Homepage of Scott A. Wolpert
Distinguished Scholar-Teacher
University of Maryland
Mathematics Department
saw (followed by: at math dot umd dot edu)
Hyperbolic Space Tiled with Dodecahedra, 2 by
Charlie Gunn.
Copyright 1990 by The Geometry Center at the
This was no time for play.
This was no time for fun.
This was no time for games.
There was work to be done.
-S.
Education philosophy.
I am actively involved in support of the University's undergraduate programs.
Our goal is to offer quality educational opportunities for every student.
Research interests.
I continue to be fascinated by the beauty
and intricacy of Riemann surfaces, especially descriptions by Fuchsian groups,
by cut-and-paste constructions and as solutions of algebraic equations. Klein's
Riemann surface with 168 symmetries is a beautiful example. Large
symmetry tessellations
of the hyperbolic plane are just as fascinating. My early and most recent research concerns
the variation of Riemann surfaces. Variations are prescribed by analytic
means or by varying the geometric assembly. The Fenchel-Nielsen
construction of surfaces with hyperbolic metric from hyperbolic
right-hexagons leads to a description of the Teichmueller space of Riemann
surfaces by specifying twist-length parameters. Thurston’s
earthquakes provide a variation by shearing along a measured geodesic
lamination. Analytic variations in the style of Kodaira-Spencer are
provided by solving the Beltrami differential equation.
I continue to study the Weil-Petersson (WP) geometry for the Teichmueller space. Understanding convexity and curvature are basic matters. The WP metric completion of the Teichmueller space, is a stratified CAT(0) metric space: a space where the distance and angle measurements for a triangle are bounded by the corresponding measurements for a Euclidean triangle with the corresponding edge-lengths. CAT(0) metric spaces are spaces of generalized non positive curvature. An overview of selected results on the WP metric is given in the research brief and in the introduction of the Geometry of the WP completion. In the second reference, the CAT(0) geometry and the behavior of geodesics in-the-large are studied in detail. An asymptotic expansion for the WP metric is provided; a classification of flats is provided and a (pre)compactness theorem for the space of geodesics is provided. A direct proof is also provided that the mapping class group is the full group of WP isometries. Jeff Brock has studied the WP geometry at large-scale and provided an approximate combinatorial model for the geometry. Teichmueller space with the WP metric is quasi-isometric to the Hatcher-Thurston pants graph. Maryam Mirzakhani has studied WP volumes of moduli spaces of bordered Riemann surfaces with fixed boundary lengths. She has presented a recursive scheme for determining the volumes and established that the recursion satisfies the string equation and dilaton equation. The recursion is the Witten-Kontsevich conjecture. She has also studied for a Riemann surface the counting function for simple closed curves in terms of length. A highlight of more recent progress is presented in Weil-Petersson perspectives. An overview of results and research questions are presented on comparison of classical metrics for Teichmueller space, on the WP synthetic geometry and on characteristic classes. The intrinsic local WP geometry is examined in Behavior of geodesic-length functions on Teichmueller space. Geodesic-length function expansions are developed for the pairing of WP gradients, Hessian and covariant derivative of the gradient. Comparability models for the metric are presented including for Fenchel-Nielsen coordinates. The behavior of geodesics near the completion locus is analyzed and a simple model is presented for the Alexandrov tangent cone at the completion. Convexity properties of Weil-Petersson geodesics are further investigated in Extension of the Weil-Petersson connection. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces.
Leon Takhtajan and coauthors have of late studied the infinite dimensional universal Teichmueller space T(1), showing that T(1) carries a new structure of a complex Hilbert manifold and that the connected component of the identity is a topological group. They define the WP metric on T(1) by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that the metric is Kaehler-Einstein with negative Ricci and sectional curvatures. The universal Liouville action of a cocycle plays an important role in the considerations. They further introduce and compute Mumford-Miller-Morita characteristic forms for the vertical tangent bundle of the universal Teichmueller curve fibration over the T(1). As an application, they obtain our earlier curvature formulas for the finite-dimensional Teichmueller spaces from the formulas for the universal Teichmueller space. Curt McMullen has recently shown that the WP metric on Teichmueller space can be reconstructed from the dimensions of dynamical quantities, such as measures on the unit circle and limit sets on the sphere. His approach reveals a connection between Hausdorff dimension, norms of holomorphic forms, and the central limit theorem for geodesic flows, especially the variance of observables of mean zero. The elements of consideration are mediated by the thermodynamic formalism, which leads to parallel results for Julia sets, polynomials and Blaschke products.
