|
Organizers Michael Brannan Matthew Kennedy Nico Spronk Kateryna Tatarko Andy Zucker |
| Date | Speaker, Title and Abstract |
|---|---|
| May 9 ***4:30PM-5:20PM in MC 5417 | Sascha Troscheit, University of Oulu Dynamical Self-similar Covering Sets A classical problem in dynamical systems is known as the shrinking target problem: given a sequence of 'target' subsets A_n \subset X and a dynamic T: X \to X we ask how 'large' the set of all points R \subset X is whose n-th iterate hits the target, T^n (x) \in A_n, infinitely often. Much progress has been made on understanding this type of 'recurrent' set and I will highlight some recent results on this and the related 'dynamical covering problem' which is a dynamical generalisation of the Dvoretzky covering problem. The talk is based on joint results with Balázs Bárány, and Henna Koivusalo and Balázs Bárány. |
May 9 ***3:30PM-4:20PM in MC 5417 | Alex Chirvasitu, University at Buffalo Strange manifolds, small cohomotopy and Baire classes Pr¨ufer surfaces are non-metrizable separable 2-manifolds originally defined by Calabi and Rosenlicht by doubling the upper half-plane along a continuum’s worth of real-line boundary components. The construction and variations on it have since been studied by Gabard, Baillif and many others for the purpose of probing the pathologies of non-paracompact manifolds. The fundamental groups of such surfaces and higher-dimensional cousins are known to be (essentially) free on the sets S of connected boundary components, so their first cohomotopy groups (i.e. sets of homotopy classes of continuous maps to rather than from the circle) are identifiable with maps from S to the integers. Which functions S → Z arise in this manner is a natural question, with (perhaps) a surprising answer. The goal will be to discuss that problem, but the manifolds themselves might provide some entertainment value on their own. |
April 25 | Adam Fuller, Ohio University C*-algebra Inclusions and Regular Ideals Let $B \subseteq A$ be an inclusion of C$^*$-algebras. In this talk we will discuss the relationship between the regular ideals of $A$ and a sub-lattice of regular ideals of $B$. This will be used to see what properties are preserved when quotienting by regular ideals. In particular, assume the elements of $A$ which normalize $B$ span a dense subset of $A$; and if there is a unique faithful map from $A$ to the injective envelope $I(B)$ which extends the inclusion of $B$ in $I(B)$ (i.e. with a faithful unique pseudo-expectation). Let $J$ be any regular ideal of $A$. Then there is a faithful unique pseudo-expectation from $A/J$ to $I(B/(J\cap B))$. Examples of faithful unique pseudo-expectations include Cartan inclusions, and the inclusion of the cycline algebra in a $k$-graph C$^*$-algebra. This is joint work with Jon Brown, David Pitts, and Sarah Reznikoff. |
April 11 | Jananan Arulseelan, McMaster University Quantum Complexity, Computability and QWEP By the recent MIP*=RE result from quantum complexity, Kirchberg's QWEP conjecture is known to be false. I will show how this implies that there is no effective axiomatization of QWEP C*-algebras nor an effective axiomatization of QWEP W*-probability spaces. As an application we show that there is a ultraproduct of non-QWEP C*-algebras with QWEP. This is joint work with Isaac Goldbring and Bradd Hart. |
April 4 | Corey Jones, North Carolina State University Constructing actions of fusion categories on C*-algebras A fusion category is an algebraic object that simultaneously generalizes finite groups and their representation categories. Fusion categories can ``act" on C*-algebras by bimodules, extending the familiar concept of a group acting by automorphisms to a non-invertible setting. Building actions of specific fusion categories on specific C*-algebras is hard. In this talk, we will discuss a general method that allows for the construction of actions of fusion categories on interesting C*-algebras with minimal algebraic input. As an application, we construct actions of exotic fusion categories on noncommutative tori. Based on joint work with David Evans. |
March 28 | Katarzyna Wyczesany, Carnegie Mellon University Dualities on sets and how they appear in optimal transport. In this talk, we will discuss order reversing quasi involutions, which are dualities on their image, and their properties. We prove that any order reversing quasi-involution is of a special form, which arose from the consideration of optimal transport problem with respect to costs that attain infinite values. We will discuss how this unified point of view on order reversing quasi-involutions helps to deepen the understanding of the underlying structures and principles. We will provide many examples and ways to construct new order reversing quasi-involutions from given ones. This talk is based on joint work with Shiri Artstein-Avidan and Shay Sadovsky |
