Adriana Fernández Quero, University of Iowa

Aldo Garcia Guinto, Michigan State University

Arianna Cecco, University of Houston

Bill Helton, UCSD

Brandon Seward, University of California San Diego

Caleb Barnett, University of Houston

Connor Thompson, Purdue University

David Gao, UCSD

Dulanji Amaraweera, University of Iowa

Dumindu Sandakith Kasiwatte Kankanamge, Vanderbilt University

Fabian Salinas, Vanderbilt

Gregory Patchell, UC San Diego 

Hans Wenzl, UCSD

Ishan Ishan, UC Riverside

Jennifer Pi, UC Irvine

Juan Felipe Ariza Mejia, The University of Iowa

Jun Yang, Harvard University 

Juniper Bahr, UCLA

Kai Toyosawa, Vanderbilt University

Koichi Oyakawa, Vanderbilt University

Krishnendu Khan, Purdue University

Michael Davis, University of Iowa

Nicholas Boschert, Ucla

Patrick DeBonis, Purdue University 

Patrick Hiatt, UCLA

Pradyut Karmakar, Ohio University

Priyanga Ganesan, UC San Diego

Qingyuan Chen, UCSD

Rolando de Santiago, Purdue

Sayan Das, UC Riverside

Tanvi Telang, University of Houston

Tianyi Zheng, UCSD

Vincent Villalobos, University of Illinois at Urbana-Champaign

Xin Ma, University of Memphis

Yoonkyeong Lee, Michigan State University

Zhen-Chuan Liu, Baylor University

Friday, February 10: 

9:00-9:40      Ben Hayes

9:50-10:30    Hui Tan

10:30-11:00  coffee break

11:00-11:40  Isaac Goldbring 

11:50-12:30  Gil Goffer

12:30-2:30    lunch break

2:30-3:10     Robin Tucker-Drob

3:10-3:40     coffee break

3:40-4:20     Bin Sun

4:30-5:30     problem session

Saturday, February 11: 

9:30-10:10   Jingyin Huang

10:10-10:40 coffee break 

10:40-11:20  Alex Furman 

11:30-12:10  Dima Shlyakhtenko

12:10-2:00 lunch break

2:00-2:40    Sorin Popa

2:40-3:10    coffee break

3:10-3:50    David  Jekel

4:00-4:40    Daniel Drimbe

Sunday, February 12: 

9:00-9:40    Srivatsav Kunnawalkam Elayavalli

9:50-10:30  Mehrdad Kalantar

10:30-11:00 coffee break 

11:00-11:40 Jesse Peterson

11:50-12:30 Changying Ding

End of the workshop

Changying Ding (Vanderbilt University)

Title: Biexact von Neumann algebras (part II)

Abstract: The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying indecomposability properties of von Neumann algebras.  We will introduce the notion of a biexact von Neumann algebra, which allows us to place many previous indecomposability results in a more systematic context, and naturally leads to extensions of these results. We will give explicit connections between biexactness, malleable deformations, and closable derivations, allowing us to unify three separate approaches to solidity that were developed initially 15-20 years ago. The techniques also give a new characterization of weakly exact von Neumann algebras, answering a problem of Brown and Ozawa. 

Daniel Drimbe (KU Leuven)

Title: Rigidity for von Neumann algebras of graph product groups 

Abstract: In this talk I will discuss some rigidity results of group von Neumann algebras L(G), where G is a graph product group whose underlying graph is a certain cycle of cliques and the vertex groups are the wreath-like product property (T) groups introduced recently by Chifan, Ioana, Osin and Sun. By combining methods from Popa's deformation/rigidity theory with new techniques pertaining to graph product groups, we describe all the symmetries of these von Neumann algebras. This is joint work with Ionut Chifan and Michael Davis.

 Alex Furman (University of Illinois at Chicago)

Title:  Marked length spectrum and arithmeticity in negative curvature 

Abstract: Given a closed manifold $M$ equipped with a Riemannian metric $g$ of negative sectional curvature, one can associate a positive real number $\ell_g(c)$ to each conjugacy class $c$ of the fundamental group $\Gamma=\pi_1(M)$, namely the $g$-length of the shortest loop on $M$ in the free homotopy class $c$.

In the talk we will discuss some conjectures and results, old and new related to the function $\ell_g(c)$, arithmeticity of rank one lattices, and commensurators. 

The talk is based on a work by Yanlong Hao, Furman and Hao, and some related work by Bader and Furman.

 Gil Goffer (UCSD)

Title: compact URS and compact IRS

Abstract: I will discuss compact uniformly recurrent subgroups and compact invariant random subgroups in locally compact groups, and present results from ongoing projects with Pierre-Emanuel Caprace and Waltraud Lederle, and with Tal Cohen.

