In Spring 2004 the seminar will meet Wednesdays 3:00 to 4:00 PM in Room 5417 at the CUNY Graduate Center. The graduate center building is 365 5th Avenue (34th St) in Manhattan. This seminar is organized by Yiannis Petridis, who can be contacted at petridis@comet.lehman.cuny.edu. For a link to the previous semester's talk go to Fall 2003
Mahler measure of knot polynomials and hyperbolic volume
Discovered 20 years ago, the Jones polynomial is still not understood in terms of the topology of the knot complement. Recent interest in the Jones polynomial has focused on the geometry of the knot complement. Notably, the hyperbolic volume conjecture proposes that the volume of a hyperbolic knot is the logarithmic limit of colored Jones polynomials evaluated at roots of unity. The Mahler measure of the Alexander polynomial and A-polynomial has been related to the volume of the knot complement. We show that the Mahler measure of the Jones polynomial and of the colored Jones polynomials converges under twisting in any link diagram. In this respect, the Jones polynomial, like the Alexander polynomial, behaves like hyperbolic volume under Dehn surgery. We discuss related results for pretzel knots, torus knots, and the simplest hyperbolic knots. The main proofs combine the representation theory of braid groups with linear skein theory. This is joint work with Abhijit Champanerkar (Barnard College, Columbia University).
Minimal deformations versus complex deformations of subvarieties of Kaehler manifolds
Wirtinger's inequality implies that smooth complex subvarities of a Kaehler manifold are area minimizers, hence minimal submanifolds. A classical problem is to understand under which conditions the converse holds. I will talk about how to construct non-holomorphic compact minimal submanifolds of compact Kaehler manifolds (with emphasis on the more interesting case of positive curvature) and to relate the classical theory of deformations of holomorphic submanifolds due to Kodaira-Spencer and Bloch, to the one for minimal submanifolds.
Construction of Random Riemann Surfaces (joint work with Robert Brooks)
What are the geometric properties of a typical Riemann surface of large genus? We address this problem by constructing compact Riemann surfaces from random cubic graphs. The resulting surfaces form a dense set in the set of all Riemann surfaces. This construction enables us to investigate the geometry of Riemann surfaces using the geometry of random graphs. The resulting picture is an interesting mix of techniques from Riemannian geometry, complex analysis, and probability theory, and contains some surprising answers.
Integrated density of states for random metrics on manifolds (joint with D.Lenz and N.Peyerimhoff)
"We study ergodic random Schrdinger operators on a covering manifold, where the randomness enters both via the potential and the metric. We prove measurability of the random operators, almost sure constancy of their spectral properties, the existence of a self-averaging integrated density of states and a Pastur-Shubin type trace formula."
Quantum Random Walks on graphs
Quaternion-Kaehler manifolds
Abstract: Quaternion-Kaehler (QK) geometry is one of the geometries determined by the shortlist of possible holonomy groups for (irreducible non-ocally symmetric) Riemannian manifolds given by Berger. This geometry, at first, has certain similarities with Kaehler geometry which is the reason for its name. We shall review these similarities and the construction of the twistor space. The manifolds turn out to be Einstein, thus giving a trichotomy by the sign of the scalar curvature. When the scalar curvature is positive (and the manifold is compact), it is conjectured that these manifolds are symmetric and belong to the list given by Wolf. We shall describe briefly how this claim has been settled in low (quaternionic) dimensions.
The Decomposition of global conformal invariants.
Abstract: The work I will present deals with the integrals ${\int}_{M^n}P(g^n)dV_{g^n}$ of scalar Riemannian quantities over compact even-dimensional manifolds. We consider such quantites (of weight $-n$) for which the integral above is invariant under conformal transformations of the metric $g^n$. It had been conjectured that such a quantity must be a linear combination of a scalar conformal invariant, a divergence of a vector field, and of the Chern-Gauss-Bonnet integrand. In this lecture we will outline the proof of that conjecture.
Introduction to Random Matrix Theory
We will start by explaining briefly the main physical motivation for Random Matrix Theory (RMT), namely that it suggests a model that describes the statistical behavior of energy levels of complex systems. There are three main types of ensembles of random matrices that are physically motivated: unitary, orthogonal, and symplectic. After that we define the Unitary Ensemble of random matrices, introduce the basic probabilistic quantities of interest, and show how these quantities can be expressed in terms of orthogonal polynomials (OP's). We will then explain the idea of universality in RMT, and in particular introduce the appropriate scaling limit. Universality means that the statistical behavior predicted by RMT should not depend on a particular choice of distribution of the matrix elements (which has no physical meaning), but should depend only on the type of symmetry imposed on the ensemble (in this case, unitary) which is physically meaningful. At this point it will be apparent that the proof of universality for unitary ensembles reduces to a study of asymptotics of the OP's. Such a study is possible due to the fact that the OP's solve a certain Riemann-Hilbert problem (RHP). Finally, we mention the appropriate RHP, and the proof of universality for the unitary case. This talk can serve as a preparation for our second talk that deals with the proof of universality for the orthogonal and symplectic ensembles. (The second talk is on our joint work with P.Deift.)
