David Glickenstein, Department of Mathematics, The University of Arizona, will speak on “Combinatorial Yamabe Flow” at 4:00 PM in Math 402.
We shall study a piecewise-linear geometry which lies somewhere between the geometry of graphs and the geometry of Riemannian manifolds. In our context, the geometry comes from a simplicial complex whose vertices are given weights which determine the lengths of edges (so the vertices and edges form a weighted graph), and hence the area and volume of higher dimensional simplices. Combinatorial Yamabe flow is a way to deform the geometry into something less complicated via an ordinary differential equation, an analogue of the Ricci or Yamabe flow in Riemannian geometry designed for a piecewise-linear object instead of a smooth manifold. Such equations may be helpful in applying the successful methods of geometric evolution equations to new realms of problems in physics, topology, algebraic geometry, numerical analysis, graph theory, and other fields. The methods will involve basic Euclidean geometry as well as the application of simple ideas from partial differential equations to functions on graphs. This talk will be self-contained and should be easily accessible to graduate students and those in other fields.
Pani Konstantinou, Department of Mathematics, The University of Arizona, will speak on “The space of homomorphisms from the fundamental group of a surface into PSL(2,R)” at 4:00 PM in Math 402.
This will be a survey talk on the space of homomorphisms from the fundamental group of a compact surface into PSL(2,R). This space divides up into a finite number of connected components. Goldman proved that one of these components is Teichmuller space. The mapping class group acts properly discontinuously on this component. The nature of the action on the other components is unknown. We will discuss some preliminary results.
Michael Otto, Department of Mathematics, The University of Arizona, will speak on “An introduction to the Duistermaat-Heckman formula” at 4:00 PM in Math 402.
The Duistermaat-Heckman formula provides an interesting example of a
localization formula. It describes how an integral related to a
Hamiltonian torus on a symplectic manifold M can be evaluated by just
looking at the points of M fixed under the torus action. The formula
thus obtained essentially describes how the Liouville measure on M is
pushed forward under the associated moment map. Students are especially
encouraged to attend.
Derek Habermas, Department of Mathematics, The University of Arizona, will speak on “Compact Symmetric Spaces, Triangular factorization, and Cayley coordinates” at 4:00 PM in Math 402.
Symmetric spaces are a special class of Riemannian manifolds, and
such a space X can be realized as a quotient space of a Lie
group G that acts isometrically, by GΘ, the
fixed point set of an involution of G. The embedding
φ: G/GΘ → G, φ: gGΘ → gg–Θ
(due to Cartan) is totally geodesic, and its image satisfies a simple
algebraic equation. Another fact is that a generic element in G
can be decomposed uniquely into a product g=ldu relative
to a triangular factorization of the Lie algebra. When we intersect φ(G/GΘ)
with this triangular factorization, interesting things happen, and when
X is compact, Cayley coordinates can be used to make explicit
calculations.
Arlo Caine, Department of Mathematics, The University of Arizona, will speak on “Compact symmetric spaces, the diagonal distribution, and the Duistermaat-Heckmann formula” at 4:00 PM in Math 402.
Doug Pickrell recently produced a formula for the Fourier transform of the diagonal distribution for the push-forward of the invariant measure of a compact type symmetric space U/K under the Cartan embedding U/K -> U. I will set up the statement of the problem and its answer and discuss the techniques from Poisson and symplectic geometry involved in the proof. The talk will emphasize examples and all the necessary notions from Poisson geometry will be introduced. Familiarity with the Duistermaat-Heckman formula as presented in Michael Otto's recent lecture is all that will be assumed from symplectic geometry.
Lennie Friedlander, Department of Mathematics, The University of Arizona, will speak on “Non-commutative residue” at 4:00 PM in Math 402.
I will give an overview of the notion of non-commutative residue
that was introduced by Guillemin and Wodzicki about twenty years ago.
The non-commutative residue is used for analytic continuation of
generalized zeta functions. This is a preparation for my follow up talk
on determinants of zero order operators (my joint work with Victor
Guillemin.)
Leonid Friedlander, Department of Mathematics, The University of Arizona, will speak on “Determinants of zeroth order operators” at 4:00 PM in Math 402.
I will discuss the definition and the properties of the determinant
of an elliptic pseudodifferential operator of order zero. The results
are a part of my joint project with Victor Guillemin.
Adam Spiegler, Department of Mathematics, The University of Arizona, will speak on “Stability of generic equilibrium of the 2n-dimensional free rigid body” at 4:00 PM in Math 402.
The trajectories of the motion of the 2n-dimensional free rigid body are geodesics with resepect to a left-invariant metric on the Lie group SO(2n). The phase space for the system is a symplectic manifold, namely the cotagent bundle of SO(2n). Using the underlying geometry of the phase space, one can show the corresponding restriction to coadjoint orbits in the dual Lie algebra is a Hamiltonian system as well. I will discuss the stability of generic equilibrium to the 2n-dimensional free rigid body, which can be obtained using the energy-Casimir method.
Kirti Joshi, Department of Mathematics, The University of Arizona, will speak on “Infinite dimensional vector bundles in algebraic geometry” at 4:00 PM in Math 402.
This is a report on a paper of Drinfeld available on the preprint archive. I will try to keep the talk as down to earth as possible.