Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (University of Windsor)

"On representation theory of integrals for Hopf algebras"

Time: 14:30

Room: MC 108

The study of integrals is an important topic in the theory of Hopf algebras. This notion was introduced by Sweedler in 1969 motivated by the uniquenness of the Haar integral on locally compact groups. In this talk we focus on the representation theory of integrals and explain how special types of integrals can produce (Anti)-Yetter-Drinfeld modules. This can be used to classify total integrals and (cleft) Hopf Galois extensions.

Analysis Seminar

Speaker: Ilia Binder (University of Toronto)

"The rate of convergence of Cardy-Smirnov observable"

Time: 15:30

Room: MC 108

Convergence of the Cardy-Smirnov observables is the crucial part of the famous proof of existence of the scaling limit of critical percolation on hexagonal lattice. I will discuss a proof of the power law convergence of Cardy-Smirnov observables on arbitrary simply-connected planar domains. The proof works for the usual critical percolation on hexagonal lattice, as well as for some modified versions. In the heart of the proof lies a careful study of the fine boundary properties of arbitrary planar domains. I will also explain the relevance of this result for the investigation of the rate of convergence of the critical percolation to its scaling limit. This is a joint work with L. Chayes and H. K. Lei.

Index Theory Seminar

Speaker: Sean Fitzpatrick (Western)

"Index theory for twisted Dirac operators on Spin$^c$ manifolds"

Time: 12:00

Room: MC 107

After a brief introduction to the "quantization commutes with reduction" credo in symplectic geometry (and what index theory has to do with quantization), I will report on recent results of Paradan and Vergne on the multiplicities of the equivariant index for twisted Spin$^c$-Dirac operators. (In this case, "recent" means that the results appeared on the arXiv the day before the talk.)

Algebra Seminar

Speaker: Kirsten Wickelgren (Georgia Tech)

"Splitting varieties for triple Massey products in Galois cohomology"

Time: 15:00

Room: MC 107

The Brauer-Severi variety a $x^2 + b y^2 = z^2$ has a rational point if and only if the cup product of cohomology classes with $F_2$ coefficients associated to $a$ and $b$ vanish. The cup product is the order-$2$ Massey product. Higher Massey products give further structure to Galois cohomology, and more generally, they measure information carried in a differential graded algebra which can be lost on passing to the associated cohomology ring. For example, the cohomology of the Borromean rings is isomorphic to that of three unlinked circles, but non-trivial Massey products of elements of $H^1$ detect the more complicated structure of the Borromean rings. Analogues of this example exist in Galois cohomology due to work of Morishita, Vogel, and others. This talk will first introduce Massey products and some relationships with non-abelian cohomology. We will then show that $b x^2 =$ $(y_1^2 - a y_2^2$ $+ c y_3^2 - ac y_4^2)^2 - c(2 y_1 y_3 - 2 a y_2 y_4)^2$ is a splitting variety for the triple Massey product $\langle a,b,c \rangle$ with $F_2$ coefficients, and that this variety satisfies the Hasse principle. It follows that all triple Massey products over global fields vanish when they are defined. Jan Minac and Nguyen Duy Tan have extended this result to all $\mathbb{F}_p$ and with $p=2$ to all fields of characteristic different from $2$. The method discussed in the talk could produce splitting varieties for higher order Massey products. This is joint work with Michael Hopkins.