Subfactor Seminar
Fall 2011

---

Organizers: Dietmar Bisch, Vaughan Jones, and Jesse Peterson

Fridays, 4:10-5:30pm in SC 1432

---

• Date: 8/26/11
  • Darren Creutz, Vanderbilt University
  • Title: Normal Subgroups of Commensurators and Rigidity of SAT Actions
  • Abstract: I will present some results of myself and Y. Shalom in a pair of talks.
    
    During the first talk, I will focus on our Normal Subgroup Theorem for Commensurators of lattices: any normal subgroup of a (dense) commensurator of a lattice in a locally compact group necessarily contains the lattice. Consequences of this theorem will also be discussed: classification of normal subgroups of commensurators; an improved form of Bader-Shalom’s normal subgroup theorem for lattices in products; and a partial answer to a question of Lubotzky, Mozes and Zimmer on tree automorphisms.
    
    The second talk will focus on our results on group dynamics for quasi-invariant actions that are the main new ingredient required to prove the normal subgroup theorem. I will discuss strongly approximately transitive actions and their various structural and rigidity properties. The talk will conclude with a discussion of a potential structure theory for quasi-invariant actions.
    
    The second talk should be understandable even without the background presented in the first though it will be helpful.

• Date: 9/2/11
  • Darren Creutz, Vanderbilt University
  • Title: Normal Subgroups of Commensurators and Rigidity of SAT Actions
  • Abstract: I will present some results of myself and Y. Shalom in a pair of talks.
    
    During the first talk, I will focus on our Normal Subgroup Theorem for Commensurators of lattices: any normal subgroup of a (dense) commensurator of a lattice in a locally compact group necessarily contains the lattice. Consequences of this theorem will also be discussed: classification of normal subgroups of commensurators; an improved form of Bader-Shalom’s normal subgroup theorem for lattices in products; and a partial answer to a question of Lubotzky, Mozes and Zimmer on tree automorphisms.
    
    The second talk will focus on our results on group dynamics for quasi-invariant actions that are the main new ingredient required to prove the normal subgroup theorem. I will discuss strongly approximately transitive actions and their various structural and rigidity properties. The talk will conclude with a discussion of a potential structure theory for quasi-invariant actions.
    
    The second talk should be understandable even without the background presented in the first though it will be helpful.

• Date: 9/9/11
  • Darren Creutz, Vanderbilt University
  • Title: Normal Subgroups of Commensurators and Rigidity of SAT Actions
  • Abstract: I will present some results of myself and Y. Shalom in a pair of talks.
    
    During the first talk, I will focus on our Normal Subgroup Theorem for Commensurators of lattices: any normal subgroup of a (dense) commensurator of a lattice in a locally compact group necessarily contains the lattice. Consequences of this theorem will also be discussed: classification of normal subgroups of commensurators; an improved form of Bader-Shalom’s normal subgroup theorem for lattices in products; and a partial answer to a question of Lubotzky, Mozes and Zimmer on tree automorphisms.
    
    The second talk will focus on our results on group dynamics for quasi-invariant actions that are the main new ingredient required to prove the normal subgroup theorem. I will discuss strongly approximately transitive actions and their various structural and rigidity properties. The talk will conclude with a discussion of a potential structure theory for quasi-invariant actions.
    
    The second talk should be understandable even without the background presented in the first though it will be helpful.

• Date: 9/16/11
  • Scott Carter, University of South Alabama
  • Title: Group families of Quandles and invariants of knotted graphs, embedded foams, and so forth.
  • Abstract: Given a group and its automorphism group it is possible to define a family of quandles which are indexed by the elements of the group. There is an associated homology theory that incorporates both group cocycles and quandle cocycles and is suited to the study of knotted graphs, knotted handle bodies, and knotted foams in 4-space. In the talk, the homology theory will be outlined, and the singularities of foams and their higher dimensional generalizations will be discussed.

• Date: 9/23/11
  • James Tener, UC Berkeley
  • Title: Manifestly unitary conformal field theory.
  • Abstract: We will present a manifestly unitary construction of a conformal field theory (in the sense of Segal). In order to do this, we will discuss a unitary version of the determinant line bundle of a polarized Hilbert space, and its connection to CAR algebras and fermionic second quantization. We will also discuss the relationship between this conformal field theory and loop group representations on fermionic Fock space.

• Date: 9/24/11 – 9/25/11

• Wabash Miniconference, at IUPUI.

