Subfactor Seminar
2012-2013

• Date: 8/31/12
  • Vaughan Jones, Vanderbilt University
  • Title: AF-algebras, Cuntz algebras, free Gaussian functor and several algebra structures associated with a planar algebra
  • Abstract: We will begin by reminding people of AF-algebras, Cuntz algebras and Voiculescu's free Gaussian functor. Then we will introduce planar algebras with three examples and show how the previous algebras can be generalised and what new phenomena, and many open questions, occur with the generalisation.

• Date: 9/7/12
  • Natasha Blitvic, Vanderbilt University
  • Title: Two-parameter non-commutative Gaussian processes
  • Abstract: The setting for this talk is the deformed commutation relations and Fock space representations of deformed quantum harmonic oscillator algebras. Building on the work of Bozejko and Speicher in the single-parameter case, we will begin by constructing the (q,t)-Fock space, a two-parameter deformation of the bosonic and fermionic Fock spaces. We will focus on the probabilistic interpretation of the algebras of bounded linear operators on this space with particular emphasis on the operators playing the role of the Gaussian random variables. We will discuss the combinatorial structure underlying these objects, their role in a generalized non-commutative Central Limit Theorem, and several surprising connections to well-studied mathematical objects.

• Date: 9/13/12, Mathematics Colloquium (4:10-5:00pm in SC 5211)
  • Nigel Higson, Penn State University
  • Title: Contractions of Lie Groups and Representation Theory
  • Abstract: Let K be a closed subgroup of a Lie group G. The contraction of G to K is a Lie group, usually more elementary in structure than G itself, that approximates G to first order near K. The terminology is due to the mathematical physicists, who examined the group of Galilean transformations as a contraction of the group of Lorentz transformations. My focus will be on a related but different class of examples, the prototype of which is the group of isometric motions of Euclidean space, viewed as a contraction of the group of isometric motions of hyperbolic space. It is natural to expect some sort of limiting relation between representations of the contraction and representations of G. But in the 1970s George Mackey carried out a few calculations pointing to an interesting rigidity phenomenon: as the contraction group is deformed back to G, the representation theory remains in some sense unchanged. In particular the irreducible representations of the contraction group parametrize the irreducible representations of G. I shall formulate a reasonably precise conjecture that was inspired by subsequent developments in C*-algebra theory and noncommutative geometry, and describe the evidence in support of it, which is by now substantial. However a conceptual explanation for Mackey's rigidity phenomenon remains elusive.

• Date: 9/14/12
  • Natasha Blitvic, Vanderbilt University
  • Title: Two-parameter non-commutative Gaussian processes, continued
  • Abstract: see talk from 9/7/12.

• Date: 9/21/12
  • Michael Brandenbursky, Vanderbilt University
  • Title: Link and braid invariants via counting surfaces
  • Abstract: A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this talk I will present simple formulas for an infinite family of invariants in terms of counting surfaces of certain genus and number of boundary components in a Gauss diagram associated with link/closed braid. I will identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.

• Date: 9/28/12
  • No Meeting.

• Date: 10/5/12
  • No Meeting, Fall Break.
• Date: 10/12/12
  • Nicolas Monod, EPFL Lausanne
  • Title: Non-amenable free group free groups

• Date: 10/19/12
  • Michael Brandenbursky, Vanderbilt University
  • Title: Link and braid invariants via counting surfaces, continued
  • Abstract: see talk from 9/21/12.

• Date: 10/26/12
  • Kate Juschenko, Vanderbilt University
  • Title: Small spectral radius and percolation constants on non-amenable Cayley graphs
  • Abstract: Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $\Gamma$, does there exist a generating set $S$ such that the Cayley graph $(\Gamma,S)$, without loops and multiple edges, has non-unique percolation, i.e., $p_c(\Gamma,S) < p_u(\Gamma,S)$? We show that this is true if $\Gamma$ contains an infinite normal subgroup $N$ such that $\Gamma/ N$ is non-amenable. Moreover for any finitely generated group $G$ containing $\Gamma$ there exists a generating set $S'$ of $G$ such that $p_c(G,S') < p_u(G,S')$. In particular this applies to free Burnside groups $B(n,p)$ with $n \geq 2, p \geq 665$. We also explore how various non-amenability numerics, such as the isoperimetric constant and the spectral radius, behave on various growing generating sets in the group. Some application of the above results to group $C^*$-algebras will be given. This is joint with T. Nagnibeda-Smirnova.

• Date: 11/2/12
  • Feng Xu, UC Riverside
  • Title: On questions about intermediate subfactors
  • Abstract: In this talk I will describe a few questions about intermediate subfactors motivated by the theory of finite groups, and report on some recent progress related to Hopf algebras, fusion categories, quantum groups and conformal field theory. I will also discuss similar problems in Vertex Algebras and recent results which are joint work with V. Kac, P. Moseneder and P. Papi.

