Mathematical Physics -Physical Mathematics
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Tuesday 26th May 4.30 pm UK time
Tobias Osborne (Hannover)
The search for a Haagerup CFT: a microscopic approach
A fascinating open problem is to determine if, corresponding to every subfactor, there is a counterpart conformal field theory (CFT). There is
already some promising evidence for this conjecture, with considerable focus falling on the Haagerup-type subfactors, for which there are currently no known counterpart CFTs. Since subfactors give rise to unitary fusion categories with algebra object, one can imagine attempting to construct counterpart CFTs via physical models built directly arising from these categories. In this talk, I report on progress generalising due to F. Feiguin, S. Trebst, Z. Wang, M. Freedman, A. A. W. Ludwig, and A. Kitaev where this technique was successfully applied to a simple example, namely Fibonacci anyons. I will employ this approach to build microscopic models of CFTs from fusion category data via such anyon chains. Such a model may be formulated for the H3 fusion category (corresponding to the Haagerup subfactor). Furthermore, I explain how it can be used to search for a critical model constructed from fusion categories that correspond to the Haagerup subfactor, and report on several numerical investigations we have done in this direction. I will report on the methods used to study such models and our (so far negative) progress in extracting a nontrivial CFT from the Haagerup chain.
Tuesday 2nd June 4.30 pm UK time
Stefan Hollands (Leipzig)
Physical operations on quantum states correspond to channels, i.e. completely positive maps.Such operations are typically not invertible. Given that a state having gone through a channel cannot be completely recovered, it is an important question — both theoretically but alsofor practical purposes such as quantum error correction — under what circumstances the state can perhaps berecovered with a high fidelity, and how. As is well known, channels may only reduce the relative entropybetween the given state and some reference state, a fact expressed by the famous “data processing inequality”.In this talk, I present a strengthened version of this inequality for arbitrary channels on v. Neumann algebras and explain how this inequality characterizes the efficiency of state recovery. (Based on joint work with T. Faulkner.)
Tuesday 9th June 11.00 am UK time
Zhengfeng Ji (UTS, Sydney)
A complexity-theoretic solution to Connes’ Embedding Problem
This talk aims to introduce how ideas and techniques from quantum information theory and complexity theory help to resolve Tsirelson’s problem in physics and Connes’ embedding problem in mathematics. In this complexity-theoretic approach, the central problem is to understand the complexity of approximating the entangled value of nonlocal games, a model used to study constraint satisfaction problems and interactive proofs in computer science and Bell inequalities in quantum mechanics. The problem is shown to be as hard as the Halting problem and this implies a negative answer to Tsirelson’s problem via known connections. At the core of the proof is a recursive gap-preserving compression lemma, which in turn leverages many recent ideas from the study of nonlocal games. I will not assume any background in quantum information or complexity theory in this talk.
Tuesday 16th June 4.30 pm UK time
Ilijas Farah (York, Ontario)
Tuesday 23rd June 10.00 am. UK time
Gus Lehrer (Sydney)
The second fundamental theorem of invariant theory for the orthosymplectic and periplectic groups
Tuesday 30th June 4.30 pm UK time
Constantin Teleman (Berkeley)
Tuesday 7th July 10.00 am UK time
James Tener (ANU, Canberra)
Tuesday 14th July 4.30 pm UK time
Mike Hartglass (Santa Clara)
Tuesday 28th April 4.00 pm UK time
Paul Fendley (All Souls and Physics, Oxford)
Statistical Mechanics and Fusion Categories
Lattice models of statistical mechanics have a deep relation with tensor categories. Most of the central examples, e.g. the Temperley-Lieb algebra, were understood in the lattice-model context before they arose in topological field theory. I give an overview and several useful consequences of these connections. I explain how lattice topological defects can be constructed using category data, and show how they point to a new way of understanding integrability. I will also explain quickly how to define a topological Tutte polynomial, naturally extending the Potts model to the torus.
Tuesday 5th May 4.00 pm UK time
Mikael Rordam (Copenhagen)
Around the Connes Embedding Problem: from operator algebras to groups and quantum information theory
In his famous classification paper from 1976, Alain Connes suggested that it “ought to be true” that all finite von Neumann algebras admit an approximate embedding into the so-called hyperfinite type II_1 factor R. One can replace the hyperfinite II_1 factor by a matrix algebra, and so the Connes Embedding Problem asks if any von Neumann algebra (or C*-algebra) with a tracial state can be approximated by matrix algebras, with respect to the norm arising from the trace. The depth of this problem is witnessed by its many reformulations (and applications) in different areas of mathematics, including in group theory (Are all groups hyperlinear?, or sofic?) and later, via deep theorems of Kirchberg, in quantum information theory (in the form of Tsirelson’s conjecture). Very recently, a negative answer to the Connes Embedding Problem has been announced by Ji, Natarajan, Vidick, Wright and Yuen in their 165 pages-long paper titled MIP*=RE, using quantum complexity theory.
In this overview talk, I will explain some of the several facets of the Connes Embedding Problem, with particular emphasis on the related interplay between operator algebras and quantum information theory.
Tuesday 12th May 4.00 pm UK time
Roland Speicher (Saarbrücken)
Random Matrices and Their Limits
The free probability perspective on random matrices is that the large size limit of random matrices is given by some (usually interesting) operators on Hilbert spaces and corresponding operator algebras. The prototypical example for this is that independent GUE random matrices converge to free semicircular operators, which generate the free group von Neumann algebra. The usual convergence in distribution has been strengthened in recent years to a strong convergence, also taking operator norms into account. All this is on the level of polynomials. In my talk I will recall this and then go over from polynomials to rational functions (in non-commuting variables). Unbounded operators will also play a prominent role. In particular, we will try to characterize the
division closure of polynomials of our operators in the algebra of affiliated unbounded operators. This is joint work with Tobias Mai and Sheng Yin.
Tuesday 19th May 4.30 UK time
Makoto Yamashita (Oslo)
Categorical quantization of symmetric spaces and reflection equation
Reflection equation is a powerful guiding principle to quantize Poisson homogenous spaces into actions of quantum groups. I will explain a universality property of module categories from cyclotomic Knizhnik-Zamolodchikov equations in the formal `multiplier’ algebra setting, where the reflection operator becomes a complete invariant of categorical structure. As an application we obtain a Kohno-Drinfeld type result comparing the type B braid group representations from the KZ equations on the one hand, and the universal R-matrix and the universal K-matrix of Balagovic and Kolb on the other. The proof relies on elementary but curious combination of Lie algebra cohomology, Hochschild cohomology, and formality machinary from noncommutative geometry inspired by works of Calaque and Brochier. Based on joint works with Kenny De Commer, Sergey Neshveyv, and Lars Tuset.
Slides are here