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Organizers Michael Brannan Matthew Kennedy Nico Spronk Kateryna Tatarko |
| Date | Speaker, Title and Abstract |
|---|---|
| September 8 | Sang-Gyun Youn, Seoul National University Information-theoretic analysis of covariant quantum channels It has a long history in quantum information theory to study quantum channels with symmetries, which we call covariant quantum channels. Some standard examples are the depolarizing quantum channels and Werner-Holevo quantum channels arising from the fundamental symmetries of the unitary group. There have been lots of studies for more general group symmetries, but with non-unified approaches and limitations to the fundamental group symmetries without the representation theory. This talk divides into two different topics. One part is to present a universal representation-theoretic framework to understand the structure of covariant quantum channels, and the other part is to discuss detailed information-theoretic properties of SU(2)-covariant quantum channels in some low-dimensional situations. More precisely, we present the complete characterization of their entanglement-breaking property, degradability, Holevo information, and (almost) super-activation of the coherent information |
| January 12 | No Speaker |
| January 19 | Jeremy Hume, University of Glasgow The K-theory of a rational function The dynamics of iterating a rational function exhibits complicated and interesting behaviour when restricted to points in its Julia set. Kajiwara and Watatani constructed a C*-algebra from a rational function restricted to its Julia set in order to study its dynamics from an operator algebra perspective. They showed the C*-algebras are Kirchberg algebras that satisfy the UCT, and are therefore classified by K-theory. The K-theory groups of these algebras have been computed in some special cases, for instance by Nekrashevych in the case of a hyperbolic and post-critically finite rational function. We compute the K-theory groups for a general rational function using methods different to those used before. In this talk, we discuss our methods and results, including new topological conjugacy invariants for a rational function restricted to its Julia set. |
| January 26 | Erik Seguin, University of Waterloo Amenability and stability for discrete groups The notion of a representation of a group on a Hilbert space can be generalized to that of an "approximate representation", in which the usual homomorphism condition is replaced by some bound on the norm distance between the operators φ(xy) and φ(x) φ(y). It is natural to ask about the stability of this class of maps: namely, when the defect of an approximate representation is small, is the approximate representation well-approximated by a genuine representation of the group? In this talk, we explore the connection between amenability and the stability of approximate representations for discrete groups. |
| February 3, 3:00PM (SPECIAL TIME!) | Robert Martin, University of Manitoba Non-commutative measure theory Measure theory on the complex unit circle and analytic function theory in the unit disk, in particular the theory of Hardy spaces, are fundamentally connected. Several celebrated theorems due to P. Fatou, G. Herglotz, F. and M. Riesz and G. Szeg¨o describe the relationship between these theories. We will show that many of these classical results have natural extensions to the multivariate and non-commutative settings of the full Fock space, or free Hardy space of square–summable power series in several non-commuting variables and positive non-commutative (NC) measures. Here a (positive) NC measure is any positive linear functional on the free disk system, the operator system generated by the left creation operators, which act as left multiplication by the independent NC variables on the free Hardy space. We will focus on a recently established NC Szeg¨o theorem and its consequences. |
| February 9 | Camila Sehnem, University of Waterloo C*-envelopes and semigroup C*-algebras The C*-envelope of an operator algebra C is the smallest C*-algebra generated by a completely isometric copy of C. In this talk I will consider the C*-envelope of the non-selfadjoint operator algebra generated by the canonical isometric representation of a semigroup P on $\ell^2(P)$, where $P$ is assumed to be a submonoid of a group. I will show that the C*-envelope of this algebra is canonically isomorphic to the boundary quotient of the Toeplitz algebra of P. If time permits, I will discuss a similar result in a more general setting, in which the semigroup of isometries is replaced by a product system of C*-correspondences. |
| February 16 | Jason Crann, Carleton University Values of quantum non-local games The theory of non-local games, rooted in Bell’s seminal work on non-locality, is at the forefront of entanglement theory and has revealed profound connections with non-commutative analysis. Examples of quantum non-local games have been studied, wherein quantum states and/or measurements are used in place of classical questions and/or answers, but a general theory has only recently begun to emerge. This developing theory and its potential applications increase the necessity to avail of a systematic way of comparing essential attributes of distinct quantum games, in particular, their values, i.e., maximum success probabilities. In this work, we give operator space tensor norm expressions for the local, quantum, and quantum commuting values for general quantum non-local games. This is joint work with Rupert Levene, Ivan Todorov and Lyudmila Turowska. |
