Fall 2017

- Date:
**8/25/17 (joint with Geometry Seminar, 3:10-4:00pm, SC 1310)****Christian Fleischhack, University of Paderborn**- Title:
**Loop Quantization of Geometry** - Abstract: Loop quantum gravity aims at a mathematically rigorous quantization of general relativity. It relies on a reformulation of gravity as a gauge field theory with constraints, which is canonically quantized. The classical configuration space A is formed by connections in some principal fibre bundle and gets compactified during quantization. The resulting space can be seen as the spectrum of an appropriate C*-algebra of bounded functions on A or as a projective limit of powers of the structure group. Moreover, it exhibits a canonical measure induced by the Haar measure. In my talk, I am going to present the background and the basic structures of the theory. If time permits, I will also discuss applications to the quantization of geometric entities like area or how diffeomorphism invariance restricts the freedom in quantizing the full theory.

- Date:
**8/25/17****Robert McRae, Vanderbilt University**- Title:
**Vertex operator algebras: Theory, examples, and problems** - Abstract: Vertex operator algebras are a mathematical approach to two-dimensional chiral conformal field theory. In these talks, I will introduce the definition of vertex operator algebra with motivation from the Segal picture of conformal field theory and discuss examples coming from the Virasoro algebra, affine Lie algebras, and lattices. Where possible, I will indicate connections with more analytic approaches to conformal field theory. Further topics will include tensor structures on representations of vertex operator algebras, major open problems in the field, and, time permitting, some of my work on tensor categories of affine Lie algebra representations and vertex operator algebra extensions.

- Date:
**9/1/17****Robert McRae, Vanderbilt University**- Title:
**Vertex operator algebras: Theory, examples, and problems (continued)** - Abstract: see above.

- Date:
**9/8/17****Robert McRae, Vanderbilt University**- Title:
**Vertex operator algebras: Theory, examples, and problems (continued)** - Abstract: see above.

- Date:
**9/15/17****Bin Gui, Vanderbilt University**- Title:
**Unitarity of the modular tensor categories associated to unitary vertex operator algebras** - Abstract:
A tensor product theory for modules of a vertex operator algebra
V was developed by Huang and Lepowsky. In particular, given two
V-modules W
_{i}, W_{j}, a tensor product module W_{i}⊞ W_{j}was defined. In this talk, I will define a unitary structure on W_{i}⊞ W_{j}when V, W_{i}, and W_{j}are unitary, and when certain nice analytic conditions are satisfied. I will show that the structural maps (the associativity maps and the braid operators) are unitary operators.

- Date:
**9/22/17****Scott Atkinson, Vanderbilt University**- Title:
**Graph products of completely positive maps** - Abstract: In operator algebras, graph products unify the notions of free products and tensor products. In this talk, we will establish the graph product of unital completely positive maps on a universal graph product of C*-algebras and show that it is unital completely positive itself. The proof is an adaptation of Boca’s argument for the free version of this result; it utilizes an alternative length function specifically for words in a graph product and a Stinespring construction for concatenation. If time permits, we will discuss applications of this result to positive-definite functions on groups, unitary dilation, and a graph product version of von Neumann’s inequality. No prior knowledge of graph products will be assumed.

- Date:
**9/29/17****Ralph Kaufmann, Purdue University**- Title:
**Arcs, planar algebras and 2-representations** - Abstract: As we have previously observed there are formal similarities between planar algebras and algebras over the so-called arc-operad. In newer developments, using 2-representations, this correspondence can be made more precise. We will report on these results and their relationship to defects. If time permits, we furthermore plan to discuss the possible application of higher genus methods known from the arc side.

- Date:
**9/30/17 & 10/1/17**,**Shanks Workshop "Subfactors and Applications"** - Date:
**10/5/17**,**Mathematics Colloquium**(4:10-5:00pm in SC 5211)**Dan Voiculescu, UC Berkeley**- Title:
**Perturbations of operators and commutants mod normed ideals** - Abstract: Normed ideals of compact operators are the infinitesimals of Alain Connes' noncommutative geometry. In the study of operators modulo perturbations from these ideals a numerical invariant plays often a key role. Recently more structure is appearing in these questions from the commutant modulo the normed ideal. Connections with dynamical entropy, K-theory of operator algebras, supramenable groups and Banach space duality aspects will also be discussed.

