Geometry and Topology

Speaker: David Anderson (Ohio State)

"Operational equivariant $K$-theory"

Time: 13:30

Room: MC 107

Given any covariant homology theory on algebraic varieties, the bivariant machinery of Fulton and MacPherson constructs an "operational" bivariant theory, which formally includes a contravariant cohomology component. Taking the homology theory to be Chow homology, this is how the Chow cohomology of singular varieties is defined. I will describe joint work with Richard Gonzales and Sam Payne, in which we study the operational $K$-theory associated to the $K$-homology of $T$-equivariant coherent sheaves. Remarkably, despite its very abstract definition, the operational theory has many properties which make it easier to understand than the $K$-theory of vector bundles or perfect complexes. This is illustrated most vividly by singular toric varieties, where relatively little is known about $K$-theory of vector bundles, while the operational equivariant $K$-theory has a simple description in terms of the fan, directly generalizing the smooth case.

Analysis Seminar

Speaker: Javad Mashreghi (Laval)

"A numerical mapping theorem"

Time: 15:30

Room: MC 108

The spectral mapping theorem is one of the most important results in operator theory. However, the same statement fails for the numerical range. In this talk, we provide several examples to reveal the relation between spectrum and numerical range. Then we discuss a weak version of the spectral mapping theorem that can be extended for the numerical range. This point of view leads us to Halmos conjecture and theorems of Berger-Stampfli and Drury.

Speaker's homepage: https://www.mat.ulaval.ca/departement-et-professeurs/direction-personnel-et-etudiants/professeurs/fiche-de-professeur/show/mashreghi-javad/

Homotopy Theory

Speaker: Aji Dhillon (Western)

"Mapping spaces in higher categories (II)"

Time: 13:00

Room: MC 107

The goal of these two talks is to introduce models for mapping spaces in higher categories and show that they are all equivalent. In the process we will discuss straightening and unstraightening, infinity analogue of the Grothendieck construction. We will conclude with a discussion of cartesian fibrations.