| February 23 | Reading Week |
| March 2 | Jintao Deng, University of Waterloo The K-theory of relative group C*-algebras The relative Baum-Connes conjecture claims that a certain relative Baum-Connes assembly map is an isomorphism. It provides an algorithm of the computation of the K-theory of relative group C*-algebras. In my talk, I will present several cases when the relative assembly maps are isomorphic (or injective). The strong relative Novikov conjecture states that the relative assembly map is injective. I will also talk about the applications of the strong Novikov conjecture in geometry and topology, especially about the relative higher signatures of manifolds with boundary. |
| March 9 | Yuming Zhao, University of Waterloo An operator algebraic formulation of self-testing Suppose we have a physical system consisting of two separate labs, each capable of making a number of different measurements. If the two labs are entangled, then the measurement outcomes can be correlated in surprising ways. In quantum mechanics, we model physical systems like this with a state vector and measurement operators. However, we do not directly see the state vector and measurement operators, only the resulting measurement statistics (which are referred to as a "correlation"). There are typically many different models achieving a given correlation. Hence it is a remarkable fact that some correlations have a unique quantum model. A correlation with this property is called a self-test. In this talk, I'll introduce the standard definition of self-testing, discuss its achievements as well as limitations, and propose an operator algebraic formulation of self-testing in terms of states on C*-algebras. This new formulation captures the standard one and extends naturally to commuting operator models. I'll also discuss some related problems in operator algebras. Based on arXiv:2301.11291, joint work with Connor Paddock, William Slofstra, and Yangchen Zhou. |
| March 16 | Roberto Hernandez-Palomares, University of Waterloo K-theoretic classification of inductive limit actions of fusion categories on AF C*-algebras I will introduce a K-theoretic complete invariant of inductive limits of finite dimensional actions of unitary fusion categories on unital AF-algebras. This framework encompasses all such actions by finite groups on AF-algebras. Our classification result essentially follows from applying Elliott's Intertwining Argument adapted to this equivariant context, combined with tensor categorical techniques. Time allowing, we will discuss some applications. This is joint work with Quan Chen and Corey Jones. |
| March 23 | Bartlomiej Zawalski, Polish Academy of Sciences On affine bodies with rotationally invariant sections We will prove that an origin-symmetric affine body $K\subset\mathbb R^n$ with sufficiently smooth boundary and such that every hyperplane section of $K$ passing through the origin is an affine body of revolution, is itself an affine body of revolution. This will give a positive answer to the recent question asked by G. Bor, L. Hern\'andez-Lamoneda, V. Jim\'enez de Santiago, and L. Montejano-Peimbert (arXiv:1905.05878 [Remark 2.9]), though with slightly different prerequisites. |
| March 30 | Paul Skoufranis, York University An Overview of Free and Bi-Free Probability In this talk, we will provide an overview of the definitions, structures, examples, results, and applications in free probability and its recent generalization known as bi-free probability. |
| April 6 | No Speaker |
| April 13 | Padraig Daly, University of Waterloo Maps on the space of quantum channels Quantum superchannels are maps whose input and output are completely positive trace preserving (CPTP) maps. Rather than taking the domain to be the space of all linear maps I motivate and state a new definition of superchannel acting on the operator system spanned by CPTP maps. Arvesons extension theorem allows us to show that a Stinespring-like characterisation theorem for superchannels applies to this class of maps. I then discuss some implications of this new approach and how it differs from the usual definition. |
| April 20 | No Speaker |
| April 27 | Tomasz Tkocz, Carnegie Melon University Slicing l_p balls I shall present recent progress on sharp bounds on volume of hyperplane sections of unit balls in l_p spaces, as well as their stability. |
| May 4 | Sergii Myroshnychenko, Lakehead University How far apart can centroids be? The orthogonal projection of the centroid (barycenter, center of mass) of a convex body K onto a hyperplane H, and the centroid of projection of K onto H coincide if K is centrally-symmetric. In general, this is not the case for non-symmetric convex bodies. In this talk, we investigate how far apart these points can be with respect to the width in the direction of the segment connecting them. The optimizers are described as well. The talk is based on the joint work with K. Tatarko and V. Yaskin (https://arxiv.org/abs/2212.14456 |