March 14 | Roberto Hernandez Palomares, University of Waterloo C* Quantum Dynamics A subfactor is a unital inclusion of simple von Neumann algebras, which can be presented as a non-commutative dynamical system governed by a tensor category. Popa established that in ideal scenarios, dynamical data is a strong invariant for hyperfinite subfactors. These reconstruction results in a way give an equivariant version of Connes' classification for amenable factors. On the topological side, after the recent culmination of the classification program for amenable C*-algebras, whether there is an analogue of Popa's Reconstruction results is not clear. In this talk, I will describe the transfer of subfactor techniques to C*-algebras, introducing the largest class of inclusions of C*-algebras admitting a quantum dynamical invariant akin to subfactors. Examples include the cores of Cuntz algebras, certain semicircular systems, and crossed products by actions of tensor categories. Time allowing, we will discuss some interactions with the C* classification program. This is based on joint work with Brent Nelson |
March 7 | Peter Pivovarov, University of Missouri A probabilistic approach to Lp affine isoperimetric inequalities In the class of convex sets, the isoperimetric inequality can be derived from several different affine inequalities. One example is the Blaschke-Santalo inequality on the product of volumes of a convex body and its polar dual. Another example is the Busemann--Petty inequality for centroid bodies. In the 1990s, Lutwak and Zhang introduced a related functional analytic framework with their notion of Lp centroid bodies, for p>1. Lutwak raised the question of encompassing the non-convex star-shaped range when p<1 (including negative values). I will discuss a probabilistic approach to establishing isoperimetric inequalities in this range. It uses a new representation of star-shaped sets as special averages of convex sets. Based on joint work with R. Adamczak, G. Paouris, and P. Simanjuntak. |
February 29 | Matthijs Vernooij, TU Delft Derivations for symmetric quantum Markov semigroups Quantum Markov semigroups describe the time evolution of the operators in a von Neumann algebra corresponding to an open quantum system. Of particular interest are so-called symmetric semigroups. Given a faithful state, one can define the GNS- and KMS-inner product on the von Neumann algebra, and a semigroup is GNS- or KMS-symmetric if it is self-adjoint w.r.t. the inner product. GNS-symmetry implies KMS-symmetry, and both coincide if the state is a trace. It was shown in 2003 that the generator of a tracially symmetric quantum Markov semigroup can be written as the 'square' of a derivation, i.e. d* after d, where d is a derivation to a Hilbert bimodule. This result has proven to be very influential in many different directions. In this talk, we will look at this problem in the case that our state is not tracial. We will start by discussing how a computer can be used to decide whether such a derivation exists in finite dimensions, and work our way up to a general result on KMS-symmetric quantum Markov semigroups. This is joint work with Melchior Wirth |
February 15 | Andy Zucker, University of Waterloo Ultracoproducts and weak containment for flows of topological groups. We develop the theory of ultracoproducts and weak containment for flows of arbitrary topological groups. This provides a nice complement to corresponding theories for p.m.p. actions and unitary representations of locally compact groups. For the class of locally Roelcke precompact groups, the theory is especially rich, allowing us to define for certain families of G-flows a suitable compact space of weak types. When G is locally compact, all G-flows belong to one such family, yielding a single compact space describing all weak types of G-flows. |
February 1 (Special Time and place: 2:30pm in MC 5417) | Michael Francis, Western University Local normal forms in complex b^k geometry The b-tangent bundle (terminology due to Melrose) is defined so that its sections are smooth vector fields on the base manifold tangent along a given hypersurface. Complex b-manifolds, studied by Mendoza, are defined just like ordinary complex manifolds, replacing the usual tangent bundle by the b-tangent bundle. Recently, a Newlander-Nirenberg theorem for b-manifolds was obtained by Francis-Barron, building on Mendoza's work. This talk will discuss the extension of the latter result to the setting of b^k-geometry for k>1. The original approach to b^k-geometry is due to Scott. A slightly different approach that allows for global holonomy phenomena not present in Scott's framework was introduced by Francis and, independently, by Bischoff-del Pino-Witte. |
January 25 | Eric Culf, University of Waterloo Approximation algorithms for noncommutative constraint satisfaction problems Constraint satisfaction problems (CSPs) are an important topic of investigation in computer science. For example, the problem of finding optimal k-colourings of graphs, Max-Cut(k), is NP-hard, but it is easy to approximate in the sense that it is possible to find a colouring that satisfies a large fraction of the constraints of an optimal one. We study a noncommutative variant of CSPs that is central in quantum information, where the variables are replaced by operators. In this context, even approximating general CSPs is known to be much harder than the classical case, in fact uncomputably hard. Nevertheless, Max-Cut(2) becomes efficiently solvable. We introduce a framework for designing approximation algorithms for noncommutative CSPs, which allows us to find classes of CSPs that are efficiently approximable but not efficiently solvable. To determine the quality of our approximation algorithm, we make use of results from free probability to characterise a distribution arising from random matrices. This talk is based on work with Hamoon Mousavi and Taro Spirig (arxiv.org/abs/2312.16765). |
