# Trimester Seminar

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45

Organizers: Dietmar Bisch, Vaughan Jones, Sorin Popa, Dima Shlyakhtenko

## Tuesday, May 10

14:30 - 15:00 Round of Introductions

15:00 - 16:00 Narutaka Ozawa (RIMS, Kyoto University): A functional analysis proof of Gromov's polynomial growth theorem

Abstract: The celebrated theorem of Gromov in 1980 asserts that any finitely generated group with polynomial growth is virtually nilpotent, i.e., it contains a nilpotent subgroup of finite index. Alternative proofs have been given by Kleiner (2007), etc. In this talk, I will give yet another proof of Gromov's theorem, based on functional analysis and random walk techniques.

## Thursday, May 12

15:00 - 16:00 Roberto Longo (Università di Roma Tor Vergata): Infinite spin, infinite volume and infinite index

Abstract: In 1939 E. Wigner classified the unitary, irreducible, positive energy representations of the Poincaré group (particles). Infinite spin particles were basically disregarded henceforth as never observed in nature, although they are compatible with all physical principles. We prove that infinite spin particles cannot be observed in a bounded spacetime region, yet there are many infinite spin particle states localized in unbounded regions, or associated with charges with infinite Jones index. (Joint work with V. Morinelli and K.H. Rehren.)

## Tuesday, May 31

15:00 - 16:00 Sebastian Palcoux (IMSc): On distributive subfactor planar algebras

Abstract: A theorem of Oystein Ore states that a finite group is cyclic iff its subgroups lattice is distributive. A subfactor planar algebra is called distributive if its biprojections lattice is distributive. We define an Euler totient for such planar algebras, which exactly extends Euler's totient of natural numbers. We then extend one side of Ore's theorem to distributive subfactor planar algebras with a nonzero Euler totient, i.e. we prove the existence of a (2-box) minimal projection generating the identity biprojection. We plan to prove that the Euler totient of any distributive subfactor planar algebra is nonzero.

## Thursday, June 2

15:00 - 16:00 Rémi Boutonnet (CNRS, Université de Bordeaux): Amenability VS amalgamated free products: the state of the art

Abstract: I will present recent joint work with Cyril Houdayer about amenable subalgebras of arbitrary amalgamated free product von Neumann algebras. We proved that amenable subalgebras of M_{1} *_{B} M_{2} with large enough intersection with M_{1} are actually contained in M_{1}. This result generalizes previous results of Houdayer-Ueda and Leary. Our proof combines a central-state approach inspired from previous work with Alessandro Carderi and an intertwining criterion due to Ozawa-Popa.

## Tuesday, June 7

15:00 - 16:00 Roman Sasyk (Universidad de Buenos Aires): On the classification of II_{1} factors with the McDuff property

## Thursday, June 9

15:00 - 16:00 Rolando de Santiago (The University of Iowa): Product rigidity for the von Neumann algebras of hyperbolic ICC groups.

Abstract: Let G be an n-fold direct product of hyperbolic ICC groups G1, …,Gn, and suppose L(G) is isomorphic to L(H) for some discrete group H. We show H decomposes into an n-fold product such that L(Gi) is isomorphic to L(Hi) up to amplification; i.e. the group von Neumann algebra remembers the product structure.

Building on these techniques, we show a similar phenomena occurs for groups in Quot(Crss), a class of non-amenable groups introduced by I. Chifan, A. Ioana and Y. Kida. Specifically, if G is an ICC group in this class and L(G) is not prime, then G decomposes into a k-fold product of groups, each of which lie in Quot(Crss). (These are joint works with I. Chifan and T. Sinclair, and S. Pant, respectively)

## Tuesday, June 14

15:00 - 16:00 Ian Charlesworth (University of California, Los Angeles): Regularity in Free Probability

Abstract: Given an n-tuple of non-commuting random variables y_{1}, …, y_{n} and a non-constant self-adjoint polynomial P in n indeterminates, we set y = P(y_{1}, …, y_{n}) and ask how the behaviour of y is affected by properties of y_{1}, …, y_{n} and P. It turns out that if y_{1}, …, y_{n} are free, algebraic, and have finite free entropy, so too does y. If instead we assume that y_{1}, …, y_{n} have a dual system, then the spectral measure of y has support which is not Lebesgue null, and if P is homogeneous (e.g,. a monomial) then the spectral measure of y is Lebesgue absolutely continuous. This is joint work with Dimitri Shlyakhtenko.

## Thursday, June 16

15:00 - 16:00 Wojciech Szymanski (University of Southern Denmark): On MASAs in infinite C*-algebras

Abstract: I will discuss some of the recent results and open questions pertaining MASAs and automorphisms of purely infinite, simple C*-algebras, trying to make an (ultraweak) point of contact with von Neumann algebra theory.

