## Operator Algebras Seminar Fall 2017

Organizers: Marius Dadarlat, Andrew Toms, and Thomas Sinclair

Tuesday, 2:30-3:30pm, UNIV 319

• Date: 8/29
• Title: On AF-embedability of residually finite dimensional C*-algebras

• Date: 9/5 *Special Logic Seminar*
• Turbo Ho (Purdue)
• Title: On the Scott Sentences of Finitely Generated Structures
• Scott showed that for every countable structure A, there is an formal sentence, called the Scott sentence, which "describes" A up to isomorphism. Thus, the quantifier complexity of the sentence is one invariant that measures the complexity of the "description" of the structure. Knight et al. have studied the Scott sentences of many structures, in particular, Knight and Saraph showed that a finitely generated structure always have a $\Sigma_3$ Scott sentence, giving an upper bound for complexities. In this talk, we will start with the basic definitions, building up toward a joint work with Matthew Harrison-Trainor, and another independent work by Alvir, Knight, and McCoy, where we give characterizations of finitely generated structures where the $\Sigma_3$ upper bound is attained. Using these results, we also give a construction of a finitely generated group where the $\Sigma_3$ Scott sentence is attained. No background in logic is required.

• Date: 9/16-9/17
• Wabash Miniconference, IUPUI
• Conference webpage

• Date: 9/26
• Paul McKenney (Miami University)
• Title: Set theory and automorphisms of corona algebras
• In the 70's, Shelah found a model of set theory where every homeomorphism of the Cech-Stone remainder of the natural numbers is induced by a bijection between cofinite sets of natural numbers. In the late 2000's, Farah showed that there is a model where every automorphism of the Calkin algebra is inner. I will discuss to what extent these results find a common generalization for coronas of separable C*-algebras. This discussion will lead to some interesting connections to other areas of operator algebra. No knowledge of set theory will be assumed, beyond the very basics.

• Date: 10/3, *Special Logic Seminar*
• Alex Kruckman (Indiana)
• Title: Independence in generic structures
• "Generic structures" are a fruitful source of examples in model theory: Start with a base theory T (e.g. graphs, fields, abelian groups), expand it by adding extra structure (e.g. a predicate, an automorphism, an order), and take the model companion T*. Assuming T is inductive and T* exists, it axiomatizes the class of "generic" (existentially closed) models of T (e.g. the random graph with a generic subset, algebraically closed fields with a generic automorphism, divisible abelian groups with a generic order). Model theory seeks to classify first-order theories based on their relative complexity. The class of "simple" theories can be characterized by the existence of a well-behaved notion of independence (called forking independence) in models of the theory, and generic theories can often be shown to be simple by characterizing independence in T* in terms of independence in T. Recently, there has been increased interest in the class of NSOP1 theories, which contains the simple theories, spurred by the work of Chernikov, Kaplan, and Ramsey, who showed that NSOP1 theories can be similarly characterized by the existence of a well-behaved notion of independence (called Kim independence). Many examples of generic theories which fail to be simple have been shown to be NSOP1 by this method. In this talk, I will present a number of new examples of this phenomenon: In joint work with Nicholas Ramsey, we study the theory of the generic L-structure in an arbitrary language L. More generally, starting with a base L-theory T satisfying some modest assumptions, we consider the generic expansion of T to an arbitrary language containing L and the generic Skolemization of T. In joint work with Gabe Conant, we study the generic projective plane, considered as an incidence structure. More generally, we consider the generic bipartite graph omitting a fixed complete bipartite graph K_{m,n}. We show that all of these examples are NSOP1, and we characterize various notions of independence in these theories.

• Date: 10/7-10/8
• ECOAS 2017, University of Louisiana at Lafayette
• Conference webpage

• Date: 10/9-10/10
• Fall Break

• Date: 10/17
• Lauren Ruth (UC Riverside)
• Title: Two new settings for examples of von Neumann dimension
• Let $G = PSL(2,\mathbb{R})$, let $\Gamma$ be a lattice in $G$, and let $\mathcal{H}$ be an irreducible unitary representation of $G$ with square-integrable matrix coefficients. A theorem in Goodman--de la Harpe--Jones (1989) based on Atiyah's work on $L^2$-index states that the von Neumann dimension of $\mathcal{H}$ as a $W^*(\Gamma)$-module is equal to the formal dimension of the discrete series representation $\mathcal{H}$ times the covolume of $\Gamma$, both calculated with respect to the same Haar measure. We will present two results which take inspiration from this theorem. In the first part of the talk, we will show that there is a representation of $W^*(\Gamma)$ on a subspace of cuspidal automorphic functions in $L^2(\Lambda \backslash G)$, where $\Lambda$ is any other lattice in $G$ (and $W^*(\Gamma)$ acts on the right); and this representation is unitarily equivalent to one of the representations in [GHJ]. In the second part of the talk, we will explain how the proof of their theorem carries over to a wider class of groups, including the situation where $G$ is $PGL(2,F)$, for $F$ a local non-archimedean field, and $\Gamma$ is a torsion-free lattice in $PGL(2,F)$, which, by a theorem of Ihara, is a free group. We will conclude with computations of von Neumann dimensions when $\mathcal{H}$ is the Steinberg representation, and when $\mathcal{H}$ is a depth-zero supercuspidal representation. In particular, these spaces afford representations of free group factors that are not unitarily equivalent to the representations obtained in the setting of $PSL(2,\mathbb{R})$.

• Date: 10/23, 4:30, MATH 731 *Special Date and Time*
• Daniel Drimbe (UC San Diego)
• Title: Prime II$_1$ factors arising from irreducible lattices in products of rank one simple Lie groups
• In this talk I will present joint work with Daniel Hoff and Adrian Ioana in which we obtain that II$_1$ factors associated to icc irreducible lattices in products of simple Lie groups of rank one are prime, i.e. they cannot be decomposed as a tensor product of II$_1$ factors. This gives the first examples of prime II$_1$ factors arising from lattices in higher rank semisimple Lie groups.

• Date: 10/28
• Wabash Seminar, Wabash College
• Seminar webpage

• Date: 10/31
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• Date: 11/7
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• Date: 11/14
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• Date: 11/21
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• Date: 11/22-11/25
• Thanksgiving Break

• Date: 11/28
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• Date: 12/5
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