University of Regina

Topological K-theory is an invariant of spaces based on vector bundles. We will review vector bundles and their representability by Grassmannians. We will then introduce the group K^0(X), which measures to what extent vector bundles on X can be non...

In this second part, we will further discuss Stiefel manifolds and Grassmannians, as well as stabilization of vector bundles. Next, we will introduce the group K^0(X) along with its reduced version, and compute some examples. Time permitting, we will...

Now that we have the K-theory group K(X) at our disposal, our next goal will be to compute it for the circle S^1 and the sphere S^2. We will then discuss Bott periodicity and extend the group K(X) to a cohomology theory.

In this talk we will first review some of the properties of Chern classes and Steifel Whitney classes. Next, we will introduce the Chern character, now that we have made K(X) into a cohomology theory using Bott periodicity, in the earlier talk...

At this point we have two cohomology theories in hand – HZ (singular cohomology) and KU (complex K-theory) – and the Chern character ch establishes a comparison between them. Even the first properties of ch can already give rise to interesting...

Equivariant operads provide a powerful framework for encoding homotopic algebraic structures hidden in the non-equivariant setting, specifically those incorporating norm and transfer maps. These structures are organized by combinatorial objects...

Transfer systems are combinatorial objects that encode information about equivariant operations. More precisely, a transfer system encodes the transfers (or wrong-way maps) carried by algebras over certain equivariant operads. Thus, transfer systems...

A network of interacting qubits, usually subatomic particles, can be modelled by a connected weighted undirected graph G. A quantum walk on G is a deterministic process that describes the propagation of quantum states in the network represented by G...

Let F be a homogeneous polynomial of degree n in at least d 2 + 1 variables over the p-adic numbers, Q p . Artin conjectured that such F always have nontrivial zeros in any p -adic field. Although this has been shown to be false in general, the...

Continuing our exploration of formal connections, we will see how the bar construction gives us a way to interpret it as a morphism between homotopy associative algebras. We will also see how its adjoint, the cobar construction, is used to prove that...