Mathematical Physics -Physical Mathematics
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Upcoming seminars in 2022
Tuesday 1 February 4.30 pm UK time
Claus Koestler (Cork)
Spreadability and Partial Spreadability Slides
Distributional symmetries and invariance principles provide deep structural results in classical probability, also known as de Finetti theorems. I will introduce to some recent developments in the transfer of these principles to noncommutative probability.
First, I will discuss spreadability of an infinite sequence of noncommutative random variables. This property is about the invariance of distributions when passing from the given sequence to a subsequence. It may be regarded to be the fundamental distributional invariance principle from the viewpoint of algebraic homology as it emerges via a functor from the semisimplicial category into a category of noncommutative probability spaces.
Furthermore, my talk will address partial spreadability, a recently introduced generalization of spreadability. This invariance principle provides a connection between certain representations of the Thompson monoid F^+ and Markovianity in noncommutative probability.
Finally, as time permits, I will address some open problems when applying these two invariance principles to Jones-Temperley-Lieb algebras.
This talk is based on joint work with Gwion Evans, Rolf Gohm, Arundhathi Krishnan and Steven Wills.
- D. Gwion Evans, Rolf Gohm, Claus Köstler. Semi-cosimplicial objects and spreadability. Rocky Mountain J. Math. 47 (6) 1839 – 1873, 2017.
- Claus Köstler, Arundhathi Krishnan, Stephen J. Wills. Markovianity and the Thompson Monoid $F^+$, arXiv:2009.14811.
Tuesday 8 February 4.30 pm UK time
Guoliang Yu (Texas A&M)
A new index theory for noncompact manifolds and Gromov’s compactness conjecture
I will introduce a new index theory for noncompact manifolds based on my joint work with Stanley Chang and Shmuel Weinberger. This index theory encodes dynamics of the fundamental groups at infinity. I will then discuss my recent work with Shmuel Weinberger and Zhizhang Xie answering Gromov’s compactness question on scalar curvature using this index theory.
Tuesday 15 February 4.30 pm UK time
Jesse Peterson (Vanderbilt)
Properly proximal von Neumann Algebras
Properly proximal groups were introduced recently by Boutonnet, Ioana, and the speaker, where they generalized several rigidity results to the setting of higher-rank groups. In this talk, I will describe how the notion of proper proximality fits naturally in the realm of von Neumann algebras. I will also describe several applications, including that the group von Neumann algebra of a non-amenable inner-amenable group cannot embed into a free group factor, which solves a problem of Popa. This is joint work with Changying Ding and Srivatsav Kunnawalkam Elayavalli.
Tuesday 22 February 4.30 pm UK time
Alexandru Chirvasitu (Buffalo)
Flavors of rigidity
I will discuss a number of results which, though to my knowledge not mutually related in any direct manner, nevertheless have a recognizably common flavor: “most” objects of such-and-such a type have “few” symmetries. Examples abound; the objects in question might be
– finite graphs, where “most” is interpreted probabilistically and asymptotically as the graphs grow;
– quantum graphs (i.e. appropriately well-behaved subspaces of matrix algebras), where “most” means “along a Zariski-dense subset”;
– measured metric spaces, with “most” = “over a residual set in the measured Gromov-Hausdorff topology”;
– Riemannian manifolds, “most” meaning “over an open dense set in the smooth topology”.
“Few symmetries” is also subject to rich interpretation: it might mean a trivial automorphism group, or a quantum-group version thereof, or that the automorphism group can be prescribed beforehand.
(partly joint w/ Mateusz Wasilewski)
Tuesday 1 March 4.30 pm UK time
Amanda Young (TU Munich)
A bulk gap in the presence of edge states for a Haldane pseudopotential
In this talk, we discuss a recent result on a bulk gap for a truncated Haldane pseudopotential with maximal half filling, which describes a strongly correlated system of spinless bosons in a cylinder geometry. For this Hamiltonian with either open or periodic boundary conditions, we prove a spectral gap above the highly degenerate ground-state space which is uniform in the volume and particle number. Our proofs rely on identifying invariant subspaces to which we apply gap-estimate methods previously developed only for quantum spin Hamiltonians. In the case of open boundary conditions, the lower bound on the spectral gap accurately reflects the presence of edge states, which do not persist into the bulk. Customizing the gap technique to the invariant subspace, we avoid the edge states and establish a more precise estimate on the bulk gap in the case of periodic boundary conditions. The same approach can also be applied to prove a bulk gap for the analogously truncated 1/3-filled Haldane pseudopotential for the fractional quantum Hall effect. Based off joint work with S. Warzel.
Tuesday 8 March 4.30 pm UK time
David Reutter (Bonn)
Fusion 2-categories, their Drinfeld centers, and the minimal modular extension conjecture
A modular tensor category is a ribbon category without any non-trivial transparent object, while a super-modular category is a ribbon category with a single transparent fermion. In this talk, I will sketch a proof of the “minimal modular extension conjecture” stating that any super-modular category admits an index-2 extension to a modular category. Along the way, I will introduce various key players of this proof, such as fusion 2-categories and their Drinfeld centers.
