Mathematical Physics -Physical Mathematics
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Tuesday 15th September 4.30 pm UK time
Barry Simon (Cal Tech)
The Tale of a Wrong Conjecture: Borg’s Theorem for Periodic Jacobi Matrices on Trees
I will begin by reviewing work on removal of eigenvalue degeneracy and its relevance to gap splitting. I’ll next discuss Borg’s theorem. I’ll then describe a framework for discussing periodic Jacobi matrices on trees and possible versions of Borg’s theorem and a recent note that there are counterexamples. Finally, I’ll discuss possible modified conjectures. This includes joint work with Nir Avni and Jonathan Breuer.
Tuesday 22nd September 4.30 pm UK time
Sergey Neshveyev (Oslo)
Dual cocycles and quantization of locally compact groups
Although the problem of quantization of Lie bialgebras in the purely algebraic (formal) setting was solved in full generality in the 1990s by Etingof and Kazhdan, the list of noncompact Lie bialgebras admitting a quantization in the analytic (operator algebraic) setting is still quite short. In my talk I will review the history of the problem and its connection to classification of group actions on von Neumann algebras. I will then consider a particular class of semidirect products. The simplest example in this class is the ax+b group over the reals. An operator algebraic quantum analogue of this group was defined by Baaj and Skandalis, and as an application of a general theory it is now possible to show that it is obtained by the dual cocycle twisting from the von Neumann algebra of the ax+b group. (Based on a joint work with Pierre Bieliavsky, Victor Gayral and Lars Tuset.)
Tuesday 29th September 4.30 pm UK time
Ezra Getzler (Northwestern)
Tuesday 6th October 4.30 pm UK time
Jon Keating (Oxford)
Tuesday 13th October 4.30 pm UK time
Inna Entova-Aizenbud (Ben Gurion)
Tuesday 20th October 11.00 am UK time
Narutaka Ozawa (RIMS, Kyoto)
Tuesday 27th October 4.30 pm UK time
Lai-Sang Young (NYU Courant)
Tuesday 3rd November 4.30 pm UK time
Richard Thomas (Imperial)
Tuesday 10th November 4.30 pm UK time
Theo Johnson-Freyd (Dalhousie and Perimeter)
Tuesday 17th November 4.30 pm UK time
Cris Negron (U of North Carolina)
Tuesday 24th November 4.30 pm UK time
Philippe Di Francesco (U of Illinois at Urbana-Champaign)
Tuesday 1st December 4.30 pm UK time
Ralf Meyer (Göttingen)
Tuesday 8th December 11.00 am UK time
Tuesday 15th December 4.30 pm UK time
Feng Xu (UC Riverside)
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
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)
Recent applications of set theory to operator algebras
Certain questions in the theory of operator algebras have been recently resolved using set theory. In this talk I’ll concentrate on representation theory of simple C*-algebras and on the structure of the Calkin algebra as well as the other coronas (i.e., outer multiplier algebras) of separable, non-unital, C*-algebras. No previous knowledge of set theory or logic will be assumed.
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)
Go and No-Go in Chern-Simons theory
The 3-dimensional Chern-Simons theory for a compact Lie group G is an intriguing topological field theory which admits a distinguished 2D boundary theory, the conformal WZW model. No topological boundary theories are known except in very special cases. (“Gapped” and “ungapped” are sometimes used in lieu of conformal and topological.) Recently, Dan Freed and myself showed that, in the setting of fully extended TQFTs, no such boundary theories can exist, unless the Chern-Simons theory is equivalent to a combinatorial state-sum theory (of Turaev-Viro type). After recalling that “no-go” result, I will outline its logical development into a “go” theorem, which constructs a universal target for Chern-Simons theories (and more general Reshetikhin-Turaev ones) in which they are all generated by an object attached to a point, whose Drinfeld center is the relevant modular tensor category.
Tuesday 7th July 10.00 am UK time
James Tener (ANU, Canberra)
Thin cobordisms and operator algebras in conformal field theory
The mathematics of conformal field theory means different things to different people. The three most common ways of studying 2d chiral conformal field theories are i) vertex aglebras, ii) nets of operator algebras, and iii) geometric field theories which assign invariants to complex cobordisms. In this talk I will present an extension of the notion of geometric field theory in which the Riemann `surfaces’ are allowed to have incoming and outgoing boundary overlap. I will also discuss how these `surfaces’ give new realizations of nets of observables, and how these ideas have been applied to answer operator algebraic questions in conformal field theory.
Tuesday 14th July 4.30 pm UK time
Mike Hartglass (Santa Clara)
Realizations of Rigid C*-tensor categories as bimodules over GJS C*-algebras
Given a countably generated rigid C*-tensor category, C, I will construct a separable, simple, unital C*-algebra B with unique trace, along with a fully faithful functor F from C into the finitely-generated projective modules over B. This is joint work with Roberto Hernandez Palomares.
Tuesday 8th September 4.30 pm UK time
Ian Charlesworth (Berkeley)
Free Stein Dimension
Regularity questions in free probability ask what can be learned about a tracial von Neumann algebra from probabilistic-flavoured qualities of a set of generators. Broadly speaking there are two approaches — one based in microstates, one in free derivations — which with the failure of Connes Embedding are now known to be distinct. The non-microstates approach is not obstructed by non-embeddable variables, but can be more difficult to work with for other reasons. I will speak on recent work with Brent Nelson, where we introduce a quantity called the free Stein dimension, which measures how readily derivations may be defined on a collection of variables. I will spend some time placing it in the context of other non-microstates quantities, and sketch a proof of the exciting fact that free Stein dimension is a *-algebra invariant.