- Date:
**10/6/17****Cain Edie-Michell, Australian National University**- Title:
**Planar algebras for the Drinfeld centres of the even parts of the ADE subfactors** - Abstract: Consider a subfactor whose even and dual even parts are both equivalent to some unitary tensor category C. Such subfactors are classified by braided auto-equivalences of the Drinfeld centre of C. In this talk I'll give a presentation of the (unshaded) planar algebras associated to the centres of the even parts of the ADE subfactors. I'll then explain how we can use these planar algebras to compute the braided auto-equivalences of the corresponding centres.

- Date:
**10/7/17 & 10/8/17**,**ECOAS 2017 at the University of Louisiana at Lafayette** - Date:
**10/13/17****No Meeting, Fall Break.**

- Date:
**10/19/17**,**Mathematics Colloquium**(4:10-5:00pm in SC 5211)**Adrian Ioana, UCSD**- Title:
**Rigidity for group von Neumann algebras** - Abstract: Any countable group $\Gamma$ gives rise to a von Neumann algebra $L(\Gamma)$. The classification of these group von Neumann algebras is a central theme in operator algebras. I will survey recent rigidity results which provide instances when various algebraic properties of groups, such as the existence or absence of a direct product decomposition, are remembered by their von Neumann algebras.

- Date:
**10/20/17 (Note: Seminar in SC 1310 today)****Adrian Ioana, UCSD**- Title:
**Rigidity for von Neumann algebras of amalgamated free product groups** - Abstract:
I will explain recent work with Ionut Chifan in which we provide a
large family of amalgamated free product groups whose amalgam structure can
be completely recognized from their von Neumann algebras. Our result
significantly strengthens some of the previous Bass-Serre rigidity results
for von Neumann algebras. As a consequence, we obtain the first examples of
amalgamated free product groups which are W
^{∗}-superrigid.

- Date:
**10/27/17****Bin Gui, Vanderbilt University**- Title:
**Unitarity of the modular tensor categories associated to unitary vertex operator algebras (continued)** - Abstract: See talk from 9/15/17.

- Date:
**11/3/17****Bin Gui, Vanderbilt University**- Title:
**Unitarity of the modular tensor categories associated to unitary vertex operator algebras (continued)** - Abstract: See talk from 9/15/17.

- Date:
**11/9/17**,**Mathematics Colloquium**(4:10-5:00pm in SC 5211)**Jesse Peterson, Vanderbilt University**- Title:
**Connes' Character Rigidity Conjecture for Lattices in Higher Rank Groups** - Abstract: A character on a group is a class function of positive type. For finite groups, the classification of characters is closely related to the representation theory of the group and plays a key role in the classification of finite simple groups. Based on the rigidity results of Mostow, Margulis, and Zimmer, it was conjectured by Connes that for lattices in higher rank simple Lie groups, the space of characters should be completely determined by their finite dimensional representations. In this talk, I will discuss the solution to this conjecture, and I will discuss its relationship to ergodic theory, invariant random subgroups, and von Neumann algebras.

- Date:
**11/10/17****Sandeepan Parekh, Vanderbilt University**- Title:
**Maximal amenable subalgebras in q-Gaussian factors** - Abstract: For q in (-1,1), Bozejko and Speicher’s q-Gaussian factors can be thought of as q-deformed versions of the free group factor. Indeed they are known to share several properties in common with the free group factors like being non-injective, strongly solid, isomorphic to LF_n (for |q| small enough). Continuing this line of investigation, in a joint work with K. Shimada and C. Wen, we show the generator masa in these factors are maximal amenable by constructing a Riesz basis in the spirit of Radulescu.

- Date:
**11/17/17****No Meeting**

- Date:
**11/24/17****No Meeting, Thanksgiving Break.**

- Date:
**12/1/17****Modjtaba Shokrian Zini, UCSB**- Title:
**CFTs as scaling limit of anyonic chains** - Abstract:
We provide a mathematical definition of a low energy scaling limit of a
sequence of Hilbert spaces and Hamiltonians (W
_{n},H_{n}), and apply our formalism to anyonic chains. We conjecture that certain linear expressions in Temperley-Lieb generators e_{j}have scaling limit the Virasoro lie algebra generators L_{m}. We will provide a proof of this conjecture for the Ising model and try to recover the CFT observables from different frameworks (VOA point-like fields, local conformal nets bounded observables and Wightman's smeared fields).

- End of Fall Semester.