Geometry and Topology

Speaker: Marcy Robertson (Western)

"Operads, multicategories, and higher dimensional deformations"

Time: 15:30

Room: MC 107

Operads, and the more general multicategories, are combinatorial devices originally used in algebraic topology as a ``bookkeeping'' devices that described the internal operations of iterated loop spaces. The basic idea of an operad, however, is quite flexible and can be adapted to problems in algebra, mathematical physics, and computer science. The goal of this talk is to give a quick introduction to the Grothendieck-Teichm\"{u}ller group, as introduced by Drinfeld and Ihara, describe some of the conjectures relating this group to quantized deformations, and explain how this conjecture is being understood through the machinery of operads (up to homotopy).

Colloquium

Speaker: Boris Khesin (University of Toronto)

"Symplectic fluids and point vortices"

Time: 15:30

Room: MC 107

We describe the motion of symplectic fluids as an Euler-Arnold equation for the group of symplectic diffeomorphisms. We relate it to the Lagrangian study of symplectic fluids by D.Ebin, describe a symplectic analog of vorticity and the finite-dimensional Hamiltonian systems of symplectic point vortices.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"Ricci Flow in Differential and Noncommutative Geometry (2)"

Time: 10:30

Room: MC 108

Studying the Ricci flow of a smooth, closed manifold M equipped with a Riemannian metric g involves the process of allowing the metric g to evolve over time under the PDE g_{t} = -2Ric(g). Ricci flow was, in fact, the main tool used by Perelman to prove the Poincare conjecture. The purpose of this talk will be to discuss what is Ricci flow, to present where it comes from and to provide examples of Ricci flow of certain manifolds. Our discussion will then lead into an analysis of a paper written by Bhuyain and Marcolli, who constructed a version of Ricci flow for noncommutative two-tori. The Ricci flow is a fundamental tool used to understand the geometry and topology of manifolds, and understanding it well will help us understand how we can classify other noncommutative spaces such as noncommutative tori in higher dimensions.

Algebra Seminar

Speaker: Ali Moatadelro (Western)

"Spectral geometry of noncommutative two torus"

Time: 14:30

Room: MC 107

Recently an analogue of the Gauss-Bonnet theorem has been proved by Connes-Tretkoff and Fathizadeh-Khalkhali for noncommutative two torus. The idea is based on the direct computation of the value at origin of the zeta function associated to the corresponding Laplacian. In this talk we will briefly discuss the above theorem and explain a related problem.

Geometry and Topology

Speaker: Sean Tilson (Wayne State University)

"Power operations in the Kunneth Spectral Sequence"

Time: 15:30

Room: MC 107

Power operations have been constructed and successfully utilized in the Adams and Homological Homotopy Fixed Point Spectral Sequences by Bruner and Bruner-Rognes. It was thought that such results were not specific to the spectral sequence, but rather that they arose because highly structured ring spectra are involved. In this talk, we show that while the Kunneth Spectral Sequence enjoys some nice multiplicative properties, there are no non-zero operations on the $E_2$ page of the spectral sequence. Despite the negative results we are able to use old computations of Steinbergers with our current work to compute operations in the homotopy of some relative smash products.

Noncommutative Geometry

Speaker: Farzad Fathizadeh (York University)

"Determinant of Laplacians on Noncommutative Two Tori"

Time: 11:00

Room: MC 106

The noncommutative two torus $A_theta$ equipped with a general complex structure and Weyl conformal factor, is a noncommutative Riemannian manifold where the metric information is encoded in the Dirac operator $D$ of a spectral triple over this C*-algebra. In a recent joint work with M. Khalkhali, we computed a local expression for the scalar curvature of $A_theta$. This was achieved by finding an explicit formula for the value at the origin of the analytic continuation of the spectra zeta function $\Zeta_a(s) := Trace (a|D|^{-s}) (Real(s) >> 0)$ as a linear functional in $a \in A_theta$ . This local expression was also computed by Connes and Moscovici independently. In this talk, I will explain how they have used this local formula and variational methods to compute the determinant of the Laplacian D2 on $A_theta$.

Ph.D. Presentation

Speaker: Gaohong Wang (Western)

"The generating hypothesis for the stable module category"

Time: 13:00

Room: MC 107

We review the study of the generating hypothesis (GH) in derived categories and stable module categories. For $p$-groups, only $C_2$ and $C_3$ satisfy the GH. We present results on the ghost numbers of $p$-groups, which test the failure of the generating hypothesis in stable module categories. We also introduce a strong version of GH for stable module categories.

Algebra Seminar

Speaker: Lex Renner (Western)

"Quotients of algebraic group actions"

Time: 14:30

Room: MC 107

Let $G \times X \rightarrow X$ be an action of the algebraic group $G$ on the affine, algebraic variety $X$. There are two quite different notions of quotient associated with this situation. If we embrace the conventional approach, we just accept the object $Y =$ Spec$(k[X]^G)$, along with the natural map $\pi : X \rightarrow Y$, as the inevitable thing to study. If $G$ is reductive then this "quotient" is a variety and it has the anticipated universal property, even if $Y$ is not an orbit space. Similar results hold if $X$ is a projective variety. Many important moduli spaces have been constructed using this approach. But maybe there is another approach, where the emphasis is on orbits of maximal dimension rather than on closed orbits. In this scenario we consider sufficiently small open, $G$-subsets $U$ of $X$ such that each $G \times U \rightarrow U$ has as many desirable properties as the situation will tolerate. If we define $U/G$ by the equation $k[U/G] = k[U]^G$, then we can ask for the following. (1) $k[U/G]$ is finitely generated. (2) $k[U/G]$ is a regular ring. (3) $\pi : U \rightarrow U/G$ is surjective. (4) $\pi : U \rightarrow U/G$ separates orbits of maximal dimension. (5) $\pi : U \rightarrow U/G$ has no exceptional divisors. (6) $\pi : U \rightarrow U/G$ is flat. We do this so as to discard only a small portion of $X$. We then try to glue all these $U/G$'s together to get a separated quotient variety, without the help of semi-invariants.

Ph.D. Presentation

Speaker: Mike Rogelstad (Western)

"Quadratic Forms, Quaternion Algebras and the Investigation of"

Time: 13:30

Room: MC 108

E. Witt observed in the 1930's that suitable classes of quadratic forms form what is now known as the Witt ring of quadratic forms. After the work of C.M. Cordes, A. Fr$\ddot{o}$hlich, D.K. Harrison, H. Koch and R. Ware, finally M. Spira and J. Min$\acute{a}\breve{c}$ established in the 1980's and published in the 1990's a rather precise link between the Witt ring of quadratic forms and Galois theory. This connection has subsequently led to many important developments in this area. We will discuss quadratic forms, quaternion algebras, the Witt ring and Brauer group as well as these fascinating interconnections and the prospects for future research.

Analysis Seminar

Speaker: Paul Gauthier (Universite de Montreal)

"Approximating the Riemann zeta-function by strongly recurrent functions"

Time: 14:30

Room: MC 107