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Organizers Michael Brannan Matthew Kennedy Jesse Peterson Nico Spronk Kateryna Tatarko Andy Zucker |
| Date | Speaker, Title and Abstract |
|---|---|
| December 11 | Joaco Prandi, University of Waterloo When the weak separation condition implies the generalize finite type in R^d Let S be an iterated function system with full support. Under some restrictions on the allowable rotations, we will show that S satisfies the weak separation condition if and only if it satisfies the generalized finite-type condition. To do this, we will extend the notion of net intervals from R to R^d. If time allows, we will also use net intervals to calculate the local dimension of a self-similar measure with the finite-type condition and full support. |
December 4 | Noé de Rancourt, Université de Lille Big Ramsey degrees of metric structures Distortion problems, from Banach space geometry, ask about the possibility of distorting the norm of a Banach space in a significant way on all of its subspaces. Big Ramsey degree problems, from combinatorics, are about proving weak analogues of the infinite Ramsey theorem in structures such as hypergraphs, partially ordered sets, etc. Those two topics, coming back to the seventies, have quite disjoint motivations but share a surprisingly similar flavour. In a ongoing work with Tristan Bice, Jan Hubička and Matěj Konečný, as a step towards the unification of those two topics, we developped an analogue of big Ramsey degrees adapted to the study of metric structures (metric spaces, Banach spaces...). Our theory allows us to associate to certain metric structures a sequence of compact metric spaces quantifying their default of Ramseyness. In this talk, I'll present our theory and its motivations and illustrate it on the examples of the Banach space $\ell_\infty$ and the Urysohn sphere. If time permits, links with topological dynamics will also be discussed. |
November 27 | Guillaume Dumas, University of Maryland Boundedness of weak quasi-cocycles for higher rank simple groups If G is a second countable locally compact group, the Delorme-Guichardet theorem states that Kazhdan property (T) is equivalent to the fixed-point property for continuous affine isometric actions on Hilbert spaces—that is, every 1-cocycle with values in a Hilbert space is bounded. Many rigidity statements rely on property (T): for example, morphisms of G into R are trivial. However, it does not provide tools for studying quasi-homomorphisms, since these maps do not respect the group structure. In order to study this class of maps, Ozawa introduced wq-cocycles, which respect a cocycle identity up to a bounded error. A group is said to have property (TTT) if all wq-cocycles are bounded. In this talk, I will discuss the relationship between this property and other more analytical forms of “almost” property (T). I will also explain how to prove that a group possesses this property, with a focus on simple groups and their lattices. |
November 20 | Jennifer Zhu, University of Waterloo Limits of Quantum Graphs Quantum graphs were originally introduced as confusability graphs of quantum channels by Duan, Severini, and Winter. Weaver generalized a quantum graph to any weak-* closed operator system V in B(H) that is bimodule over the commutant of some von Neumann algebra M in B(H). To date, there seem to be two notions of quantum graph morphism. Weaver introduced and Daws extended a notion of CP morphism of quantum graphs. Musto, Reutter, and Verdon have also defined classical morphisms of quantum graphs in finite dimensions which agrees with CP morphisms in finite dimensions. Notably, however, these morphisms are not UCP maps between operator systems of the respective quantum graphs. Using a characterization of quantum relations as left ideals in the extended Haagerup tensor product, we will obtain a notion of quantum graph morphism (and hence limit) using the categories of von Neumann algebras and operator spaces. Time permitting, we will show that this limit recovers profinite classical graphs. |
November 13 | Thomas Sinclair, Purdue University Model theory of metric lattices We propose a general first-order framework for studying geometric lattices within the model theory of metric structures. As an application we develop a novel continuous limiting theory for finite partition lattices and discuss potential implications to their asymptotic combinatorics. This is joint work with Jose Contreras-Mantilla. |
November 6 | Brent Nelson, Michigan State University Closable derivations are anticoarse, of course The anticoarse space of an inclusion $N\subset M$ of tracial von Neumann algebras is an $N$-subbimodule of $L^2(M)$ whose size is sensitive to several structural properties of the inclusion. It has become a staple of so-called microstates techniques in free probability, where it allows one to parlay finite dimensional approximations into algebraic properties. On the other hand, non-microstates techniques, which exploit the regularity of certain derivations on a von Neumann algebra, have not made use of the anticoarse space, until now. In this talk, I will discuss how deformations given by closable derivations provide a natural connection to anticoarse spaces and consequently yield new applications of free probability. This is based on joint work with Yoonkyeong Lee. |
