Dept Oral Exam

Speaker: Youlong Yan (Western)

"Tilting Sheaves on Brauer-Severi Schemes and Arithmetic Toric Varieties"

Time: 13:00

Room: MC 107

The derived category of coherent sheaves on a smooth projective variety is an important object of study in algebraic geometry. One import device relevant for this study is the notion of tilting sheaf. We first show the existence of tilting bundles on some Brauer-Severi schemes, and as an application, we construct tilting bundles on some arithmetic toric varieties.

Index Theory Seminar

Speaker: Matthias Franz and Sean Fitzpatrick (Western)

"The Weyl and Kirillov character formulas"

Time: 12:00

Room: MC 107

This will be a two-part presentation on character formulas for representations of compact Lie groups. In the first part, Matthias Franz will present Weyl's formula. In the second part, Sean Fitzpatrick will present Kirillov's formula.

Homotopy Theory

Speaker: Dan Christensen (Western)

"An introduction to Type Theory"

Time: 14:00

Room: MC 107

The theory of types was introduced by Bertrand Russell as a foundation for mathematics that avoids Russell's paradox. A more refined theory was introduced by Martin-Löf in 1972, and this theory is the basis for much modern research on type theory. It was later realized that type theory also has an intrinsically homotopical interpretation, and describing this will be the long term goal of the seminar series.

In this talk, I will give an introduction to (non-homotopical) type theory, explaining how types can be thought of either as sets or as propositions, and give examples of how to do mathematics using type theory. I will also show how type theory is used in the proof assistant Coq, which has been used to formalize many non-trivial results. I will also discuss the intuitionistic logic that arises in type theory.