I am Professor in the
Mathematics Department
at the University of Maryland.
 Office: 3106 William E. Kirwan Hall, 4176 Campus Drive,
College Park, MD 207424015
 TEL: (301)4055124, FAX: (301)3140827
 EMAIL: wmg AT umd DOT edu
 Most people refer to me by my nickname "Bill".
I am teaching Math 431
(Geometry for Computer Applications),
a course I've been developing for many years.
In 2021 it was taught by my former doctoral student
Justin WyssGallifent and his notes are a good reference
for some of the quantitative aspects of the course.
A description of the course may be found here.
I serve on the faculty of both the
MATH and
AMSC
graduate programs at the University of Maryland and have supervised
several doctoral dissertations in both programs.
Recent and upcoming lectures:
Selected research projects:
 Here is a recent preprint discussing
how the deformation space of complete affine 2manifolds relates to a
twisted cubic cone in 4space. This result was suggested by Pierre Deligne
after I mentioned it in a lecture I gave at the Institute for Advanced
Study during my 2021 sabbatical there.
It gives an explicit proof of Oliver Baues's theorem that this deformation
space is homeomorphic to a plane. This paper will appear in a Contemporary Mathematics volume celebrating Ravi Kulkarni's 80th birthday.
 My first paper with my colleague Giovanni Forni,
Mixing Flows on moduli spaces of flat bundles over sruraces appeared in Volume 2 of ``Geometry and Physics,'' dedicated
to Nigel Hitchin's 70th birthday conferences in Oxford, Aarhus and Madrid.
(I could only attend the conference in Oxford.)
This paper outlines a program to study dynamical problems arising from
the classification of geometric structures on manifolds by introducing a
cocycle of symplectic moduli spaces over the Teichmueller geodesic flow
and SL(2,R)action.
A second paper, with Forni and Carlos Matheus and Sean Lawton,
may be found here.

The paper,
by Rachel (Nakyung) Lee and Karin Melnick,
extends Lee's 2023 awardwinning doctoral dissertation
classifying Lorentzian conformal structures with unipotent holonomy on closed manifolds. This is the first major progress on the classifiction of
arabolicgeometry structures on closed manifolds with nilpotent holonomy
since my 1983 paper on conformally flat manifolds with nilpotent holonomy.

A part
of the introduction for the Festschrift for Toshiyuki Kobayashi
dealing with his work on proper actions in pseudoRiemannian geometry.

A ZblMath review of Michael Magee's
article "Random Unitary Representations of Surface Groups I:Asymptotic
Expansions."
 Current and past experimental projects
 I am currently investigating a
remarkable family of real affine cubic
surfaces which arise as relative SL(2)character varieties for a oneholed
torus.
See if you can run the
interactive Mathematica notebook
I wrote with Ajeet Gary when he was an undergraduate at Maryland.
 Codirector (with Rodrigo Trevino),
Lab for Experimental Mathematics at MAryland (LEMMA).
I cofounded this experiment with
Richard Schwartz
in 2000, when it was called the
Experimental Geometry Lab.
You can read about it here.
I was one of the founding members of
Geometry Labs United,
and coorganized the
Geometry Labs United Conference
which took place at
ICERM,
1617 July 2020.

Computational Aspects of Discrete Subgroups of Lie groups,
ICERM, 1418 June 2021.
Here are the slides from my talk,
and written version here.
In that talk I discuss mathematical developments arising from
a 1992 REU project with thenundergraduate Robert Benedetto,
when we discovered compact components of the SL(2,R)character variety
of a fourholed sphere.

Another project in 1992 with Sergey Brin
in the Undergraduate Apprecenticeship in Research and Scholarship,
concerned Visualizing Pappus's Theorem in Projective Geometry.
Recent Administrative Activities:
 MemberAtLarge of the Council of the
American Mathematical Society
and member of the
Committee on Publications.

Brin Mathematics Research Center at the University of Maryland.
 Member, Scientific Advisory Board.
 I recently coorganized (with Steve Bradlow and Richard Wentworth)
Brin MRC
a workshop ``Advances in Higgs bundles."
there.
 Former member (with Patrick Brosnan and Abba Gumel),
Department Colloquium Committee.
 Administrative activities before 2015
 Network Executive Committee (original coPI, and director
of Maryland hub),
GEAR Network, an NSF Research Network in the Mathematical Sciences,
20112022; with Steve Bradlow, Steve Kerkchff, Richard Wentworth and
Anna Wienhard.
 Scientific Advisory Board (20202022),
Institute for Computational and Experimental Research in Mathematics
(ICERM), Brown University.
Curriculum Vitae
Research papers
Selected reviews
Previous research students;
also see my listing on the
Math Genealogy Project.
Check out Evan Goldman's art
A photograph of my brother and me with
President Harry S. Truman
at his home in Independence, Missouri in 1964.
Link to Ameridan Mathematical Society Bookstore for
Geometric Structures on Manifolds,
published in the AMS Graduate Studies in Mathematics volume 227.
(li+409 pp. ISBN: [9781470471033]; [9781470471989]; [9781470471972] 57 (22 5301))
Check out the recent
review by the Mathematical Association of America and
in Math Reviews (MathSciNet) from the American Mathematical Society.
Recent draft of
Proper actions of discrete groups of affine transformations,
coauthored with Jeffrey Danciger, Todd Drumm and Ilia Smilga,
from the Margulis Festschrift,
``Dynamics, Geometry, Number Theory: The Impact of Margulis in Modern Mathematics," University of Chicago Press.
See also http://arxiv.org/abs/2002.09520.
A preliminary draft of
a recent survey, from ``Geometries in History" (S. Dani and A. Papadopoulos, eds.
Springer 2019)
Exotic projective triangle tesselation
on the cover of the November 2002 issue of the Notices
of the American Mathematical Society, drawn by Bill Casselman.
Last updated: Tuesday 22 October 2024.