This is based on arXiv:2105.15167 and is joint work with Theo Johnson-Freyd.
Tuesday 15 March 4.30 pm UK time
Elizabeth Gillaspy (Montana)
K-theory for real k-graph C*-algebras
Purely infinite simple real C*-algebras, like their complex counterparts, are classified by their K-theory. Indeed, there are purely infinite simple real C*-algebras (e.g. the exotic Cuntz algebra E_n) whose existence is only known thanks to K-theory computations. Our long-term goal, in this joint research project with Jeff Boersema, is to construct more concrete models for such C*-algebras. We begin by showing how k-graphs, or higher-rank graphs (which are a higher-dimensional generalization of directed graphs), can give rise to purely infinite simple real C*-algebras. To evaluate whether this class of real k-graph C*-algebras includes E_n, we need to compute the K-theory of real k-graph C*-algebras. To that end, we adapt the spectral sequence studied by D.G. Evans, which converges to the K-theory of a complex k-graph C*-algebra, to the setting of real C*-algebras. Using this spectral sequence, we compute K-theory for several examples of real k-graph C*-algebras.
This is joint work with Jeff Boersema.
Tuesday 22 March 4.30 pm UK time
Alexis Virelizier (Lille)
State sum invariants of homotopy classes of maps Slides
Homotopy quantum field theories (HQFTs) generalize topological quantum field theories (TQFTs) by replacing manifolds by maps from manifolds to a fixed target space X. In particular, such an HQFT associates a scalar invariant under homotopies to each map from a closed manifold to X. In this talk, I will explain how to generalize the state sum Turaev-Viro-Barett-Westburry TQFT to an HQFT with target X in the following two cases. First when X is a 1-type using fusion categories graded by a group (joint work with Vladimir Turaev). Second when X is a 2-type using fusion categories graded by a crossed module (joint work with Kursat Sozer).
Tuesday 10 May 4.30 pm UK time
Bethany Marsh (Leeds)
An introduction to tau-exceptional sequences.
Joint work with Aslak Bakke Buan (NTNU).
We introduce the notion of a tau-exceptional sequence for a finite dimensional algebra, which can be regarded as the generalisation of a classical exceptional sequence considered in the hereditary case. The new sequences behave well for both non-hereditary and hereditary algebras. The work is motivated by the signed exceptional sequences introduced, in the hereditary case, by Igusa-Torodov, and by tau-tilting theory.
We also introduce a notion of a signed tau-exceptional sequence and show that there is a bijection between the set of complete signed tau-exceptional sequences and ordered basic support tau-tilting objects.
Tuesday 17 May 4.30 pm UK time
Jamie Vicary (Cambridge)
Higher categories and quantum computation Slides
I will show how some fundamental computational processes, including encrypted communication and quantum teleportation, can be defined in terms of the higher representation theory of defects between 2d topological cobordisms, giving insight into fundamental questions in classical and quantum computation. Everything will be explained from first principles, with lots of pictures!
Tuesday 24 May
No seminar due to annual Colloquium
Tuesday 31 May 4.30 pm UK time
George Elliott (Toronto)
Classification of C*-algebras – simple vs. non-simple
Well-behaved simple C*-algebras (very simple axioms) have now been classified. This now leaves the less well behaved simple case (some evidence!), and the non-simple case, of course well behaved to begin with! There are already a number of results in the well-behaved non-simple case.
Tuesday 7 June 4.30 pm UK time
Hannes Thiel (Kiel)
Are C*-algebras determined by their linear and orthogonality structure?
It is well-known that every C*-algebra is determined by its linear and multiplicative structure: Two C*-algebras are *-isomorphic if and only if they admit a multiplicative, linear bijection. We study if instead of the whole multiplicative structure it suffices to record when two elements have zero product. While it is not clear if every C*-algebra is determined this way, we obtain many positive results. In particular, two unital, simple C*-algebras are *-isomorphic if and only if they admit a linear bijection that preserves zero products.
This is joint work with Eusebio Gardella.
Tuesday 4 October 4.30 pm UK time
Christopher Schafhauser (Lincoln, Nebraska)
Subalgebras of simple AF-algebras
A long-standing open question, formalized by Blackadar and Kirchberg in the mid ‘90s, asks for an abstract characterization of C*-subalgebras of AF-algebras. I will discuss some recent progress on this question: every separable, exact C*-algebra which satisfies the UCT and admits a faithful, amenable trace embeds into an AF-algebra. Moreover, the AF-algebra may be chosen to be simple and unital with unique trace and the embedding may be taken to be trace-preserving. Modulo the UCT, this characterizes C*-subalgebras of simple, unital AF-algebras. As an application, for any countable, discrete, amenable group G, the reduced C*-algebra of G embeds into a UHF-algebra. I’ll also discuss how these techniques have played a role in a new proof to the classification theorem of separable, simple, nuclear, Z-stable, UCT C*-algebras, which is based on joint work with Jos\’e Carri\’on, James Gabe, Aaron Tikuisis, and Stuart White.
