Subfactor Seminar
Spring 2022
Organizers: Dietmar Bisch, Jesse Peterson
Fridays, 4:10-5:30pm on Zoom. Zoom
Meeting ID: 943 9395 6397. Email Dietmar Bisch for the passcode.
- Date: 2/4/22
- No meeting due to faculty meeting
Date: 2/11/22
Adrian Ioana, UCSD
Title: Almost commuting matrices and stability for product groups
Abstract:
I will present a result showing that the direct product group
$G=\mathbb F_2\times\mathbb F_2$, where $\mathbb F_2$ is the free group on
two generators, is not Hilbert-Schmidt stable. This means that G admits a
sequence of asymptotic homomorphisms (with respect to the normalized
Hilbert-Schmidt norm) which are not perturbations of genuine homomorphisms.
While this result concerns unitary matrices, its proof relies on techniques
and ideas from the theory of von Neumann algebras. I will also explain how
this result can be used to settle in the negative a natural version of an
old question of Rosenthal concerning almost commuting matrices. More
precisely, we derive the existence of contraction matrices A,B such that A
almost commutes with B and B* (in the normalized Hilbert-Schmidt norm), but
there are no matrices A’,B’ close to A,B such that A’ commutes with B’ and
B’*.
Date: 2/18/22
Jesse Peterson, Vanderbilt University
Title: Properly proximal von Neumann Algebras
Abstract:
Properly proximal groups were introduced recently by Boutonnet,
Ioana, and the speaker, where they generalized several rigidity results to
the setting of higher-rank groups. In this talk, I will describe how the
notion of proper proximality fits naturally in the realm of von Neumann
algebras. I will also describe several applications, including that the
group von Neumann algebra of a non-amenable inner-amenable group cannot
embed into a free group factor, which solves a problem of Popa. This is
joint work with Changying Ding and Srivatsav Kunnawalkam Elayavalli.
Date: 2/25/22
Hans Wenzl, UCSD
Title: Categories for Quantum Group Fusion Categories
Abstract:
We present a method to construct large
classes of module categories from deformations of
embeddings of Lie groups $Sp(N)\subset SU(N)$ (N even) and
$Sp(N-1)\subset SU(N)$ (N odd). Moreover, we can give detailed
descriptions of these module categories. In particular,
if the categories are unitarizable,
we obtain finite depth subfactors whose first principal graphs
are trunctions of the induction-restriction graphs of representations
for the embeddings listed above.
Date: 3/4/22
Emily Peters, Loyola University Chicago
Title: Conway's Rational Tangles and the Thompson group
Abstract:
In the process of studying Thompson's group F (of piecewise
linear homeomorphisms from the closed unit interval [0,1] to itself,
which are differentiable except at finitely many dyadic rational
numbers), Vaughan Jones observed a map from F to knots. He proved
that every knot is in the image of this map -- that is, that every
knot can be seen as the "knot closure" of a Thompson group element.
Jones' algorithm to achieve this is rather piecemeal, and he asked if
there was a better one.
In a project with undergraduate student Ariana Grymski, we approach
this question through the lens of Conway's rational tangles. We are
able to give methods to construct any product or concatenation of
simple tangles, and we hope these are seeds for a more skein-theoretic
approach to the construction question.
Date: 3/11/22
Date: 3/18/22
Dima Shlyakhtenko, UCLA
Title: An Inequality for Non-Microstates Free Entropy Dimension for Crossed Products by Finite Abelian Groups
Abstract:
For certain generating sets of the subfactor inclusions coming from a crossed product by a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of a large class of von Neumann algebras by the direct sum of an infinite number of copies of the two-element group.
Date: 3/25/22
Special time: 3:00-5:00pm, on Zoom (same
Zoom link).
Srivatsav Kunnawalkam Elayavalli, Vanderbilt University
Title: Free entropy theory and rigid von Neumann algebras
Abstract: Thesis defense.
Date: 4/1/22
Matthew Kennedy, University of Waterloo
Title:
The ideal intersection property for essential groupoid C*-algebras
Abstract:
I will discuss recent work characterizing the ideal intersection property
for essential C*-algebras of étale groupoids with locally compact Hausdorff
space of units. For Hausdorff groupoids, this C*-algebra coincides with the
reduced C*-algebra. In the minimal case, the ideal intersection property is
equivalent to simplicity, so as a consequence we obtain a characterization
of étale groupoids that are "C*-simple.”
This is joint work with Se-Jin (Sam) Kim, Xin Li, Sven Raum and Dan Ursu.
Date: 4/8/22
This talk will be in SC 1431 (in person)!
Hui Tan, UCSD
Title: Spectral gap characterizations of property
(T) for II1 factors
Abstract:
For property (T) II1 factors, any inclusion into a tracial von
Neumann algebra has spectral gap, and therefore weak spectral gap. I will
discuss characterizations of property (T) for II1 factors by weak
spectral gap in inclusions. I will explain how this is related to the
non-weakly-mixing property of the bimodules containing almost central
vectors, from which we also obtain a characterization of property (T).
Date: 4/15/22
Quan Chen, The Ohio State University
Title: Q-system realization and applications
Abstract:
Q-systems are unitary versions of Frobenius algebra objects which
appeared in the theory of subfactors. I will discuss higher unitary
idempotent completion for C*/W* 2-categories called Q-system completion and
its inverse 2-functor called realization. We will explain several
applications of Q-system realization, including the coned construction to
describe the inductive limit II_1 factor of a Jones tower and induced
actions of unitary tensor categories on C*-algebras. Time permitting, we
will explain the relationship between the equivariant K-theory and the
K-theory of the realization.
Date: 4/19/22
Math Colloquium by Aldroubi-Azhari Prize Winner Cain Edie-Michell, 4:10pm to 5:00pm in SC 1206.
Cain Edie-Michell, UCSD
Title: Representations of Quantum Group Categories
Abstract:
Given any algebraic object, it is important to study the representations
of that object. This is particularly true for the tensor categories
constructed from quantum groups. In this setting the representations
classify certain conformal field theories, and give rise to highly non-trivial
subfactors of many Von Neumann algebras. In this talk I will present some
progress in the classification of these representations. Our results show
the existence of several infinite families, along with a handful of sporadic
examples.
Date: 4/22/22
Peter Huston, The Ohio State University
Title: Composing topological domain walls and anyon mobility
Abstract:
Topologically ordered phases of matter are a subject which can be
approached from a number of perspectives, including the perspective of
lattice models, where topological order arises from systems of local
operators on a Hilbert space. I will introduce lattice models of 2+1D
topological order and 1+1D topological boundaries between them, and outline
the relationship between these models and tensor categories which classify
their topological order. A pair of parallel topological boundaries can
decompose as a direct sum, a phenomenon which can be seen concretely in
lattice models. In recent join work with Fiona Burnell, Corey Jones,
and Dave Penneys, we have used the graphical calculus of 3-categories of
enriched fusion categories to understand how this decomposition can be
computed algebraically.
Date: 4/29/22
This talk will be in SC 1431 (in person)!
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 $\cS\leq \cR$ of
discrete, probability measure-preserving 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 $H\leq G$ of groups. 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. I will mention some analogies
and connections with operator algebras and subfactor theory, as well as some
applications.
End of Spring Semester.
Past Subfactor and NCGOA seminars
NCGOA home page
Dietmar Bisch's home page
Jesse Peterson's home page