October 30 | Josse van Dobben de Bruyn (Charles University) Asymmetric graphs with quantum symmetry Quantum isomorphisms of graphs form a bridge between noncommutative geometry (NCG) and quantum information theory (QIT), as they connect quantum automorphism groups of graphs with nonlocal games. This makes it possible to use techniques from QIT in NCG and vice versa. In this talk, I will present a striking application of this connection, where we use ideas from QIT to prove a surprising result in NCG. Using a construction similar to the Mermin–Peres magic square from QIT, we construct graphs with trivial automorphism group and non-trivial quantum automorphism group, which shows that even graphs with no symmetry at all can have hidden quantum symmetries. These are the first known examples of any kind of commutative spaces in NCG with this property. This talk is based on joint work with David E. Roberson (Technical University of Denmark) and Simon Schmidt (Ruhr University Bochum). |
October 23 | Junichiro Matsuda, University of Waterloo Quantum graphs violate the classical characterization of the existence of $d$-regular graphs. Quantum graphs are a non-commutative analogue of classical graphs that replace the function algebra on vertices with C*-algebras. It is known that classical simple $d$-regular graphs on $n$ points exist if and only if $dn$ is even. This is false for quantum graphs in both directions. We provide a necessary condition on the number of quantum edges between quantum vertices (matrix summands) to make it a $d$-regular quantum graph. Using this technique, we also describe $1$-regular or $2$-regular quantum graphs on general tracial quantum sets. $1$-regular quantum graphs have quantum edges only between summands of the same size. Centrally connected $2$-regular quantum graphs are classified into 8 families by their central skeleton. This is a joint work with Matthew Kennedy and Larissa Kroell. |
October 9 | Roberto Hernandez Palomares, University of Waterloo Quantum graphs and spin models Spin models for singly-generated Yang-Baxter planar algebras are known to be determined by certain highly-regular classical graphs such as the pentagon or the Higman-Sims graph. Examples of spin models include the Jones and Kauffman polynomials, as well as certain fiber functors. We will explore the notion of higher-regularity for quantum graphs as well as their potential to encode spin models. Time allowing, we will give examples of non-classical graphs with these properties. |
October 2 | September 25 | September 18 | Jashan Bal, University of Waterloo Projectivity in topological dynamics A compact space is defined to be projective if it satisfies a certain universal lifting property. Projective objects in the category of compact spaces were characterized as exactly the extremally disconnected compact spaces by Gleason (1958). Analogously, if we fix a topological group G, then one can consider projectivity in the category of G-flows or affine G-flows. We present some new results in this direction, including a characterization of amenability or extreme amenability for closed subgroups of a Polish group via a certain G-flow being projective in the category of affine G-flows or G-flows respectively. Lastly, we introduce a new property, called proximally irreducible, for a G-flow and use it to prove a new dynamical characterization of strong amenability for closed subgroups of a Polish group. In doing so, we answer a question of Zucker by characterizing when the universal minimal proximal flow for a Polish group is metrizable or has a comeager orbit. |
September 4 | Aareyan Manzoor, University of Waterloo. There is a non-Connes embeddable equivalence relation Connes embeddability of a group is a finite dimensional approximation property. It turns out this property depends only on the so-called group von Neumann algebra. The property can be extended to all von Neumann algebras. The fact that there is a von Neumann algebra without this property was proved in 2020 using the quantum complexity result MIP*=RE. It is still open for group von Neumann algebras. I will discuss the best-known partial result, which is that there is a group action without this property. In particular, this implies the negation to the Aldous-Lyons conjecture, a big problem in probability theory |
August 26 **4:00PM in MC 5501** | Beatrice-Helen Vritsiou, University of Alberta On the Hadwiger-Boltyanski illumination conjecture for convex bodies with many symmetries. Let us think of a convex body in R^n (convex, compact set, with non-empty interior) as an opaque object, and let us place point light sources around it, wherever we want, to illuminate its entire surface. What is the minimum number of light sources that we need? The Hadwiger-Boltyanski illumination conjecture from 1960 states that we need at most as many light sources as for the n-dimensional hypercube, and more generally, as for n-dimensional parallelotopes. For the latter their illumination number is exactly 2^n, and they are conjectured to be the only equality cases. The conjecture is still open in dimension 3 and above, and has only been fully settled for certain classes of convex bodies (e.g. zonoids, bodies of constant width, etc.). In this talk I will briefly discuss some of its history, and then focus on recent progress towards verifying the conjecture for all 1-symmetric convex bodies and certain cases of 1-unconditional bodies. |