Isaac Goldbring (UCI) 

Title:  II$_1$ factors with the Brown property are generic

Abstract:  A separable II$_1$ factor $M$ is said to have the Brown property if for any separable subalgebra $N$ of the ultrapower $M^U$, there is a separable subalgebra $P$ of $M^U$ containing $N$ such that the relative commutant $P’\cap M^U$ is a factor.  We show that the Brown property is generic for II$_1$ factors in the sense that every so-called infinitely generic II$_1$ factor has the Brown property.  This is achieved by relating the Brown property to a uniform version of the super McDuff property (which states that the relative commutant of the II$_1$ factor in its own commutant is a factor) combined with the recent result by Chifan, Drimbe, and Ioana that II$_1$ factors with property (T) are embedding universal.  In particular, infinitely generic factors are uniformly super McDuff, generalizing another recent result of Chifan, Drimbe, and Ioana.  Time permitting, we will discuss the question of whether or not so-called finitely generic II$_1$ factors have the Brown property and why this problem is more difficult than the infinitely generic case.  This is joint work with David Jekel, Srivatsav Kunnawalkam Elayavalli, and Jennifer Pi.

Ben Hayes (University of Virginia)

Title: Co-spectral radius, equivalence relations and the growth of unimodular random rooted trees

Abstract: I will speak on joint work with Mikolas Abert and Mikolaj Fraczyk. In it, we define the co-spectral radius of inclusions of discrete, probability measure-preserving orbit equivalence relations, using the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on G/H for inclusion of groups G,H. Unlike the group case the almost sure existence of this pointwise limit is nontrivial to establish, and we provide a novel general method for proving almost sure existence of such limits based on the mass-transport principle. Time permitting, I will mention some analogies and connections with operator algebras and subfactor theory, as well as some applications.

Jingyin Huang (Ohio State University)

Title: Measure equivalence superrigidity for some generalized Higman groups

Abstract: In the 1950s, Higman introduced the first class of examples of infinite finitely presented groups without any non-trivial finite quotient. We study Higman groups from the viewpoint of measure equivalence -  a notion introduced by Gromov as a measurable counterpart to quasi-isometry. For most Higman groups and some generalizations, we prove a strong measure equivalence rigidity theorem, and obtain W*-superrigidity results for all free, ergodic, probability measure-preserving actions of these groups. Moreover, our main result is a consequence of a more general theorem on measure equivalence invariants of certain classes of groups acting acylindrically on CAT(-1) spaces. In this talk, I'll sketch the proof, and possibly compare to some other measure equivalence rigidity/flexibility results in the literature if time allows. This is joint work with Camille Horbez.

Mehrdad Kalantar (University of Houston)

Title: A conjecture on type I locally compact groups

Abstract: The notion of type I, hailing from the very origins of operator algebras and representation theory, can be seen as a rigorous way to define the class of groups for which unitary representations can be classified in any meaningful manner. By a celebrated result of Thoma, a discrete group is type I if and only if it is virtually abelian. In the non-discrete case, the current state of the art is not nearly as complete, despite numerous results ensuring that various important families of groups (e.g. every connected semisimple Lie group) are type I. What is completely lacking, in contrast to Thoma’s theorem, is a definite structural consequence of type I. This talk is around the following conjecture: Every second countable locally compact group of type I admits a cocompact amenable subgroup. We motivate the conjecture, provide some supporting evidence for it, and prove it for type I hyperbolic locally compact groups admitting a cocompact lattice.

This is joint work with Pierre-Emmanuel Caprace and Nicolas Monod.

Srivatsav Kunnawalkam Elayavalli (IPAM, UCLA)

Title: Small at infinity compactification of a von Neumann algebra. 

Abstract: In this talk I will describe a generalized notion of a small at infinity compactification a la Ozawa of a finite von Neumann algebra introudced by the speaker, Ding and Peterson. The key technical novelty here was to bypass a well known obstruction, involving the absence of interesting derivations from B (L^2 M) into the compact operators, by considering the closure of the compacts in a topology of Magajna from the theory of strong operator bimodules. By effectively working in this topological framework we clarify old problems of Anantharaman-Delaroche involving various viewpoints of the Haagerup property for II_1 factors, and also the notion of mixingness for bimodules. This work additionally allows for several kinds of rigidity results via a generalized notion of proper proximality (a la Boutonnet-Ioana-Peterson) for von Neumann algebras. Some of the applications I will talk about from this work include: absense of weakly compact cartan subalgebars for a wide class of II_1 factors; a structure result for subfactors of L (G) where G is non amenable bi-exact, settling a problem of Popa for the case of F_2 and solid ergodicity for various Gaussian actions, extending works of Boutonnet and Chifan-Ioana. By developing an abstract upgrading result for the notion of relative proper proximality, the speaker and Ding also recently obtained a new application of this framework, involving the structure of free products. I will discuss these results and touch on some key new ideas.

David Jekel (UCSD)

Title:  Free entropy theory, model theory, and embeddings into matrix ultraproducts

Abstract:  We define a new variant of free entropy using ideas from continuous model theory, and use it to study embeddings into matrix ultraproducts; in particular, we show that every non-strongly-1-bounded tracial von Neumann algebra admits an embedding into a matrix ultraproduct with trivial relative commutant.   Voiculescu's free entropy and Jung and Hayes' 1-bounded entropy are measurements of the amount of matrix approximations of the generators X_1, ..., X_n of a tracial von Neumann algebra M, where 'matrix approximations' is interpreted as the matrix tuples with approximately the same non-commutative moments.  We define analogous versions of entropy where the 'matrix approximation' is understood in terms of the model-theoretic type, meaning that instead of polynomials our test functions are formulas involving suprema and infima over auxiliary variables.   The model-theoretic 1-bounded entropy has many natural properties in how it relates to the original 1-bounded entropy, change of coordinates, definable closures, and more.

