Algebraic Geometry

Speaker: Deepak Sadanandan (Western)

"WAG: Ideals"

Time: 15:30 - 15:30

Room: MC 107

Graduate Seminar

Speaker: Blake Whiting (Western)

"Acyclic models"

Time: 16:30 - 17:30

Room: MC 107

Acyclic models, as it's commonly seen today, is a proof technique used to show when two chain complexes are chain equivalent or have isomorphic homology. It originated as a theorem by Eilenberg and MacLane (1953), where it was immediately used to show the Eilenberg-Zilber theorem (1953). This theorem, proven directly via acyclic models, gives us a Künneth theorem and defines the cup product, which turns cohomology into a graded ring.

This talk will be an exposition on (one version of) the acyclic models theorem, as given by Michael Barr in 2002. I will give the necessary definitions to understand Barr's modern formulation of acyclic models, and then prove it. Time permitting, I will also discuss how the Eilenberg-Zilber theorem follows directly from it and potential avenues to generalizing acyclic models.

Algebraic Geometry

Speaker: Mac Martin (Western)

"WAG: Grobener Bases"

Time: 15:30 - 15:30

Room: MC 107

Transformation Groups Seminar

Speaker: Tao Gong (Western)

"Affine toric varieties and cones"

Time: 10:30 - 11:30

Room: MC 204

In this lecture, I will give an introduction to toric varieties and cones, and propositions and relations about them.

Algebraic Geometry

Speaker: Kumar Shukla (Western)

"WAG: Degree and dimension"

Time: 15:30 - 15:30

Room: MC 107

Geometry and Topology

Speaker: Sterling Ebel (Western)

"Synthetic approach to the Quillen model structure on spaces"

Time: 15:30 - 16:30

Room: MC 107

Quillen's construction of a model structure on the category of topological spaces is a fundamental result in homotopy theory. This construction has since been applied to several related categories, such as k-spaces, and the importance of many model categories is justified by their equivalence with Quillen's structure on spaces.

In this talk, we will present an axiomatic approach to constructing Quillen's model structure on spaces to apply it to a wider range of settings. As special cases we recover several existing model structures, such as on the categories of sober spaces and of pseudotopological spaces. We also use this approach to construct a novel model structure on the category of locales, making the coreflection to sober spaces a Quillen adjunction.

This is joint work with Chris Kapulkin (arXiv:2310.14235).