November 30 | Junichiro Matsuda, Kyoto University Algebraic connectedness and bipartiteness of quantum graphs Quantum graphs are a non-commutative analogue of classical graphs related to operator algebras, quantum information, quantum groups, etc. In this talk, I will give a brief introduction to quantum graphs and talk about spectral characterizations of properties of quantum graphs. We introduce the notion of connectedness and bipartiteness of quantum graphs in terms of graph homomorphisms, and these properties have algebraic characterizations in the same way as classical cases. We also show the equivalence between bipartiteness and two-colorability of quantum graphs defined by two notions of graph homomorphisms: one respects adjacency matrices, and the other respects edge spaces. This talk is based on arXiv:2310.09500. |
| November 23 | Yuming Zhao, University of Waterloo Positivity and sum of squares in quantum information A multivariate polynomial is said to be positive if it takes only non-negative values over reals. Hilbert's 17th problem concerns whether every positive polynomial can be expressed as a sum of squares of other polynomials. Many problems in math and computer science are closely connected to deciding whether a given polynomial is positive and finding certificates (e.g., sum-of-squares) of positivity. In quantum information, we are interested in noncommutative polynomials in *-algebras. A well-known theorem of Helton states that an element of a free *-algebra is positive in all *-representations if and only if it is a sum of squares. The theorem provides an effective way to determine if a given element is positive, by searching through sums of squares decompositions. In this talk, I'll present joint work with Arthur Mehta and William Slofstra in which we show that no such procedure exists for the tensor product of two free *-algebras: determining whether an element of such an algebra is positive is coRE-hard. Consequently, tensor products of free *-algebras contain elements which are positive but not sums of squares. I will also discuss the connetions to quantum information theory. |
| November 21 | Ian Charlesworth, Cardiff University Permutation matrices, graph independence over the diagonal, and consequences Graph products were first introduced by Green in the context of groups, giving a mixture of direct and free products. They have recently been studied in the contexts of operator algebras and of non-commutative probability theory by M\l{}otkowski, Caspers and Fima, Speicher and Wysocza\'nski, and others. It is interesting to ask how properties of a family of von Neumann algebras are witnessed in a graph product; while free products and tensor products are well understood, their interplay can be quite subtle in this more general setting. With Collins, I showed how conjugation by random unitary matrices in a tensor product of matrix algebras creates asymptotic graph independence, when the unitaries are independent and uniformly distributed but only on particular subalgebras. In this talk, after spending some time introducing the setting, I will discuss how techniques inspired by the work of Au, C\'ebron, Dahlqvist, Gabriel, and Male can be used to make a similar statement about random permutations leading to asymptotic graph independence over the diagonal subalgebra; the combinatorial techniques required involve some interesting subtleties which are not apparent at first glance. I will also discuss some consequences for von Neumann algebras. For example, suppose that $(M_v)_v$ is a collection of finite dimensional algebras. $M_v$ can be embedded into a larger matrix algebra in such a way that it is constant on the diagonal, and the standard matrix units of $M_v$ are embedded as elements whose entries are roots of unity. Then if $M$ is a graph product of the $M_v$, we can find matricial approximations of a generating set which enjoy the same properties, and this in turn allows us to show (using techniques of Shlyakhtenko) that if the if $M$ is diffuse and algebra generated by the $M_v$ within $M$ has vanishing first $L^2$ Betti number then $M$ is strongly 1-bounded in the sense of Jung. This is joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson. |
| November 16 | Benjamin Anderson-Sackaney, University of Saskatchewan Amenability of Fusion Modules and Coideals The coideals of a quantum group offer a quantum analogue of a subgroup of a group. For certain classes of coideals there is an obvious quantum analogue of a quasi-regular representation. For a larger class of coideals recently introduced by De Commer and Dzokou Talla, namely, the so-called $g$-integral coideals, we will introduce a notion of a $g$-quasi-regular representation. We will then define a notion of $g$-coamenability that generalizes the notion of a coamenable inclusion of groups. We will also introduce a notion of amenability of a fusion module equipped with a dimension function that is compatible with a dimension function on the given fusion algebra. This notion gives a characterization of $g$-coamenability at the tensor categorical level. |
