Welcome to the Geometry and Topology Seminar website. If you'd like to give a talk or suggest a possible speaker, please feel free to contact one of the organizers.
Organizers: Larry Guth, Hugo Parlier.
The seminar takes place on Thursdays in the Bahen Centre, room BA6183, 4:10–5:00, unless otherwise indicated.
Nov. 26, 2009, Alexandra Pettet, University of Michigan, Dynamics of Out(F_k)
The mapping class group of an orientable surface consists of homotopy classes of homeomorphisms of the surface; a pseudo-Anosov is an element no representative of which leaves a proper subsurface invariant. By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill the surface, every element not conjugate to a power of one of the twists is a pseudo-Anosov. The outer automorphism group Out(F_k) of a rank k non-abelian free group shares many properties with the mapping class group, although it is difficult in general to draw direct analogies between the two groups. For example, there are two analogues for a pseudo-Anosov mapping class: a hyperbolic and a fully irreducible outer automorphism of F_k. In this talk we will describe some of the spaces associated to Out(F_k) which have been modeled after those of the mapping class group. Using these spaces, together with an analogue of Thurston's theorem, we will show that every element of GL(k,Z) is induced by an automorphism which is both hyperbolic and fully irreducible.This is joint work with Matt Clay.
Nov. 19, 2009, Robert Young, Courant Institute, The Dehn function of SL(n,Z)
The Dehn function is a group invariant which connects geometric and combinatorial group theory; it measures both the difficulty of the word problem and the area necessary to fill a closed curve in an associated space with a disc. The behavior of the Dehn function for high-rank lattices in high-rank symmetric spaces has long been an open question; one particularly interesting case is SL(n,Z). Thurston conjectured that SL(n,Z) has a quadratic Dehn function when n>=4. This differs from the behavior for n=2 (when the Dehn function is linear) and for n=3 (when it is exponential). I have proved that it is quadratic when n>=5, and in this talk, I will discuss some of the background of the problem and sketch a proof that it is at most quartic when n >= 5.
Oct. 29, 2009, Yosef Yomdin, Weizmann Institute, Reconstruction of Semi-Algebraic Sets from Integral Measurements and Moment vanishing Problem
The problem of reconstruction of semi-algebraic sets and functions from integral measurements, like moments or Fourier transform naturally arises in Signal Processing on one side, and in Qualitative Theory of ODE's on the other. We plan to present some recent results in this direction, stressing the uniqueness of reconstruction. This reduces to the moment vanishing problem: give conditions for identical vanishing of the moments m_k = \int P^k(x)q(x)dx, for various classes of P and q, and various integration domains. We shall present recent progress in some special cases of this problem, as well as relations with the Mathieu conjecture in representations of compact Lie groups, and (through the recent work of Wenhua Zhao) with certain questions around the Jacobian conjecture.
Oct. 22, 2009, Armin Rainer, University of Toronto, Perturbations of polynomials and lifting mappings over invariants
Given a smooth family of complex univariate polynomials, it is natural to study the regularity of its roots. This is an old problem with important applications in PDE theory and perturbation theory of linear operators. I will give an exposition of the known results, focusing on recent progress which was made using resolution of singularities. I will also explain how the perturbation problem for polynomials fits into a more general geometric framework, namely, the problem of lifting mappings over invariants of reductive linear algebraic group representations.
Oct. 8, 2009, Rustam Sadykov, University of Toronto, Topological methods of solving differential relations
Early results by Smale, Hirsh, Nash and others prompted Gromov to introduce and develop the so-called h-principle, which is a topological method of solving differential equations and, more generally, differential relations. I will give an elementary exposition of ideas behind the h-principle. I will also introduce a b-principle which is a homology version of the h-principle. In comparison to the h-principle, the b-principle gives less information about solutions of differential relations, but on the other hand, it is considerably easier from the point of view of computations.
Oct. 1, 2009, Hugo Parlier, University of Toronto, Teichmüller spaces and simple closed geodesics
Roughly speaking, Teichmüller spaces are a parameter set for the set of constant curvature metrics one can put on a given topological surface. The goal of the seminar will be to show certain aspects of how one can understand certain aspects of Teichmüller spaces by studying simple closed geodesics on surfaces.
The Geometry & Topology seminar archives.
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