## Tuesday, June 21

15:00 - 16:00 Rui Okayasu (Osaka Kyoiku University): Haagerup approximation property for von Neumann algebras

Abstract: I will present recent joint works with Narutaka Ozawa and Reiji Tomatsu about Haagerup approximation property for arbitrary von Neumann algebras. I will introduce our definition and several characterizations of this property

## Thursday, June 23

15:00 - 16:00 Ben Hayes (Vanderbilt University): Fuglede-Kadison determinants and sofic entropy

Abstract: Abstract: Let G be a countable, discrete group, an algebraic action of G is an action by automorphisms of a compact, abelian, metrizable group X. The data of an algebraic action is equivalent, via Pontryagin duality, to a countable Z(G)-module A, where Z(G) is the integral group ring. A particular case of interest is as follows: fix f in M_{k,n}(Z(G)), and let X_{f} be the Pontraygin dual of Z(G)^{n} / Z(G)^{k }f (as an abelian group). There has been a long history of connecting the entropy of the action of G on X_{f} to the Fuglede-Kadison determinant (defined via the von Neumann algebra of G) of f in various degrees of generality. In the amenable case, this was studied by Lind-Schmidt-Ward, Deninger, Deninger-Schmidt, Li and completely settled by Li-Thom. We study the entropy of such actions when G is sofic (using sofic entropy as defined by Bowen, Kerr-Li). Generalizing work of Bowen, Kerr-Li, Bowen-Li (as well as the amenable case) we completely settle the connection between Fuglede-Kadison determinants and sofic entropy of these actions when k=n, as well as give general upper bounds when k is not n. Moreover, we show that it is completely impossible to find a general connection between entropy and torsion for sofic groups analogous to the one Li-Thom developed for amenable groups. We will comment on the techniques, which differ from the amenable case and are the first to avoid approximating the Fuglede-Kadison determinant of f by finite-dimensional determinants. No knowledge of sofic entropy or Fuglede-Kadison determinants will be assumed.

## Tuesday, June 28

15:00 - 16:00 Bas Janssens (Universiteit Utrecht): Characterization of Reflection Positivity

Abstract: Roughly speaking, a state on an algebra is called Reflection Positive (RP) if any observable on one side of the system has a nonnegative correlation with its reflection on the other side. Techniques relying on RP were introduced in constructive QFT by Osterwalder-Schrader in the 1970s, and soon found their way to (Quantum) Statistical Physics and Representation Theory. We give a concrete characterization of RP equilibrium states in terms of the Hamilton operators by which they are defined. This is applied to Parafermion algebras, which are used in the description of Anyons. (Joint work with A. Jaffe)

## Thursday, June 30

15:00 - 16:00 Hans Wenzl (University of California, San Diego): Braid representations in connection with exceptional Lie algebras

## Tuesday, July 12

15:00 - 16:00 Vlad Sergiescu (University of Grenoble): Around smoothings of Thompson's groups: how and why

Abstract: Richard Thompson's groups F, T and V are generally defined as acting on the interval, resp. the circle and the Cantor space. This talk concerns mainly F and T. We will first discuss a variety of definitions and occurrences of them. Then, we plan to consider regularity aspects of their representations in dimension 1, some applications and some questions.

## Wednesday, July 20

11:00 - 12:00 Marcel Bischoff (Vanderbilt University): Defects and Phase boundaries in Algebraic Conformal Quantum Field Theory

Abstract: I will discuss how techniques from (type III) subfactors can be used to construct and classify defects and phase boundaries between conformal nets. Based on joint work with Y. Kawahigashi, R. Longo and K.-H. Rehren.

## Tuesday, July 26

15:00 - 16:00 Pinhas Grossman (University of New South Wales): Operator algebras and graded extensions of fusion categories

Abstract: In this talk we will review Izumi's method of constructing quadratic tensor categories from endomorphisms of the Cuntz C*-algebras, and then discuss an application to constructing graded extensions of fusion categories. This method leads to a complete classification of finite graded extensions of the even parts of the Asaeda-Haagerup subfactor. This is joint work with Masaki Izumi and Noah Snyder.

## Tuesday, August 2

15:00 - 16:00 Thomas Schucker (Aix-Marseille University): Gravitational birefringence at cosmological scales

Abstract: The trajectory of massless particles with spin is computed in a spatially flat Robertson-Walker metric. At arrival, the offset between the two polarisation states of a photon emitted at redshift z is given by (z+1) times its wavelength at emission. Experimental observation of the effect is a challenge and would imply new tests of our present cosmological models.

## Thursday, August 4

15:00 - 16:00 Florin Radulescu (Institute of Mathematics of the Romanian Academy & University of Rome Tor Vergata): Operator algebras and number theory

Abstract: We describe the use operator algebra techniques to establish spectral gap for Hecke Operators on Maass forms for general discrete subgroups, of finite covolume, of PSL(2,R).