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The cobordism hypothesis gives a functorial bijection between oriented 

n-dimensional fully local topological field theories, valued in some 

higher category C, and the fully dualizable object of C equipped with 

the structure of SO(n)-fixed point.  In this talk I'll explain recent 

works of Haugseng, Johnson-Freyd and Scheimbauer which construct a 

Morita 4-category of braided tensor categories, and I'll report on joint 

work with Brochier and Snyder which identifies two natural subcategories 

therein which are 3- and 4-dualizable.  These are the rigid braided 

tensor categories with enough compact projectives, and the braided 

fusion categories, respectively.  I'll also explain work in progress by 

us to construct SO(3)- and SO(4)-fixed point structures in each case, 

starting from ribbon and pre-modular categories, respectively.

Applying the cobordism hypothesis, we obtain 3- and 4-dimensional fully 

local TFT's, which extend the 2-dimensional TFT's we constructed with 

Ben-Zvi and Brochier, and which conjecturally relate to a number of 

constructions in the literature, including: skein modules, quantum 

A-polynomials, Crane-Kauffmann-Yetter invariants; hence our construction 

puts these on firm foundational grounds as fully local TFT's.  A key 

feature of our construction in dimension 3 is that we require the input 

braided tensor category neither to be finite, nor semi-simple, so this 

opens up new examples -- such as non-modularized quantum groups at roots 

of unity -- which were not obtainable by earlier methods.

Port-based teleportation (PBT) is a variant of the well-known task of quantum teleportation in which Alice and Bob share multiple entangled states called "ports". While in the standard teleportation protocol using a single entangled state the receiver Bob has to apply a non-trivial correction unitary, in PBT he merely has to pick up the right quantum system at a port specified by the classical message he received from Alice. PBT has applications in instantaneous non-local computation and can be used to attack position-based quantum cryptography. Since perfect PBT protocols are impossible, there is a trade-off between error and entanglement consumption (or the number of ports), which can be analyzed using representation theory of the symmetric and unitary groups. In particular, without loss of generality the resource state can be assumed to have a “purified" Schur-Weyl duality symmetry. I will give an introduction to the task of PBT and its symmetries, and show how the asymptotics of existing formulas for the optimal performance for a given number of ports can be derived using a connection between representation theory and the Gaussian unitary ensemble.

Joint work with M. Christandl, C. Majenz, G. Smith, F. Speelman & M. Walter

In the first part of the talk, I will describe the new large N limit of tensor models, based on the “index” of graphs (in contrast to the standard large N expansion  based on the “degree”), and the associated new large D limit of matrix models. This new limit sheds an interesting light on the relation between disordered models à la SYK, tensor models and black holes. In the second part of the talk, I will apply these ideas to discuss the phase diagrams of some strongly coupled matrix quantum mechanics. The phase diagrams display many interesting features, including first and second order phase transitions and quantum critical points. Some of these phase transitions can be argued to provide a quantum mechanical description of the phenomenon of gravitational collapse.