Organizers: Dietmar Bisch, Marcel Bischoff, Arnaud Brothier and Ben Hayes
Fridays, 4:10-5:30pm in SC 1432
I will discuss part of ongoing joint work with Lewis Bowen and Brandon Seward. In it, we investigate structural properties of Pinsker factors for sofic entropy, as well as Rohklin entropy. The Pinsker factor is the largest zero entropy. In the nonamenable case, there is a modification of the Pinkser factor called the Outer Pinsker factor, the Outer Pinsker factor defined via extension entropy. Extension entropy turns out to fix many “pathological” monotonicity properties of entropy (namely increase of entropy under factors). We prove useful structural properties for Outer Pinsker factors. These properties imply that certain algebraic actions have completely positive entropy (which means that every factor has positive entropy). In the sofic case, our methods rely on a new property of actions called “the independent microstate lifting property” and in the Rokhlin entropy case they rely on the actions being weakly contained in Bernoulli shifts.
Furstenberg entropy is a numerical measure of how far a quasi-invariant action of a group on a probability space is from measure-preserving. The main new tool I introduce is a relative version of this entropy measuring how far a homomorphism between such spaces is from being relatively measure-preserving.
I show that it enjoys the properties one would expect such as additivity over compositions and apply this notion to develop an Òinformation theoryÓ of joinings proving analogues of many of the key theorems in the information theory of random variables.
TBD
A handful of speakers in the Subfactor research group will give short presentations on their recent work.
Is there a commutative algebra of dimension 3+\sqrt3? The answer turns out to be yes, if we replace the category of vector spaces by a braided monoidal category. Commutative algebras in braided monoidal categories describe many phenomena including quantum subgroups, braided subfactors of type I, module tensor categories, extensions in chiral conformal field theory and gapped domain walls of topological phases.
I will review the structure of commutative algebras in (unitary) braided fusion categories, give examples, and discuss relations to the Longo-Rehren inclusions.
Ultimatively, I want to introduce a notion and give a classification of certain finite (quantum) actions by Markov maps on noncommutative probability spaces with applications to chiral conformal nets. This gives a generalization of fixed points by group symmetries to what might be called quantum symmetries.
TBD
After defining a canonical “free graph algebra” associated to a weighted graph, I will explain how certain loop algebras from compact quantum metric spaces in the sense of Rieffel. I will then explain which convergence properties hold, and which properties “should” hold. This is joint work with Dave Penneys.
I will talk about the results on unique maximal extensions of certain subalgebras in a free group factor.
Because we are interested in von Neumann algebras, we work in the
ambient category B=Bim(R), the tensor category of bimodules over a
hyperfinite von Neumann factor R, which can be thought of as a
categorification of B(H). Given a unitary fusion category C inside
Bim(R), we identify its bicommutant C”, and we show that C” is a
bicommutant category. This categorifies the theorem by which a finite
dimensional *-algebra that can be faithfully represented on a Hilbert
space is actually a von Neumann algebra.
We will also discuss applications, including machinery to construct
elements of C’, the quantum double subfactor, and commutants of
multifusion categories.
We will discuss quantum isometry groups of various algebraic structures arising in quantum algebra. We then describe a general class of new examples of algebraic structures which admit a quantum isometry groups.