• Date: 11/9/12
  • Darren Creutz, Vanderbilt
  • Title: Poisson Boundaries, Harmonic Functions and Random Walks on Groups
  • Abstract: I will present the construction of the Poisson Boundary of a group, originally defined by Furstenberg, and explain its various properties and applications. The Poisson Boundary can be thought of as the exit boundary of a random walk on the group and can be identified with the space of harmonic functions on the group. The first talk will focus on the construction of the Poisson Boundary and various results due primarily to Furstenberg and Zimmer about boundaries. The second talk will focus on the dynamical behavior of the boundary and its applications to ergodic theory.

• Date: Tuesday, 11/13/12, 4pm, SC 1431
  • Scott Morrison, ANU Canberra
  • Title: Webs and skew Howe duality
  • Abstract: A ``pictures mod relations'' presentation of the representation theory of SL(n).
    The representation category of SL(n) is a pivotal tensor category. This means that one can draw planar diagrams representing morphisms, with composition corresponding to vertical stacking, and tensor products corresponding to horizontal juxtaposition. Any planar isotopy of such a diagram gives equations between the corresponding morphisms. For any such category, we'd like to be able to give a presentation via certain generators modulo local relations. For Rep(SL(n)), we've had a conjectural presentation for several years, but no good tools for showing that we have all the relations. With Sabin Cautis and Joel Kamnitzer, we now have not only a proof that this presentation is correct, but also a clear conceptual explanation of how the relations arise. This explanation uses skew Howe duality.

• Date: 11/23/12
  • No Meeting, Thanksgiving Break.

• Date: 11/30/12
  • No Meeting.

• Date: Wednesday, 12/5/12, SC 1432
  • Darren Creutz, Vanderbilt
  • Title: Poisson Boundaries, Harmonic Functions and Random Walks on Groups (continued)
  • Abstract: See abstract from 11/9.

• End of Fall Semester.

• Date: Friday, 1/25/13,
  • Darren Creutz, Vanderbilt
  • Title: Mixing on Rank-One Transformations
  • Abstract: In this talk on a more classical part of ergodic theory, that of Z-actions, I will explain the construction of rank-one transformations via cutting and stacking that goes back to von Neumann and Kakutani and has been used to create examples and counterexamples of various mixing-like properties. Following the explanation of the subject, I will present some of my work on when such transformations are mixing. Some of the results presented are joint work with Cesar Silva.

• Date: Friday, 2/1/13,
  • Kostya Medynets, US Naval Academy
  • Title: Finite Factor Representations of Higman-Thompson groups.
  • Abstract: We will talk about relations between ergodic properties of group actions and the structure of group characters. The latter is equivalent to the classification of all finite-type factor representations. The outstanding conjecture (often attributed to Vershik) is that for a large class of groups their group characters must have the form \mu(Fix(g)), where $\mu$ is a G-invariant measure for some special group action on a measure space, Fix(g) is the set of all fixed points of group element $g$. We will then establish Vershik's conjecture for the family of Higman-Thompson groups. Since these groups have no non-trivial ergodic measures, we get that they have no non-trivial factor representations. Examples of other classes of groups known to satisfy Vershik's conjecture will be also discussed. The talk will be based on two recent preprints by Dudko and Medynets, "Finite factor representations of Higman-Thompson Groups" ArXiv 1212.1230 and "On characters of inductive limits of symmetric groups" Arxiv 1105.6325.

• Date: Friday, 2/15/13,
  • Robin Tucker-Drob, Caltech
  • Title: Expressions of non-amenability in ergodic theory
  • Abstract: In this talk I will discuss how strong forms of non-amenability are reflected in the asymptotic behavior of a group's Bernoulli action. Central to the discussion will be the notion of shift-minimality: A countable group G is called shift-minimal if every non-trivial measure preserving action weakly contained in the Bernoulli shift of G is free. I will discuss the connection between shift-minimality and certain properties of the reduced C*-algebra of G, and indicate the proof that if G admits a free pmp action of cost >1 then there is a finite normal subgroup N of G such that G/N is shift-minimal.

