I will discuss compressible types and relate them to uniform definability of types over finite sets (UDTFS), to uniformity of honest definitions and to the construction of compressible models in the context of (local) NIP. All notions will be defined during the talk.
Joint with Martin Bays and Pierre Simon.

The holographic entanglement entropy formula identifies the generalized entropy of the bulk AdS spacetime with the entanglement entropy of the boundary CFT. However the bulk microstate interpretation of the generalized entropy remains poorly understood. Progress along this direction requires understanding how to define Hilbert space factorization and entanglement entropy in the bulk closed string theory.   As a toy model for AdS/CFT, we study the entanglement entropy of closed strings in the topological A model, which enjoys a gauge-string duality.   We define a notion of generalized entropy for closed strings on the resolved conifold using the replica trick.   As in AdS/CFT, we find this is dual to (defect) entanglement entropy in the dual Chern Simons gauge theory.   Our main result is a bulk microstate interpretation of generalized entropy in terms of open strings and their edge modes, which we identify as entanglement branes.   

More precisely, we give a self consistent factorization of the closed string Hilbert space which introduces open string edge modes transforming under a q-deformed surface symmetry group. Compatibility with this symmetry requires a q-deformed definition of entanglement entropy. Using the topological vertex formalism, we define the Hartle Hawking state for the resolved conifold and compute its q-deformed entropy directly from the reduced density matrix.   We show that this is the same as the generalized entropy.   Finally, we relate non local aspects of our factorization map to analogous phenomenon recently found in JT gravity.

In this talk I will discuss a number of topics at the interface between C* algebras and Geometric Group Theory, with an emphasis on Kazhdan projections, various versions of amenability and their connection to the geometry of groups. This is based on joint work with P. Nowak and J. Mackay.

We showcase some newly emerging connections between the theory of random walks and operator algebras, obtained by associating concrete algebras of operators to random walks. The C*-algebras we obtain give rise to new and interesting notions of ratio limit space and boundary, which are computed by appealing to various works on the asymptotic behavior of transition probabilities for random walks. Our results are leveraged to shed light on a question of Viselter on symmetry-unique quotients of Toeplitz C*-algebras of subproduct systems arising from random walks.