Events

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.

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

AJ Fong, Department of Pure Mathematics, University of Waterloo

"The mathematics of juggling (and perhaps a geometric application)"

Before videos could be easily transmitted over the internet, mathematical notation for juggling patterns was used by jugglers to share instructions and new patterns with each other. After introducing these, I will show that a mild generalisation of this gives a natural partial order on juggling patterns. If time permits, I will describe a relatively recent result that demonstrated that juggling patterns can be used to index a natural stratification of Grassmannians, which naturally arises in Poisson geometry, total positivity and cluster algebras.

MC 5417

Lucia Martin Merchan, Department of Pure Mathematics, University of Waterloo

"Hodge decomposition for Nearly Kähler manifolds"

Verbitsky proved that Nearly Kähler 6-dimensional manifolds satisfy Kähler-type identities. These lead to a Hodge decomposition in the compact case, and restrictions on their Hodge numbers. In this talk, we discuss a new proof for most of these results that is independent of the dimension. This is work in progress with Spiro, Michael and Aleks.

MC 5403

Aleksandar Milivojevic, Department of Pure Mathematics, University of Waterloo

"Obstructions to almost complex structures following Massey"

I will report on work in progress with Michael Albanese, in which we prove statements claimed by Massey in 1961 concerning the obstructions to finding an almost complex structure on an orientable manifold (or more generally, reducing the structure group of a real vector bundle over a CW complex to the unitary group). These obstructions involve the integral Stiefel-Whitney classes – which detect the existence of integral lifts of the mod 2 Stiefel-Whitney classes, namely putative Chern classes – and relations between the Pontryagin and Chern classes. A somewhat surprising aspect of these obstructions is that they are in fact generally proper fractional parts of what one might at first expect. For example, the obstruction in degree eleven is 1/24 of the eleventh integral Stiefel-Whitney class.

MC 5403

Anne Johnson, Department of Pure Mathematics, University of Waterloo

"Attributes and morphisms of schemes"

We start Chapter 3 of Eisenbud and Harris, discussing finiteness conditions, properness and separation. We discuss the construction of Proj S as time allows.

MC 5417

Andy Zucker, Department of Pure Mathematics, University of Waterloo

“Preliminaries on Polish Spaces”

This inaugural talk will introduce some of the background on Polish spaces that we will need in our study of Polish groups. We will mostly draw from the early chapters of Kechris's book ‘Classical Descriptive Set Theory’.

MC 5403

Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo

“The linear algebra of 2-forms in 4-dimensions”

I will present some important facts about the linear algebra of 2-forms in 4 dimensions, which everyone should know. We start with classical results about self-dual and anti-self dual 2-forms, and then proceed to discuss "hypersymplectic" structures in 4d à la Donaldson. Then we put all this on an oriented Riemannian 4-manifold.

MC 5417

Benoit Charbonneau, Department of Pure Mathematics, University of Waterloo

“Coxeter groups and Clifford Algebras”

If one wants to understand representation theory of the rotation group of the icosahedron, or of its lift to Sp(1), it is extremely useful to be able to compute things intelligently. It turns out that instead of using matrices, it is much better to play with Clifford Algebras. I’ll explain those concepts and illustrate them.

MC 5417

Alex Chirvasitu, University at Buffalo

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.

MC5417

Sascha Troscheit, University of Oulu

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.

MC5417