• Date: Friday, 2/22/13,
  • Ben Hayes, UCLA
  • Title: Extended von Neumann Dimension for Representations of Equivalence Relations
  • Abstract: In past work, we define a notion of l^{p}-Dimension for uniformly bounded Banach space representations of a sofic group. This dimension is equal to the von Neuamn dimensnion, when H is a unitary representation of G contained in a multiple of the left-regular representation. We also computed this dimension for central natural representations of a sofic group, including direct sums of the translation action on l^{p}(G), and the multiplication action on L^{p}(L(G)). In this work, we shall explain how to define this notion of l^{p}-Dimension for representations of a sofic equivalence relation. When this equivalence relation satisfies a certain "finite presentation" assumption, we define an analogue of l^{2}-Betti numbers (or really l^{2}-Betti number +1) in the l^{p}-case. We can then connect some natural questions about this dimension with the cost versus l^{2}-Betti number conjecture.

• Date: Friday, 3/1/13,
  • Yves de Cornulier, Universite Paris-Sud 11
  • Title: Commensurating actions and Property FW
  • Abstract: A group has Property FW if every action on a set commensurating a subset fixes a subset at bounded distance. This is a combinatorial weakening of Kazhdan Property T (and strengthening of Serre's Property FA), which was characterized in a similar (measurable) fashion by Robertson and Steger. I will discuss Property FW in various contexts, and notably for lattices in Lie groups.

• Date: Friday, 3/15/13, (joint with the geometry seminar)
  • Kamran Reihani, Northern Arizona University
  • Title: Noncommutative Metrics for Dynamical Systems
  • Abstract: Spectral triple is the fundamental object of the metric aspects of Connes' noncommutative geometry. A spectral metric space is a spectral triple (A, H, D) with additional properties guaranteeing that the Connes metric on the state space of A induces the weak*-topology. It is, in fact, the noncommutative analog of a complete metric space. Let (A,H,D) be a spectral metric space and G be a group of automorphisms of A. In this talk I will consider the problem of whether there is a natural spectral triple for the crossed product algebra C*(G,A) that can characterize the metric properties of the dynamical system (G,A). I will discuss a solution to this problem when a single automorphism of A generates G as an equicontinuous family of quasi-isometries. I will also address the converse problem, namely, when a spectral metric space for the crossed product gives rise to one for A. When the action is not equicontinuous (e.g., when the action is uniformly hyperbolic), following the philosophy of Diffeomorphism-Invariant Geometry of Connes and Moscovici, we suggest replacing the dynamical system (G,A) by a dynamical system (G,B), where G acts isometrically. The algebra B is called the metric bundle associated with (G,A). Some candidates for the metric bundle B will be introduced. This talk is based on a joint work with Jean Bellissard and Matilde Marcolli.

• Date: Friday, 3/29/13,
  • Marston Conder, University of Auckland, New Zealand
  • Title: The Intersection Condition for regular polytopes
  • Abstract: An abstract polytope is a generalised form of a geometric polytope, and may be viewed as a partially-ordered set (endowed with a rank function) that satisfies certain properties motivated by the geometry. A polytope is called `regular' if its automorphism group is transitive (and hence sharply-transitive) on the set of all flags -- which are the maximal chains in the poset. The automorphism group of a regular polytope is a smooth quotient of a 'string' Coxeter group (with a linear Dynkin diagram). Conversely, any finite smooth quotient of such a group is the automorphism group of a regular polytope, provided that it satisfies a condition known as the `intersection condition'. In this talk I will explain these things, and describe some recent discoveries about the intersection condition, including its application to find the smallest regular polytopes of any given rank.

• Date: Friday, 4/5/13,
  • Darren Creutz, Vanderbilt University
  • Title: Stabilizers of Actions of Product Groups and Lattices in Product Groups
  • Abstract: I will present my recent work on the stabilizers of actions of products of groups and irreducible lattices in products. The main results are a classification of all possible stabilizer groups for actions of products of Howe-Moore groups, at least one of which has (T), and a classification statement for actions of lattices in such products. In contrast to previous work (joint with J. Peterson) on stabilizers, the approach taken here does not involve writing lattices as commensurators and therefore applies even in the case when neither of the ambient groups are totally disconnected and in this sense complement the previous work.

• Date: Friday, 4/12/13,
  • Jesse Peterson, Vanderbilt University
  • Title: Non-commutative boundaries and characters on lattices and commensurators
  • Abstract: A character on a group is a function of positive type which is invariant under conjugation. The study of characters on group was initiated with the work of Thoma, and is closely connected to the study of II_1 factors, especially in the presence of rigidity phenomena. I will discuss recent joint work with Darren Creutz, where we investigate the classification of characters on lattices and commensurators in semi-simple groups via non-commutative Poisson boundaries of II_1 factors.

• Date: Friday, 4/19/13,
  • Jesse Peterson, Vanderbilt University
  • Title: Non-commutative boundaries and characters on lattices and commensurators, part 2
  • See abstract from 4/12.

• Date: Friday, 4/26/13,
  • Simon Thomas, Rutgers University
  • Title:
  • Abstract:

Past Subfactor and NCGOA seminars