Jesse Peterson (Vanderbilt University)

Title: Biexact von Neumann algebras (part I)

Abstract: The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying indecomposability properties of von Neumann algebras.  We will introduce the notion of a biexact von Neumann algebra, which allows us to place many previous indecomposability results in a more systematic context, and naturally leads to extensions of these results. We will give explicit connections between biexactness, malleable deformations, and closable derivations, allowing us to unify three separate approaches to solidity that were developed initially 15-20 years ago. The techniques also give a new characterization of weakly exact von Neumann algebras, answering a problem of Brown and Ozawa. 

Sorin Popa (UCLA)

Title: In search of maximal amenable subalgebras of $L\mathbb F_n$ that are not freely complemented. 

Abstract: Starting in early '80s, a series of absorption and in-decomposability phenomena in $L\mathbb F_n$, and more generally in (amalgamated) free product II$_1$ factors $M=M_1*_B M_2$, unravelled over the years. But a key question along these lines keeps lingering: do there exist maximal amenable subalgebras of $L\mathbb F_n$ that are not freely complemented? I will comment  on  this problem and also  present a result with Remi Boutonnet, where we exhibit a large family of distinct maximal amenable MASAs in $L\mathbb F_n$, which seem to be good candidates for such examples. 

Dimitri Shlyakhtenko (UCLA)

Bin Sun (Oxford University)

Title: Rigidity examples constructed with wreath-like

product groups

Abstract: Wreath-like product groups were introduced recently and used to construct the first positive examples of rigidity conjectures of Connes and Jones. In this talk, I will review those examples, as well as discuss some ideas to construct examples with other rigidity phenomena by modifying the wreath-like product construction.

Hui Tan (UCSD)

Title: Spectral gap characterizations of property (T) for II_1 factors

Abstract: I will discuss characterizations of property (T) for II_1 factors by weak spectral gap in extensions, and by the non-weakly-mixing property of the bimodules containing almost central vectors. I will explain how these are related and compare them to the corresponding characterizations of property (T) for groups.

Robin Tucker-Drob (University of Florida).

Title: Measure Equivalence, Schlichting Completions, and Baumslag-Solitar groups

Abstract: A subgroup H of a group Γ is commensurated by Γ if all H-orbits in Γ/H are finite. In this situation, the closure of Γ in the group of all permutations of Γ/H is a totally disconnected locally compact group called the Schlichting completion of the pair (Γ, H) and we denote it G(Γ, H). We show that if H and K are amenable commensurated subgroups of Γ and Λ respectively such that the associated Schlichting completions G(Γ, H) and G(Λ, K) are both hyperbolic with trivial amenable radical, then every measure equivalence coupling of Γ with Λ descends canonically to a measure equivalence coupling of the (possibly nonunimodular) groups G(Γ, H) and G(Λ, K). This unifies and generalizes theorems of Houdayer--Raum, Kida, and Monod--Shalom. I will discuss the consequences of this for the open problem of classifying Baumslag-Solitar groups up to measure equivalence.

The following is a list of open problems discussed during the problem session. Thanks for Greg Patchell for compiling the list.

Price Center is a few minutes away from the AP&M building. For a list of places to eat at in Price Center, including their weeekend hours, see here. This website also lists a few other dining options close to the AP&M building, under the heading Dining@Student Center.

Financial support for junior participants is available through the NSF grants FRG-DMS-1854074 (administered through UCSD) and FRG-DMS-1853989 (admninistered through Vanderbilt University). Participants who will be awarded financial support will be contacted with reimbursement instructions and policies, depending on the institution which will disburse the reimbursement.

As federal funds will be used, all flight itineraries must comply with the Fly America Act. All airfare must be booked on U.S. flag carriers and on coach/economy class.

Participants are expected to make their own hotel arrangements. Graduate students are encouraged to share a room. Some hotels to consider are:

• 1 Best Western Premier Hotel Del Mar (Del Mar Inn), located in Del Mar, close to the beach, about 5.4 miles north on campus. To get to campus, one can use the 101 bus route. To make a reservation, call the hotel ( (858) 755-9765 ), ask for the UCSD discount, and say that you are here for a conference at UCSD.

• 2 Sheraton La Jolla Hotel, next to the UCSD campus, 1.2 miles south of the AP&M building. To make a reservation, go to:  www.sheratonlajolla.com

·         Put in desired stay dates

·         Go to special rate drop down arrow

·         Go to Corporate/Promo code and type in:  UC0 (zero, not ‘O’)

If the discounted rate is available, it will populate.  If it’s not available, then the best available rate at that time will populate.

• 3 Residence Inn by Mariott, next to the UCSD campus, 0.9 miles south of the AP&M building.