Tuesday 11 October 4.30 pm UK time
Palle Jorgensen (Iowa)
The talk aims to extend tools from reflection positivity (RP) to non-commutative harmonic analysis, spectral theory, and new duality theory for unitary representations of Lie groups.
In its original form, reflection positivity (RP), also referred to as Osterwalder-Schrader positivity, came to serve as a crucial link between problems in quantum physics and in mathematics. More precisely, the original variant of RP serves to link Euclidean field theory (math) to relativistic quantum field theory (physics). Thereby linking difficult questions for relativistic quantum fields, on one side, with, on the other side, Euclidean fields. Tools and solutions available on one side have often yielded insights on the other side. The RP-axioms link the two. The basic symmetry groups are different, i.e., the Poincare group vs the Euclidian group. Hence the associated harmonic analysis and spectral theory are different of course. In recent work by the speaker and multiple co-authors have aimed to expand the framework of RP to the theory of representations of Lie group, spectral theory, and operator algebras. Among other things, this viewpoint yields insight into the role of the Markov property, as opposed to Osterwalder-Schrader positivity.
As a general principle, the Reflection positivity (RF) correspondence has proved useful in mathematics and in many neighboring areas. In its original form, RP successfully combines powerful tools from analysis, from geometry, from representation theory to questions in quantum physics. For example, due to work by many authors, the RF-correspondence has served to link abelian (commutative) properties of Gaussian processes/fields in the Euclidean setting, to the context of non-commutativity in the study of quantum fields. And by now, RP has further become a powerful tool in non-commutative harmonic analysis, and in the theory of unitary representations of Lie groups.
Tuesday 18 October 4.30 pm UK time
Sahand Seifnashri (IAS, Princeton)
Non-Invertible Symmetries and their Representations
I will discuss generalized symmetries in two-dimensional QFTs that can be non-invertible. Such finite non-invertible symmetries are described by a fusion category rather than a finite group. Using the results in the mathematical physics literature, we find the appropriate notion of representation for non-invertible symmetries needed to derive selection rules on correlation functions. We use this understanding to derive a Cardy-like formula for 2d CFTs with a finite non-invertible symmetry. More specifically, we derive a universal formula for the asymptotic density of states transforming in an irreducible representation of a fusion category symmetry.
Based on arXiv:2208.05495 and in collaboration with Ying-Hsuan Lin, Masaki Okada, and Yuji Tachikawa.
Tuesday 1 November 4.30 pm UK time
William Slofstra (Waterloo)
Bell inequalities and decision problems in C*-algebras
Bell inequalities have been in the news lately with the recent physics Nobel prize. On the mathematics side, given a Bell inequality, we’d like to be able to determine the maximum possible violation allowed in quantum mechanics. This simple problem has led to a lot of developments in the last few years. In particular, Ji, Natarajan, Vidick, Wright, and Yuen have shown that it’s impossible to calculate this violation even approximately, thereby resolving the Connes embedding problem in operator algebras. In this talk, I’ll give an overview of these developments, starting with physics, but with the aim of showing that there are lots of other interesting decision problems in operator algebras. I’ll include work with Honghao Fu and Carl Miller on the membership problem for quantum correlations, and work in progress with Arthur Mehta and Yuming Zhao on deciding positivity.
Tuesday 8 November 4.30 pm UK time
Gabor Szabo (KU Leuven)
The dynamical Kirchberg-Phillips theorem
In this talk I will present the main results of a joint work with James Gabe: Given a countable discrete group G, two amenable and outer G-actions on stable Kirchberg algebras are cocycle conjugate precisely when they are equivariantly KK-equivalent. In fact we have a suitable version of such a theorem for actions of locally compact groups. After discussing some of the core concepts, I will state the classification results and, depending on the available time, discuss some interesting corollaries.
Tuesday 15 November 4.30 pm UK time
Tatiana Gateva-Ivanova (American University in Bulgaria)
Set-theoretic solutions of the Yang-Baxter equation and their quadratic algebras
Tuesday 22 November 4.30 pm UK time
Gregory Moore (Rutgers)
Summing over bordisms in TQFT slides
I will review the contents of the paper I wrote with Anindya Banerjee 2201.00903,
including a few updates. We consider a new construction in TQFT which was motivated by recent developments in the quantum gravity community. Specifically, we aim to give mathematical precision to a model proposed by D. Marolf and H. Maxfield 2002.08950. Given a TQFT one can consider – formally – the sum over all the amplitudes associated to all bordisms between fixed ingoing and outgoing spatial slices. We discuss when this construction makes sense and a curious splitting property satisfied by the total amplitude. The discussion of the splitting property has been significantly improved by an important remark by Daniel Friedan.
There is a youtube channel for some earlier talks from this link.
Edwin Beggs, David Evans, Gwion Evans, Rolf Gohm, Tim Porter
MPPM Seminars in 2020
MPPM Seminars in 2021
There is a youtube channel for some earlier talks from this link.