September 28, 2006

Speaker: Adam Koranyi, Herbert H Lehman College (CUNY)
Title of talk: Homogeneous operators On Hilbert Spaces With Reproducting Kernel
Abstract: This will be about joint work with Gadadhar Misra. An operator T on a Hilbert space is called homogeneous (with respect to the unit disc D) if its spectrum is contained in the closure of D and g(T) is unitarily equivalent to T for every Mobius transformation g. These operators have interesting properties, but not overly many examples of them have been known until now. In the talk a family of such operators depending on countably many continuous parameters will be constructed. The construction is explicit and elementary, but in its background there are results about homogeneous holomorphic vector bundles. It will be attempted to explain these connections.

October 5, 2006

Speaker: Andrew Granville, Montreal
Title of talk: Various pretentious characters
Abstract: We explain how certain key results in analytic number theory can be rephrased in terms of pretentiousness, and discuss some joint results with K. Soundararajan motivated by this new concept.

October 12, 2006

Speaker: Weiqiang Wang, University of Virginia
Title of talk: Platonic solids, Hamilton's quaternion, and Dynkin diagrams
Abstract: We describe some classical algebraic and geometric connections among the five regular polyhedra, finite subgroups of quaternion, and Dynkin diagrams. We will then discuss a modern variation of the above themes, via Hilbert schemes of points and wreath products.

March 1, 2007
3:30pm, Room 328
Speaker: Michael Spivak
Title of talk: Physicists’ Rigid Bodies With Mathematician’s (Being Lesson 1 of Physics Without Tears)
Abstract: Newton's laws apply to "particles" or "point masses," which can also be considered to apply to the objects of astronomical problems, but you can't do most other physics problems without considering larger (rigid) bodies.

Newton never discussed rigid bodies (smart man). Euler's pioneering treatment, the basis for the elementary undergraduate hocus-pocus, regards solid bodies as continuous expanses of matter, a rather disconcerting view in the atomic age, whereas the advanced graduate hocus-pocus considers a collection of particles bound by "constraints" in a manner sufficiently abstract to hide all the difficulties in a haze of generalities.

This lecture attempts to give a coherent exposition of the subject, essentially explaining and giving meaning to some of the strange things that physics textbooks contain.

March 5, 2007 - (please note this is a Monday, also the room change)
3:30pm, Room 302
Speaker: Endre Szemeredi, Rutgers University
Title of talk: Finite and infinite arithmetic progressions in sumsets
Abstract: We prove that if A is a subset of at least cn^{1/2} elements of {1,2,...,n}, (where c is a sufficiently large constant), then the collection of sums formed from the subsets of A contains an arithmetic progression of length n. As an application, we confirm a long standing conjecture of Erdos and Folkman on complete sequences. Joint work with Van Vu.

March 6, 2007 - (please note this is a Tuesday)
3:30pm, Room 304
Speaker: Bill Goldman, University of Maryland
Title of talk: Dynamics of surface group representations
Abstract: The space of representations of the fundamental group of a surface
in a Lie group is a rich geometric object, with an algebraic structure enjoying much symmetry. The simplest examples include symplectic vector spaces, Jacobi varieties, and moduli spaces of holomorphic vector bundles.

Fricke-Teichmueller spaces also arise as representation spaces. They are a special case of deformation spaces of locally homogeneous geometric structures in the sense of Ehresmann and Thurston. The underlying algebraic structure of deformation spaces closely relates to the geometric structures they parametrize. Understanding the geometric structures is often a key for understanding the topology and dynamics of these spaces.

The mapping class group of the surface acts on this space preserving a natural Poisson geometry. Natural Hamiltonian flows on the deformation space generalize the classical Fenchel-Nielsen twist flows on Teichmueller space. For compact Lie groups, the mapping class group action is chaotic. The proof of ergodicity can be regarded as an analog of the Fenchel-Nielsen coordinates for Teichmuller space. For representations corresponding to uniformizations by geometric structures, the action is proper.

In general the dynamics falls between these two extremes. In the case of a one-holed torus, the dynamics reduces to an action of the modular group on cubic surfaces related to the Markoff equation, where both chaotic and proper dynamics coexist.

March 8, 2007
3:30pm, Room 304
Speaker: Herbert Lange, Erlangen, Germany
Title of talk: Schur and Kanev correspondences.
Abstract: Correspondences on curves are used to construct Prym-Tyurin varieties which represent a generalization of Prym varieties: special types of abelian varieties. In order to construct Prym-Tyurin varieties, several people associated to every finite Galois covering of smooth projective curves a type of correspondences, which are equivalent to Schur's character relations and which we therefore call Schur's correspondences. Another type of correpondences was introduced by Kanev using the monodromy of a spectral covering. In the talk the relation between both correspondences will be explained and several examples will be given. This is joint work with Anita Rojas.

March 20, 2007 (please note this is a Tuesday)
3:30pm, Room 304
Speaker: Yair Minsky, Yale
Title of talk: Curve complexes, surfaces and 3-manifolds
Abstract: A compact oriented surface determines an interesting combinatorial object: The complex whose vertices are homotopy classes of simple loops, and whose simplices are subsets of vertices with disjoint representatives. This finite dimensional, locally infinite complex turns out to be useful in studying the mapping class group of a surface, the Teichmuller space of hyperbolic structures on the surface, and the deformation theory of hyperbolic 3-manifolds. I will give a biased survey of this subject.

Monday, April 23, 2007
3:30pm, Room 323
Speaker: Professor R. Parthasarathy, Tata Institute
Title of talk: On the quantum analogue of a coherent family of modules at roots of 1
Abstract: This talk is about arriving at a quantum analogue ${\overline{\pi}(\mu)}_{\mu}$ for the quantum group $U_{\lambda}$ at an $\ell$-th root of 1 of a given coherent family of modules ${\pi(\mu)}_{\mu}$ of the enveloping algebra $U$ of a finite dimensional semisimple Lie algebra $\mathfrak g$. We will discuss an open problem and indicate what is involved in its solution (at present known only in low rank). Combined with the observation that one can almost surely 'put' any interesting representation as a member of a coherent family, This gives us a potential candidate which can be regarded as the quantum analogue of the given representation.