Events

Joey Lakerdas-Gayle, University of Waterloo

High incomplete c.e. set

We use an infinite injury priority tree construction to prove Sacks' theorem that there exists a high incomplete c.e. set.

MC 5403

Erik Séguin, University of Waterloo

Ulam Stability for Quantum Groups and Noncommutative Dynamics

In this talk, we discuss various "noncommutative" extensions of classical commutative phenomena. We focus on two particular cases of this: representation stability for locally compact quantum groups (extending representation stability for classical locally compact groups) and minimality for C*-dynamical systems (extending minimality for topological flows). In the first part of the talk, we discuss the question of Ulam stability for approximate representations of locally compact quantum groups. We show that compact quantum groups and amenable discrete quantum groups are representation stable. In the process, we establish a partial stability result for general amenable locally compact quantum groups, showing that if one imposes additional constraints on the approximate representation in question then the assumption of compactness or discreteness can be dispensed with. In the second part of the talk, we discuss the notion of minimality for C*-dynamical systems. We show that the accepted definition is somewhat unsatisfactory as an extension of its commutative counterpart from both a topological dynamical and an operator algebraic perspective and present an alternative notion of an extension which rectifies these issues. We demonstrate that this alternate notion also encodes operator algebraic structure which does not appear to have a direct dynamical connection, further motivating its study. We outline various characterizations and extension properties for this notion and discuss its consequences.

MC 2009

Spiro Karigiannis (University of Waterloo)

Decomposition of Riemann curvature in 4 dimensions (and beyond?)

This is a continuation of my earlier talk from May. We showed that the Riemann curvature tensor of a Riemannian metric decomposes into three orthogonal components: the scalar curvature, the traceless Ricci curvature tensor, and the Weyl curvature tensor. We will see that in dimension 4 we can say more. The Weyl curvature decomposes into two pieces, and we will explicitly determine the decomposition the curvature operator (as a self-adjoint operator on 2-forms). As an application, we will discuss the Singer-Thorpe Theorem characterizing 4-dimensional Einstein metrics in terms of this decomposition. If time permits, we will briefly discuss a generalization of these ideas to G2-geometry in seven dimensions.

MC 5417

Alex Pawelko (University of Waterloo)

Alex Pawelko (University of Waterloo)

Adiabatic Limits of Coassociative Fibrations II

We'll continue working through Donaldson's paper on coassociative Kovalev-Lefschetz fibrations, focusing on their adiabatic limits and global geometry.

MC 5417

Erik Seguin (University of Waterloo)

Minimality in noncommutative dynamics

In this talk, we discuss the notion of minimality for C*-dynamical systems. We show that the accepted definition is somewhat unsatisfactory as an extension of its commutative counterpart from both a topological dynamical and an operator algebraic perspective and present an alternative notion of an extension which rectifies these issues. We discuss various consequences of this alternative definition. 

MC 5403

Lilian Gardner (University of Waterloo)

The Chifan-Ioana-Kunnawalkam Elayavalli Lifting Lemma

The construction of a separable II_1 factor without property Gamma, and which is not elementarily equivalent to the free group factors requires a lifting lemma, for which if four orthogonal projections in an ultraproduct of II_1 factors are chosen to satisfy certain properties, then sequence representatives of each projection can be chosen to also satisfy the same properties. We go over the proof of this lifting lemma.

MC 5501

Ian Charlesworth (Cardiff University)

Mixtures of classical and free independence

\(\varepsilon\)-free probability was first introduced by Mlotkowski in the early 2000s to understand and model \(q\)-deformed Gaussian variables, which interpolate between the usual Gaussian distribution at q = 1 and the free semicircular distribution at q = 0. It describes an independence relation for tuples of algebras, some in free position and some in tensor position. A similar construction for groups (the graph product, which interpolates between the free product and the direct product) was introduced by Green in 1990, and extended to operator algebras by Caspers and Fima, independently recovering Mlotkowski's definitions. These objects have attracted quite a bit of attention over recent years, both from the viewpoint of non-commutative probability theory and in studying their operator algebraic properties directly. I will discuss the state of the art concerning \(\varepsilon\)-free independence and graph product von Neumann algebras, including work of others and some of my own. Particular focus will be given to my recent joint work with Jekel, where we are able to use a 1969 result of Cartier and Foata on the combinatorics of words as the key to describing the type I summands of a graph product of von Neumann algebras.

MC 5403