| November 2 | Christopher Schafhauser, University of Nebraska-Lincoln Finite dimensional approximations of groups Blackadar and Kirchberg introduced the class of matricial field C*-algebras, which are those which approximately embed into finite dimensional matrix algebras. Specializing this property to group C*-algebras, one defies a group to be MF if it admits approximate finite dimensional representations which are approximately faithful and and approximately weakly contained in the left regular representation. I will give a survey of known examples and properties of MF groups, including a recent result showing that free products of amenable groups amalgamated over a common normal subgroup are MF. |
| October 26 | Arthur Troupel, Université Paris Cité Free wreath product quantum groups as fundamental C*-algebras of graphs. The free wreath product of a compact quantum group by the quantum permutation group S_N^+ has been introduced by Bichon in order to give a quantum counterpart of the classical wreath product. The representation theory of such groups is well-known, but some results about their operator algebras were still open, for example stability of Haagerup property, of K-amenability or factoriality of the von Neumann algebra. I will present a joint work with Pierre Fima in which we identify these algebras with the fundamental C*-algebras of certain graphs of C*-algebras, and we deduce these properties from these constructions. |
| October 19 | Tattwamasi Amrutam, Ben Gurion University of the Negev Amenable Invariant Random Algebras We approach the study of sub-von Neumann algebras of the group von Neumann algebra L(\Gamma) for a countable group \Gamma from a dynamical perspective. We show that L(\Gamma) admits a maximal \Gamma-invariant amenable subalgebra. We introduce the notion of Invariant Random Algebra (IRA) in the space of sub-algebras, analogous to the concept of Invariant Random Subgroup (IRS). Finally, we shall show that amenable IRAs are supported on the maximal amenable \Gamma-invariant subalgebra of L(\Gamma). This is joint work with Yair Hartman and Hanna Oppelmayer. |
| October 10 (*Special Day*) | Itamar Vigdorovich, Weizmann Institute of Science The trace simplex of groups and C*-algebras We show that the space of traces of free product C*-algebras C(X)*C(Y), where X and Y are compact perfect spaces, is a Poulsen simplex i.e., the extreme points are dense. This in particular applies to the C*-algebra of the free group C*(Fn). The same statement is true for free products of matrix algebras Mn*Mn, for n at least 4. This answers a question by Musat and Rordam, and relates to quantum factorizable channels. Time permitting, I will discuss strong converses of the above, e.g. in the presence of property (T). The talk is based on a joint work with Orovitz and Slutsky. |
| October 5 (**Joint With Logic Seminar**) | Gianluca Basso, Università di Torino Chains on Peano continua, combinatorics and dynamics We generalize a theorem of Gutman, Tsankov and Zucker on the non-existence of generic chains of subcontinua in manifolds of dimension at least 3 to a large class of spaces. Among them is the Menger curve, a 1-dimensional planar continuum. Using Bing's partition theorem, we reduce the problem to a combinatorial statement about walks on finite graphs. The theorem has dynamical consequences which can be interpreted as non-rigidity results for the homeomorphism groups of the spaces involved. This is joint work with A. Codenotti and A. Vaccaro. |
| September 28 | Matteo Pagliero, KU Leuven Classification of equivariantly 𝒪2-stable amenable actions on nuclear C*-algebras. The classification of group actions on C*-algebras is an active research theme in the field of Operator Algebras. Although there has recently been substantial progress in the classification of group actions on simple, separable, nuclear, purely infinite C*-algebras, the non-simple case is not yet fully explored in the literature. In this talk, I will present joint work with G. Szabó, in which we classify well-behaved locally compact group actions on separable, nuclear C*-algebras that are not necessarily simple and tensorially absorb the Cuntz algebra 𝒪2. Our main result states that an amenable, isometrically shift-absorbing, equivariantly 𝒪2-stable action on a separable, nuclear, stable (or unital) C*-algebra is classified up to cocycle conjugacy by a topological dynamical system on the space of primitive ideals of the C*-algebra. This may be considered as a dynamical generalisation of Gabe--Kirchberg's 𝒪2-stable classification theorem. In the first part of the talk, I will introduce the aforementioned assumptions, and provide a full description of the classification invariant. In the second part, I will give an overview of our classification theorem, and elaborate on how it relates to the existing literature. |