Date: 9/30/11

• Michael Brandenbursky , Vanderbilt University
• Title: Quasi-isometric embeddings into diffeomorphism groups.
• Abstract: Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group of M, we construct quasi-isometric embeddings of either free Abelian or direct products of non-Abelian free groups into the group of volume preserving diffeomorphisms of M equipped with the Lp metric induced by a Riemannian metric on M. If time permits I will explain a relation between quasi-morphisms and L^p metrics on the group of area-preserving diffeomorphisms of the 2-disc.

Date: 10/7/11

• No Meeting, Fall Break.

Date: 10/15/11 – 10/16/11

• ECOAS 2011, at Purdue University.

Date: 10/21/11

• Bogdan Udrea, University of Iowa
• Title: New examples of von Neumann algebras with unique Cartan subalgebra.
• Abstract: In this talk we will show that any compact action of direct products of icc hyperbolic groups gives rise to a von Neumann algebra with unique Cartan subalgebra. The method we employ also allows new structural results for maximal abelian subalgebras of II₁ factors associated with products of hyperbolic groups. This is a joint work with I. Chifan and T. Sinclair.

Date: 10/27/11, Departmental Colloquium, 4:10-5:00 in SC 5211.

• Antony Wassermann, CNRS, Marseille

Date: 10/28/11

• Noah Snyder, Columbia University
• Title: The Brauer group and subfactors.
• Abstract: The Artin-Wedderburn theorem states that any semisimple algebra over a field k is a direct sum of matrix algebras over division rings over k. Furthermore, there’s an interesting algebraic structure called the Brauer group made of the central simple algebras over k. In this talk I’ll explain how Artin-Wedderburn and the Brauer group generalize to the subfactor setting (from work of Etingof, Nikshych, and Ostrik). After discussing the general theory, I’ll move on to some examples. In particular, I’ll discuss my joint work with Pinhas Grossman where we compute all algebra objects in the Haagerup fusion categories (and thus all subfactors “related to” the Haagerup subfactor), and our work in progress where we do the same for Asaeda-Haagerup. In the Asaeda-Haagerup case the analogue of the Brauer group is nontrivial.

Date: 10/29/11 – 10/30/11

• Shanks Workshop on Conformal Field Theory and Von Neumann Algebras, at Vanderbilt University.

Date: 11/4/11

• Arnaud Brothier, KU Leuven
• Title: Maximal Abelian subalgebras.
• Abstract: I will present some invariants for maximal abelian subalgebras in a II_1 factor A \subset M. I will explain some connections between them and describe their nature. In particular, I will show that the Takesaki equivalence relation can be describe by the A-bimodule structure of M.

Date: 11/11/11

• Ionut Chifan, Vanderbilt University
• Title: Some structural results for II_1 factors.
• Abstract: In this talk I will present some recent structural results for II_1 factors associated with hyperbolic group/actions. We will focus primarily on strong solidity problem and unique Cartan subalgebra problem for such factors. For instance I will show that any compact action of any hyperbolic group gives rise to a von Neumann algebra with unique Cartan subalgebra. The methods employed also allow new structural results for the measure equivalence class of a hyperbolic group. Relative and product versions of these results as well as open problems will be also discussed. This is based on two recent joint papers with T. Sinclair respectively T. Sinclair and B. Udrea.

Date: 11/18/11

• Jesse Peterson, Vanderbilt University
• Title: Unique group measure space decomposition.
• Abstract: The first examples of II_1 factors with unique Cartan subalgebra up to unitary conjugacy was given by Ozawa and Popa in 2007. There are still only a handful of such factors known, and are all constructed by using a certain amount of “smallness” of an action. In this talk I will give examples of factors which have a possibly weaker property of having unique group-measure space Cartan subalgebra. Showing this weaker property then allows us to obtain examples with more general types of actions. Part of this talk will be based on joint work with Ionut Chifan.

Date: 11/25/11

• No Meeting, Thanksgiving Break.

Date: 12/2/11

• David Sherman, University of Virginia
• Title: Model theory and II_1 factors.
• Abstract: Classical model theory studies the relation between mathematical structures and (first-order) logical properties. It does not work well for the objects of functional analysis, but there are variations that do, including a recent development called continuous model theory in which truth values are drawn not from { true, false } but from the interval [0,1]. Without assuming any background in logic, I will explain the main features of continuous model theory and how it leads to new results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. This is joint work with Ilijas Farah and Bradd Hart.

Date: 12/9/11

End of Fall Semester.