# Junior Seminar

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

Organizer: Ian Charlesworth

## Wednesday, June 1

15:00 - 16:00 Rolando de Santiago (The University of Iowa): Deformations and applications

Abstract: We discuss several examples of deformations in the sense of Popa. In particular we will cover rigidity results of group von Neumann algebras from specific deformations, such as the Gaussian and free deformations.

## Wednesday, June 8

15:00 - 16:00 Andreas Næs Aaserud (University of California, Los Angeles): Approximate Equivalence of Actions

Abstract: I will talk about various types of equivalence of probability measure preserving actions, including approximate versions of conjugacy and orbit equivalence that Sorin Popa and I introduced and studied in recent joint work (cf. arXiv: 1511.00307). In particular, I will discuss some rigidity phenomena in this context.

## Wednesday, June 22

15:00 - 16:00 Koichi Shimada (Kyoto University): On examples of maximal amenabile subalgebras of the free group factor of two generators

Abstract: In this talk, I will present some new examples of maximal amenable subalgebras of a free group factor of two generators. As Dykema showed that the free product of two hyperfinite II_{1} factors is isomorphic to the free group factor of 2 generators. Based on this identification, we construct new examples of maximal amenable subalgebras. Namely, from each hyperfinite II_{1} factor R_{i} (i=1,2), we take a Haar unitary u_{i} which generates a Cartan subalgebra of the hyperfinite II_{1} factor. Then show that the von Neumann subalgebra generated by a selfadjoint operator u_{1}+u_{1}^{-1}+u_{2}+u_{2}^{-1} is maximal amenable in the free product. Our subalgebra cannot be embedded into neither of the free components inside the free product. The proof is based on Popa's technique, asymptotic orthogonality, and on Cameron–Fang–Ravichandran–White's method for Radial MASA.

## Wednesday, June 29

15:00 - 16:00 Lauren Ruth (University of California Riverside): An introduction to the spectral theory of automorphic forms

Abstract: The first half of this talk will be interactive, drawing upon participants' shared knowledge of functional analysis: We will identify self-adjoint integral operators which commute with the Laplacian, and we will use these operators to show that decomposes as a Hilbert space direct sum of subspaces that are invariant and irreducible under the right regular representation of , where , is a discrete subgroup of containing such that is compact, and is a unitary character of . Next, we will define the adele group, explain its role in the study of automorphic forms, and sketch the work of Gelfand, Graev, and Piatetski-Shapiro generalizing the above decomposition in the case when the homogeneous space is non-compact and when is an adele group. Finally, we will relate this material to the Trimester Program by describing a new factor, the group von Neumann algebra of an irreducible lattice in an adele group modulo its center, which acts on spaces of automorphic forms. [Sources: "Automorphic Forms and Representations'' by D. Bump, "Algebraic Number Theory'' by J. Neukirch, and "Coxeter Graphs and Towers of Algebras'' by F. Goodman, P. de la Harpe, and V. Jones.]

## Wednesday, July 13

15:00 - 16:00 Scott Atkinson (The University of Virginia): Weak approximate unitary equivalence in II_{1}-factors

Abstract: Over the past forty years, there has been growing interest in considering weak approximate unitary equivalence of representations of a C*-algebra inside a von Neumann algebra. Historically the first major result concerning this problem was solved in 1976, when Voiculescu answered this question in the case when the target algebra is B(H). In just the last decade or so, the more general version of this question has received attention in the literature. In 2005, Ding and Hadwin made significant progress on this problem, and in 2007, Sherman answered this question for singly generated commutative C*-algebras in arbitrary von Neumann algebras.

The purpose of this talk is to address this question when the target algebra is a II_{1}-factor. Starting with the basic definitions and results, we will quickly see that the trace will play an important role in this problem. We will consider the interesting problem of when the trace can distinguish equivalence classes. Our discussion will involve different notions of amenability, ultrapowers/products, and stabilization while sampling results of Ding, Hadwin, Jung, Li, Sherman, and myself.

## Wednesday, August 3

15:00 - 16:00 Brent Nelson (University of California): Free monotone transport

Abstract: Free monotone transport is a powerful method in free probability for obtaining isomorphisms between von Neumann algebras (or C*-algebras) developed by Guionnet and Shlyakhtenko in 2014. The result is a non-commutative analogue of Brenier's monotone transport theorem: any probability measure on ℝ^{n} (satisfying reasonable regularity conditions) can be realized as the pushforward of the Gaussian measure by a canonical map which is monotone in the sense that it's Jacobian is positive-definite almost everywhere. In this talk, I will flush out this analogy, discuss the context in which free monotone transport can be produced, and mention some of the isomorphism results obtained from this technique.

## Tuesday, August 9

15:00 - 16:00 James Tener (Max Planck Institute for Mathematics): Segal CFT and conformal nets

Abstract: In this talk I will introduce Segal's functorial formalism for conformal field theory, and discuss the example of the free fermion. Time permitting, I will then discuss how to obtain conformal nets of von Neumann algebras as a "geometric boundary value".

## Thursday, August 18

15:00 - 16:00 Keshab Chandra Bakshi (Institute of Mathematical Sciences, Chennai): On Pimsner Popa bases

Abstract: In this talk we examine bases for finite index inclusion of II_{1} factors. These bases behave nicely with respect to basic construction towers. As application, we obtain a characterization, in terms of bases, of basic constructions. Finally we use these bases to describe the phenomenon of multistep basic constructions. If time permits, we show these can be done for connected inclusion of finite dimensional C*-algebras also.

## Wednesday, August 24

15:00 - 16:00 Chris Bruce (University of Victoria): Operator algebras and number theory

Abstract: I will discuss several operator algebraic constructions from number theory.