Distinguished Scholar-Teacher

University of Maryland

Mathematics Department

A degenerating family of elliptic

curves y^2=(1-x-t)(x^2-t).

Hyperbolic Space Tiled with Dodecahedra, 2 by Charlie Gunn.

Copyright 1990 by The Geometry Center at the University of Minnesota. Used with permission.

*This was no time for play.*

*This was no time for fun.*

*This was no time for games.*

*There was work to be done.*

* Dr.
Seuss*

*The series is divergent;*

*therefore we may be able
to do something with it.*

* Oliver Heaviside *

**Education philosophy.**

“The best teachers are usually those who are free, competent and willing to
make original researches in the library and laboratory. The best investigators are usually those who
have also the responsibilities of instruction, gaining thus the incitement of
colleagues, the encouragement of pupils, the observation of the public.” Daniel Coit Gilman - first President Johns
Hopkins University

**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)

On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. (2) 117 (1983), no. 2, 207–234.

Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), no. 1, 119–145.

The
hyperbolic metric and the geometry of the universal curve, J. Differential Geom. 31 (1990), no. 2, 417–472.

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.

Lecture:
The CAT(0) geometry of Teichmueller space, University of Chicago, April 10,
2006

Lecture:
Estimating hyperbolic Green's functions for degenerating surfaces,
University of Chicago, April 11, 2006

__Behavior of geodesic-length functions
on Teichmueller space, __Jour. Diff. Geom., 79 (2008), no. 2,
277-334.

__Grafting hyperbolic metrics and
Eisenstein series, joint with Kunio Obitsu, __Math. Annalen, 341
(2008), 685-706.

__WP Metric Geometry Quick Overview __

Extension of the Weil-Petersson connection, Duke Math. J., 146 (2009), no. 2, 281-303.

Topological dynamics of the Weil-Petersson geodesic flow, joint with Mark Pollicott and Howard Weiss, Advances in Math. 223 (2010), 1225-1235.

The Weil-Petersson metric geometry, in Handbook of Teichmueller theory, Vol. II, IRMA Lectures, European Math. Soc., 2009.

Understanding Weil-Petersson curvature, Geometry and Analysis, vol. 1. Advanced Lectures in Mathematics, Intl. Press, (2010), 495-512.

A cofinite universal space for proper actions of mapping class groups, joint with Lizhen Ji, in Contemp. Math., 510, AMS, 2010.

A Weil-Petersson sampler, a pdf presentation with a highlight of themes, current understanding and research.

Families of Riemann surfaces and Weil-Petersson Geometry, a 2008 overview on WP geometry - a preliminary plan for the following CBMS lectures is included.

Table of Contents: NSF CBMS Lectures, Families of Riemann Surfaces and Weil-Petersson Geometry, Central Connecticut State University, AMS-CBMS Regional Conference Series

Surveys
in Differential Geometry, Vol. 14 (2009): Geometry of Riemann surfaces and
their moduli spaces, a collection edited jointly with Lizhen Ji and
Shing-Tung Yau.

Geodesic-length functions and the
Weil-Petersson curvature tensor, Jour. Diff. Geom., 91 (2012), 321-359.

Lectures and notes: Mirzakhani's volume recursion and approach for the Witten-Kontsevich theorem on moduli tautological intersection numbers, lecture notes from Park City Math Institute, Graduate Summer School, 2011.

On families of holomorphic differentials
on degenerating annuli, Contemp. Math., 575, *Amer. Math., *2012.

Infinitesimal deformations of nodal stable curves, ArXiv 1204.3680

Products of twists, geodesic-lengths and Thurston shears, ArXiv 1303.0199

**Already happened:**

Workshop:
Dynamics of the Weil-Petersson geodesic flow

June 18 to June 22,
2012

American Institute
of Mathematics, Palo Alto, CA

Park City Math Institute – Graduate Summer School – Moduli Spaces of Riemann surfaces

July 3-23, 2011, Park City, UT

__Geometry and Analysis of
Riemann Surfaces and their Moduli__

September 24-26, 2010, University of Maryland, College Park

A birthday conference – group
picture – pics

A follow up to the CCSU NSF CBMS conference: Teichmueller theory

April 24-25, 2010, Central Connecticut State University

NSF CBMS Conference on Families of Riemann Surfaces and Weil-Petersson Geometry

July 20-24, 2009 Central Connecticut State University

Jeffrey K. McGowan and Eran Makover, Organizers

**And a musical
composition:**

The Mathematician by Philip Momchilovich © 2010.mp3

Philip Momchilovich is a Washington DC composer, conductor
and educator.

http://www.washingtonpromusica.org/

**Some fun math
links:**

NonEuclid- An
introduction to hyperbolic geometry and a Java-based hyperbolic geometry
drawing program.

Platonic tilings of
Riemann surfaces

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.

Penguins on the hyperbolic plane, by Misha Kapovich