Algebra Seminar

Speaker: Peter Lee (University of Toronto)

"The pure virtual braid group is quadratic"

Time: 14:30 - 15:30

Room: MC 107

If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant.

Geometry and Topology

Speaker: Phil Hackney (UC Riverside)

"Group actions on Segal operads"

Time: 15:30 - 16:30

Room: MC 107

Dendroidal simplicial sets satisfying an analogue of the Segal condition are a model for ($\infty$, 1)-colored operads, as shown by Cisinski and Moerdijk. We consider weak group actions on such a ``Segal operad'' and prove a rigidification theorem. This is joint work with Julie Bergner.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Determinant of Laplacians on Heisenberg Manifolds"

Time: 14:30 - 16:30

Room: MC 107

Analysis Seminar

Speaker: Alexander Odesskii (Brock University)

"Integrable Lagrangians and modular forms"

Time: 13:30 - 14:30

Room: MC 106

We investigate non-degenerate Lagrangians of the form $$ \int f(u_x, u_y, u_t) dx dy dt $$ such that the corresponding Euler-Lagrange equations $ (f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_t=0 $ are integrable by the so-called method of hydrodynamic reductions. The integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f, are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the `master-Lagrangian' corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball.

Algebra Seminar

Speaker: John E. Harper (Western)

"Localization and completion of nilpotent structured ring spectra"

Time: 14:40 - 15:40

Room: MC 107

Quillen’s derived functor notion of homology provides interesting and useful invariants in a wide variety of homotopical algebraic contexts. For instance, in Haynes Miller’s proof of the Sullivan conjecture on maps from classifying spaces, Quillen homology of commutative algebras (Andre-Quillen homology) is a critical ingredient. Working in the topological context of structured ring spectra, this talk will introduce several recent results on localization and completion with respect to topological Quillen homology of commutative ring spectra (topological Andre-Quillen homology), $E_n$ ring spectra, and operad algebras in spectra. This includes homotopical analysis of a completion construction and strong convergence of its associated homotopy spectral sequence. The localization and completion constructions for structured ring spectra are precisely analogous to Sullivan's localization and completion of spaces (for which he recently won the Wolf prize), and Bousfield-Kan's version of Sullivan's localization and completion called the $R$-completion of a space with respect to a ring $R$. This is joint work with Michael Ching.

Analysis Seminar

Speaker: Steven Lu (UQAM)

"Entire holomorphic curves in birational geometry"

Time: 14:30 - 15:30

Room: MC 107

After a brief review of classical function theory on $\mathbb C$, we will discuss its extension to functions with values in an algebraic variety, i.e. an entire holomorphic curve, and motivate the fact that such a curve should be constrained by birational invariants of the variety that pertains to "hyperbolicity." We will verity this conjectural fact for varieties of maximal Albanese dimension, itself a birational invariant. We will assume no prior knowledge of birational geometry. This is joint work with Joerg Winkelmann.

Pizza Seminar

Speaker: Stefan Tohaneanu (Western)

"Coding theory and a problem in plane geometry"

Time: 16:30 - 17:30

Room: MC 107

I will introduce the basic concepts and results in coding theory and I will explain how these can help determine the maximum number of collinear points from a finite set of points in the plane given by their coordinates.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Geometry of Quantum Heisenberg Manifolds"

Time: 14:30 - 16:30

Room: MC 107

Quantum Heisenberg Manifolds were first defined by M. Rieffel in 1989 as example of quantization of Heisenberg Manifolds along a Poisson bracket.(A typical Heisenberg Manifold is the quotient of Heisenberg group by a uniform lattice).They are interesting for several reasons, one being just because they are tractable examples of noncommutative manifolds.This means that , like the related but simpler noncommutative tori, Q-Heisenberg manifolds provide a nice setting in which to explore noncommutative geometry.

In these series of talks I will explore the different features of the noncommutative geometry on Q-Heisenberg manifolds. We introduce a class of spectral triples on Q-Heisenberg manifold, we introduce the space of L^2 -forms and then we characterize torsion less/Unitary connections. In addition, for a concrete family of unitary connections we compute Ricci curvature and scalar curvature.

Algebra Seminar

Speaker: German Combariza (Western)

"A few conjectures about multiple zeta values"

Time: 14:40 - 15:40

Room: MC 107

Multiple zeta values (MZV) are the numbers defined by the convergent series of the form

$$\zeta(s_1,s_2,...,s_k)=\sum_{n_1>n_2>...>n_k>0}^\infty \{1/(n_1^{s_1} >... n_k^{s_k})\}$$

for $s_i$ positive integers. For these real numbers there are some beautiful relations, some of them due to Euler, like $\zeta(2,1) = \zeta(3)$ or $\zeta(2n) = q\pi^{2n}$ for $q$ a rational number. In this lecture I will present some of the most famous conjectures about MZV and its relations. I will show how we try to see the truthfulness of this conjecture by looking at them until a small degree bounded by the capacity of the actual computers.

Geometry and Topology

Speaker: Roy Joshua (Ohio State University)

"Notions of Purity and the Cohomology of Quiver moduli"

Time: 15:30 - 16:30

Room: MC 107

We explore several variations of the notion of purity for the action of Frobenius on schemes defined over finite fields. In particular, we study how these notions are preserved under certain natural operations like quotients for principal bundles and also geometric quotients for reductive group actions. We then apply these results to study the cohomology of quiver moduli. We prove that a natural stratification of the space of representations of a quiver with a fixed dimension vector is equivariantly perfect and from it deduce that each of the l-adic cohomology groups of the quiver moduli space is strongly pure.

This is joint work with Michel Brion.

Analysis Seminar

Speaker: Ekaterina Shemyakova (Western)

"Completeness of Wronskian Formulas for Darboux transformations or order 2"

Time: 14:30 - 15:30

Room: MC 107

I shall report about my 2011 results, which is the resolution of one long standing problem in the theory of Darboux transformations. It is known that many Darboux transformations can be constructed using Darboux Wronskian formulas. The only known exceptions have been two transformations of order one - Laplace transformations, which are often used in applications. I shall show that for order one there is no other exceptions and that for order two Wronskian formulas are complete. History of the question as well as an introduction into the area will be provided.