I have also considered
questions coming from mathematical physics, first from string theory,
second from the spectral theory of the Laplacian and recently from considerations
of quantum chaos. I have been concentrating on questions in the spectral
theory of Riemann surfaces, such as isospectrality ("Can you hear the
shape of the drum?"), the existence of embedded eigenvalues for surfaces
with cusps and the nature of high-energy eigenfunctions. In a
collaboration with Carolyn Gordon and David Webb we found the first example of
different shaped domains in the Euclidean plane with the same frequency
spectrum for the Laplacian. Simulated animated drum vibrations for sound-alike
domains can be viewed at Toby Driscoll's site; a
writeup and a small catalog of pairs of sound-alike domains is available
from Peter Buser, John Conway, Peter Doyle and Klaus-Dieter Semmler at the site.
For the special Riemann surfaces which are arithmetic the above questions
overlap with considerations from classical analytic number theory, such as the
Ramanujan-Petersson conjecture on the magnitude of certain Fourier coefficients
and the Lindeloef conjecture on the magnitude of the L-function on its critical
line.
Research manuscripts and lectures (in .dvi and .pdf format)
Asymptotic
relations among Fourier coefficients of automorphic eigenfunctions,Trans. AMS 356 (2004), 427-456.
Automorphic
coefficient sums and the quantum ergodicity question, Contemp. Math., vol.
256, 2000.
The Modulus of
Continuity for $\Gamma_0(m)\backslash H$ Semi-Classical Limits, Comm. Math.
Phys., vol. 216, 2001.
Semiclassical
Limits for the Hyperbolic Plane, Duke Math. J., vol. 108, 2001.
A research brief
on the Weil-Petersson metric, in PDF Kluwer
Encyc.of Math., Supp. III, 2001. A precis current only as of March
2000. For a current discussion of the global geometry of WP geodesics see
the introduction of the following manuscript. For further results also
see the work of: Jeffrey Brock and with the coauthors Benson Farb, and Yair
Minsky; Maryam Mirzakhani; Georgis Daskalopoulos and Richard Wentworth; Zeno
Huang; Howard Masur; Michael Wolf; Sumio Yamada.
Geometry of
the Weil-Petersson completion of Teichmueller space, in PDF
Surveys in Differential Geometry, VIII: Papers in Honor of Calabi, Lawson, Siu
and Uhlenbeck, editor S. T. Yau, International Press, Nov. 2003.
Hyperbolic
3-manifolds with nonintersecting closed geodesics, in PDF joint
with Ara Basmajian, Geom. Dedicata., vol. 97, 2003.
Convexity
of geodesic-length functions: a reprise,
in PDF
Weil-Petersson perspectives, in
Problems on mapping class groups and related topics, Proc. Symp. Pure
Math., 74, 2006.
Cusps and the
family hyperbolic metric, Duke Math. Jour., vol. 138, no. 3, 423-443, 2007.
The
CAT(0) geometry of Teichmueller space, University of Chicago, April 10, 2006
Estimating
hyperbolic Green's functions for degenerating surfaces, University of Chicago,
April 11, 2006
Behavior of geodesic-length functions
on Teichmueller space
Grafting hyperbolic metrics and
Eisenstein series, joint with Kunio Obitsu
WP Metric Geometry Quick Overview
Extension of the Weil-Petersson connection
Topological dynamics of the Weil-Petersson geodesic flow, joint with Mark Pollicott and Howard Weiss
The Weil-Petersson metric geometry
The Triennial Ahlfors-Bers
Colloquium
Find out more and plan to attend: May 8-11, 2008 at Rutgers-Newark http://www.ahlfors-bers.net
Some fun math links:
NonEuclid- An
introduction to hyperbolic geometry and a Java-based hyperbolic geometry
drawing program.
Penguins on the
hyperbolic plane, by Misha Kapovich
Hyperbolic
Tessellations, by David E. Joyce
Hyperbolic
Planar Tessellations, by Don Hatch
The University of Minnesota Geometry Center
Virtual Polyhedra
- A collection of thousands of virtual reality polyhedra by George W. Hart.