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.