Seminars and Colloquia - Winter 2011
Dr. Chunhua Ou
Applied Dynamical Systems Seminar
3:30 p.m., January 26,2011
Spatial Spread of Rabies with the Impacts of the Incubation and the Spatial Movement of the Incubative
In this talk, spatial spread of rabies in Europe is re-visited with the consideration of the impacts of the incubation and its interaction with the spatial movement of the susceptible and the incubative. First, a delayed reaction diffusion model is constructed with the incorporation of the incubation only. The minimal spreading speed is derived by the classical stability analysis and it is shown to be a decreasing function of the incubation time. A completely new method based on the integration of wave pulses is proposed which yields the existence of wave patterns for all values of the incubation time. In addition, by incorporating the spatial movement of the incubative foxes, a non-local reaction diffusion system is constructed which represents the the interaction between them. Rigorous proof of the existence of the wave patterns is shown via the integration of the wave pulses coupled with the Fredholm alternative theorem. For fixed time period of incubation, the minimal speed is found to be an increasing function of the diffusion coefficient of the incubative foxes. The analysis of this paper not only presents new results concerning the spreading speed of rabies in Europe with the impact of the incubation period and the spatial movement of the incubative foxes, but also provides rigorous proofs of the existence of traveling wave patterns which has been outstanding for long time in this case.
Dr. Alexey Gordienko
1:00 p.m., January 19, 2011
On the equivalence of Lie commutators
The talk is concerned with a partial case of Yuri Bahturin's conjecture published in DNIESTER NOTEBOOK: Unsolved Problems in the Theory of Rings and Modules (available e.g. at http://www.ime.usp.br/~jcgf/MAT5728/dniester.pdf). We prove that if two multilinear Lie commutators of the same length have different number of subcommutators of the type [x,y], where x and y are variables, then they are not equivalent as polynomial identities both for associative and Lie algebras. The same is true for the number of subcommutators of the type [[x,y],z] and [[x,y],[z,u]]. Our technique involves nilpotent algebras with commutative or skew-commutative generators, constructed by graphs.
Dr. Eric Bahuaud
2 p.m., January 14, 2011
Asymptotically hyperbolic metrics and geometric flows
Consider the open unit ball in n-dimensional space. The Riemannian metric ds2 = ( 4/(1-|x|2)2 ) ( (dx1)2 + ... (dxn)2 )yields the Poincare model of hyperbolic space. An interesting feature of this metric is that if we multiply it by (1-|x|2)2/4 we obtain the usual euclidean metric, which extends over the boundary of the ball. A generalization of this gives rise to a significant notion of a "conformally compact" asymptotically hyperbolic metric. There has been growing interest in asymptotically hyperbolic metrics. The study of these metrics involves geometry and analysis on noncompact manifolds, and these metrics play an important role in conformal and Riemannian geometry. Recently asymptotically hyperbolic metrics have also been of interest to physicists both as a model for the AdS-CFT correspondence and as initial hypersurfaces for the Einstein constraint equations in general relativity. In my talk I will introduce asymptotically hyperbolic metrics and discuss their importance and some open problems. I will also describe some of my recent work using various geometric flows, including the Ricci flow, to study these metrics.
Dr. Hugh Thomas
University of New Brunswick - Fredericton
12:00 p.m., January 12, 2011
Garside structures for braid groups
As is well-known, the symmetric group Sn can be generated by the simple transpositions si = (i i+1). These satisfy the relations that si2=e (the idempotence relations), and certain additional braid relations:sisj=sjsi if |i-j|>1, sisi+1si=si+1sisi+1.If we consider a group with n-1 generators si which satisfy just the braid relations, but not the idempotence relation, we get the braid group on n strands, Bn. In order to facilitate doing calculations in Bn, it's useful to have a normal form for the elements of the group. This, and more, is provided by what is called the Garside structure for the group, the idea of which I shall explain. The first such structure for Bn was found by Garside, but I will focus on a different one, due to Birman-Ko-Lee. This structure generalizes to Artin groups associated to finite Weyl groups, as was shown by Bessis. (In fact, everything I am talking about generalizes to that setting, including Garside's original Garside structure but for simplicity I will for the most part stick to the symmetric group/braid group setting.) I will give a different description of the Birman-Ko-Lee Garside structure, due to Colin Ingalls and me, in terms of quiver representations. In order to do so, I will define quiver representations, and provide some background about them. In particular, the notion of exceptional sequence, originally studied for vector bundles by Rudakov and his school, and transported into the setting of quiver representations by Crawley-Boevey and Ringel, will be essential. Finally, I will explain a bit about ongoing efforts to extend the Birman-Ko-Lee-Bessis Garside structure to braid groups corresponding to other (not necessarily finite) reflection groups.