4:00 PM, Laszlo Lempert (Purdue University)
Analytic cohomology of a loop space
Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles
Abstract: This is a joint work with P.Deift. We give a rigorous proof of the Universality Conjecture in Random Matrix Theory for orthogonal (beta=1) and symplectic (beta=4) ensembles in the scaling limit for a class of polynomial potentials whose equilibrium measure is supported on a single interval. Our starting point is Widom's representation of the correlation kernels for the beta=1,4 cases in terms of the unitary (beta=2) correlation kernel plus a correction. In the asymptotic analysis of the correction terms we use amongst other things differential equations for the derivatives of orthogonal polynomials (OP's) due to Tracy-Widom, and uniform Plancherel-Rotach type asymptotics for OP's due to Deift-Kriecherbauer-McLaughlin-Venakides-Zhou. The problem reduces to a small norm problem for a certain matrix of a fixed size that is equal to the degree of the polynomial potential.
Singular Semi-Flat Calabi-Yau Metrics on S^2
Abstract If u is a convex function on a domain \Omega\subset R^n satisfying the Monge-Ampere equation det(u_{ij}) = 1, then there is a natural Calabi-Yau metric on the tube domain \Omega + i R^n: Extend u to be constant on the imaginary fibers and the Kahler metric u_{i\bar\j}dz^i \overline{dz^j} is Ricci-flat. We call the real metric u_{ij}dx^idx^j a semi-flat Calabi-Yau metric. Such metrics also may exist on manifolds which admit a flat affine connection the tangent bundle and a volume form \omega so that \nabla\omega=0. Recently, such metrics have been of interest in understanding mirror symmetry, according to the conjecture of Strominger-Yau-Zaslow. In particular, Gross-Wilson have constructed many such metrics on S^2 which are singular at 24 points, as real slices of limits of Calabi-Yau metrics on elliptic K3 surfaces. We construct many such metrics on S^2, singular at any n>=6 points, and compute the local affine structure near the singularities. The techniques involve affine differential geometry and a solving a semilinear PDE on S^2 minus singularities.
How to blow hyperbubbles
Abstract: We examine the differences -- both topological and analytical -- between weak and strong limits for minimizing sequences for the higher-dimensional analogue of the Dirichlet energy.
Higgs bundles and surface group representations
Abstract: On any manifold there is a well known correspondence between representations of the fundamental group and flat principal bundles over the manifold. If the manifold is complex - for instance a Riemann surface - and the representations are into a reductive Lie group, then the corresponding bundles acquire the structure of objects known as Higgs bundles. We will describe this correspondence and discuss how it can be exploited to study topological features of the representation variety, i.e. the space of all fundamental group representations modulo conjugation.
A new approach to the essential spectrum of Schroedinger, Klein-Gordon and Dirac operators
Abstract. The aim of the talk is to present a new approach to the study of the essential spectrum of the Schroedinger, Klein-Gordon and Dirac operators. We include these operators in a class of pseudodifferential operators perturbed by non-smooth potentials. For an operator under consideration we introduce a family of limit operators, and prove that the essential spectrum of the original operator is the union of spectra of limit operators. Because the limit operators have, as a rule, simpler structure than the original operator, we obtain a strong tool for investigation of the essential spectrum of differential and pseudodifferential operators. We apply this method to the study of the essential spectrum of the Schroedinger, Klein-Gordon, and Dirac operators.
9:00-10:00 am Stephanie Alexander (UIUC)
Warped Products of Metric Spaces
We give a new global criterion for warped products of metric spaces to have a given curvature bound above or below. This construction extends the known criterion for linear cones, by replacing the radial cooredinate with elements of a rich class of generalized convex functions on metric spaces. The proof considers the propogation of curvature bounds, and the geometry of vertex sets (vanishing points of the warping function). In addition to coning from 1 dimensional to arbitrary base, the construction generalizes standard gluing theorems from 0 dimensional to arbitrary fibre.
10:30-11:30 am Wei Dong Ruan (UIC)Degeneration of Kahler Einstein Spaces
In this talk we will explore the construction of a generalized special Lagrangian Torus fibration for Calabi-Yau hypersurfaces in toric varieties through the deformation method. Such construction is important in understanding the geometry of Calabi-Yau manifolds in light of the Strominger-Yau-Zaslow Conjecture from mirror symmetry.
12:00-1:00 pm Yu Ding (UC Irvine)Some existence results of harmonic functions with polynomial growth on manifolds
Abstract: We show the existence of harmonic functions on certain manifolds with nonnegative Ricci curvature. Roughly speaking, this is to solve a Dirichlet problem at the space infinity.
Vector Fields and Distributions for Optical Design
When viewing a curved mirror, it is apparent that some non-linear transformation is at work, which depends upon the mirror shape. In this talk I will address the problem of determining the mirror shape that will realize a prescribed transformation. The prescribed transformation determines a vector field which should be normal to the sought after mirror, but generally this vector field is not exact. Appealing to the Hodge theorem allows one to find "best fit" surfaces normal to the vector field. We will describe several applications, including panoramic cameras, a car mirror with no blindspot, and a means of experimentally measuring the shape of the cornea. On closer inspection one finds that the underlying object of consideration should not be a vector field, but one of several other candidates. One, for example, is a planar distribution in R^3, and the best fit functional in this model gives rise to the mean curvature equation. Finally I will describe an alternative technological approach to the problem, namely to use silicon-based micromirror arrays that can "integrate" a non-integrable distribution.