University**

- Date:
**3/20/15****David Kerr, Texas A&M University**- Title:
**Dynamics and dimension** - Abstract: The notion of dimension within the context of dynamics has become an important tool both in the classification theory of nuclear C*-algebras and in the study of the relation between K-theory and asymptotic geometry. I will present a combinatorial perspective on this dimension theory and speculate about its applications in ergodic theory and operator algebras.

- Date:
**Thursday, 4/2/15, 3:10-4:00****Rares Rasdeaconu, Vanderbilt University**- Title:
**Counting real rational curves on K3 surfaces** - Abstract: Real enumerative invariants of real algebraic manifolds are derived from counting curves with suitable signs. Based on a joint work with V. Kharlamov, I will discuss the case of counting real rational curves on simply connected complex projective surfaces with zero first Chern class (K3 surfaces), equipped with an anti-holomorphic involution. An adaptation to the real setting of a formula due to Yau and Zaslow will be presented. The proof passes through results of independent interest: a new insight into the signed counting, and a formula computing the Euler characteristic of the real Hilbert scheme of points on a K3 surface, the real version of a result due to Gottsche.

Time permitting, I will speculate on refining our signed counting of curves to a weighted counting, with certain polynomial weights. Recent results of Maulik, Oblomkov and Shende relate these weights with the HOMFLY polynomials of the links of the singularities of the curve.

- Date:
**4/3/15****Scott A. Atkinson, University of Virginia**- Title:
**Convex Sets Associated to C*-Algebras** - Abstract: Given a separable C*-algebra A, we can associate to it an invariant given by the weak approximate unitary equivalence classes of homomorphisms from A to a chosen McDuff II_1-factor M. We will see that this object takes the form of a convex set. This invariant is closely related (and sometimes affinely homeomorphic) to the trace space of A, but it is in general a finer invariant than the trace space. In addition to discussing its basic properties, we will also discuss some interesting questions arising from this new invariant. This is an ongoing project based off of a 2011 paper by Nate Brown.

- Date:
**4/10/15****Brandon Seward, University of Michigan**- Title:
**Krieger's finite generator theorem for ergodic actions of countable groups.** - Abstract: For an ergodic probability-measure-preserving action of a countable group, we define the Rokhlin entropy to be the infimum of the Shannon entropies of countable generating partitions. It is known that for free ergodic actions of amenable groups this notion coincides with classical Kolmogorov--Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Under this analogy we prove that Krieger's finite generator theorem holds for all countably infinite groups. Specifically, if the Rokhlin entropy is less than log(k) then there exists a generating partition consisting of k sets. Connections to sofic entropy, Bernoulli shifts, and Gottschalk's surjunctivity conjecture will also be discussed.

- Date:
**4/24/15****Chenxu Wen, Vanderbilt University**- Title:
**Injectivity and disjointness for the radial masa** - Abstract: The study of the inclusion of amenable subalgebras inside II_1 factors leads to many important notions in the theory such as regularity, singularity, solidity, etc.

Popa showed the first concrete examples of maximal amenable subalgebras inside a II_1 factor. Subsequent work on maximal amenable subalgebras has mostly revolved around a property due to Popa, called the asymptotic orthogonal property (AOP). Only recently, a new approach via the study of centralizers was developed by Boutonnet and Carderi.

We show a stronger version of AOP which implies the "disjointness property" that any distinct maximal amenable subalgebra cannot have diffuse intersection with the radial masa. This confirms partially a conjecture of Jesse Peterson.

- Date:
**5/1/15 - 5/7/15****The Thirteenth Annual NCGOA Spring Institute, at Vanderbilt University**

- End of Spring Semester.