| September 25 | Michael Whittaker, University of Glasgow Self-similarity of substitution tiling semigroups Substitution tilings arise from graph iterated function systems. Adding a contraction constant, the attractor recovers the prototiles. On the other hand, without the contraction one obtains an infinite tiling. In this talk I'll introduce substitution tilings and an associated semigroup defined by Kellendonk. I'll show that this semigroup defines a self-similar action on a topological Markov shift that's conjugate to the punctured tiling space. The limit space of the self-similar action turns out to be the Anderson-Putnam complex of the substitution tiling and the inverse limit recovers the translational hull. This was joint work with Jamie Walton. |
| September 14 | Sergio Giron Pacheco, University of Oxford Tensor category equivariant Z-stability In this talk I will discuss D-stability for strongly self absorbing C*-algebras D and its relevance for the classification of C*-algebras. I will then discuss an equivariant notion of D-stability with respect to an action of a tensor category and an adaptation of a classical result of Kirchberg to this setting. |
| September 7 | Tomer Zimhoni, Ben Gurion University Random Permutations from Free Products Consider a random permutation on N elements denoted by A, which is drawn uniformly from the collection {σ ∈ SN | σ3 = id}. Additionally, let B be another random permutation, uniformly selected from the set {σ ∈ SN | σ2 = id}. Let γ be a word in A and B; for instance, γ could be represented as ABAAB. This way γ defines a new random permutation, given by choosing A and B and then take the composition A◦B◦A◦A◦B. In the forthcoming talk, we will discuss several results regarding the distribution of such a γ-random permutation as N grows to infinity, such as calculating the limit of the expected number of fixed points of a γrandom permutation. We show a surprising relation between such outcomes to topological and algebraic invariants of elements in free products of groups. This talk is based on my joint work with Doron Puder from Tel Aviv University. |
| July 13 | Marcelo Laca, University of Victoria On the Matter of Numbers Over the past 35 years several C*-dynamical systems arising from number theory have been constructed and analyzed, with the expectation that their symmetries, energy spectra, and equilibrium states will (eventually) yield insight on key questions about number fields. I will start with a brief and inevitably incomplete retrospect of these systems, highlighting their main features interpreted as properties of `numerical substances'. I will then focus on a new C*-dynamical system based on the semigroup of ax+b transformations of the integers that was introduced in recent joint work with Astrid an Huef and Iain Raeburn, and finish by discussing current joint work with Tyler Schulz, in which we show that this system has an unprecedented type III phase transition in the high temperature range. |
| July 6 | Arundhathi Krishnan, Munster Technological University, Cork Markovianity and the Thompson Monoid F+ We show that representations of the Thompson monoid F+ yield a large class of unilateral stationary noncommutative Markov processes. As a partial converse, unilateral stationary Markov processes in tensor dilation form (and in particular, in the commutative setting) are shown to yield representations of F+. This is joint work with Claus Koestler and Stephen J. Wills. |
| June 29 | Florin Pop, Wagner College Detecting certain properties of C*-algebras A C*-algebra $A$ is said to detect a certain property $\mathcal{P}$ (or is a $\mathcal{P}$-detector) if, for any C*-algebra $B$, we have $A\otimes_{\min}B=A\otimes_{\max}B$ is and only if $B$ has property $\mathcal{P}$. In this talk we will survey several properties that can be detected, as well as present the algebras which play the detector's part. |
| June 8 | Ali Jabbari, Shahid Bahonar University of Kerman Affine flows, Namioka fixed points and a fixed point theorem In 1983, Isaac Namioka gave an interesting geometric characterization of minimal distal flows in a metrizable affine flow in terms of a special geometric property (which we call Namioka flows) and another geometric property regarding the extreme points of the closed convex hull of the minimal distal flow. Namioka left open the problem ofwhether or not his results are valid for an arbitrary (not necessarily metrizable) phase space or any extra condition on the semigroup. We give a positive answer to this problem by introducing a new geometrical concept named the sharp points of a flow. Also, we investigate the newly defined concept of Namioka flows, and an interesting fixed point theorem, generelizing Hahn’s fixed point theorem, for the affine Namioka flows will be proved. |
| May 18 | Mehrdad Kalantar, University of Houston Complexification of operator spaces — Revisited Complexification is a main technical tool in the study of real spaces, it allows to pass to a canonical complex space which shares almost all structural properties with the original real space. This is such a powerful tool that it is completely natural to reflect on its primal qualities, and in particular to ask is there a more general concept with some potentially similar applications? This is the main concern of this talk. We give an answer to these questions, by offering a novel framework which generalizes the complexification, as well as the less-studied notion of quaternification of real spaces. This is joint work with David Blecher. |