WILLIAM GOLDMAN
    I am Professor in the Mathematics Department at the University of Maryland.
    • Office: 3106 William E. Kirwan Hall, 4176 Campus Drive, College Park, MD 20742-4015
    • TEL: (301)-405-5124, FAX: (301)-314-0827
    • EMAIL: wmg AT umd DOT edu
      • Most people refer to me by my nickname "Bill".

Teaching

  • I recently finished teaching Math 431 (Geometry for Computer Applications), a mathematics course for Computer Science majors.
    • You can read about it here.
    • In 2021 it was taught by my former doctoral student Justin Wyss-Gallifent whose notes describe the quantitative aspects of the course.

  • I am currently teaching Math 744 (Lie Groups I).
  • 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.
  • In 2007, I was appointed a Distinguished Scholar-Teacher at the University of Maryland.
    I gave a public lecture entitled, ``Playing pool on curved surfaces and the wrong way to add fractions."

    Research

    Geometric Structures on Manifolds

    At the very end of 2022, my book Geometric Structures on Manifolds, was published in the AMS Graduate Studies in Mathematics volume 227.
    (li+409 pp. ISBN: [9781470471033]; [9781470471989]; [9781470471972] 57 (22 53-01)), and is available at the AMS bookstore.

  • Check out the recent review by the Mathematical Association of America and the review in Math Reviews (MathSciNet) from the American Mathematical Society.
  • Now I am collecting Errata and Comments.

    Related works on Geometric Structures

  • The paper, by Rachel (Nakyung) Lee and Karin Melnick, extends Lee's 2023 doctoral dissertation classifying Lorentzian conformal structures with unipotent holonomy on closed manifolds.
    This is the first major progress on the classifiction of parabolic-geometry structures on closed manifolds with nilpotent holonomy since my 1983 paper on conformally flat manifolds with nilpotent holonomy.
    Rachel's thesis won the James.C. Alexander prize for Graduate Research at the University of Maryland.
  • A part of the introduction for the Festschrift for Toshiyuki Kobayashi dealing with his work on proper actions in pseudo-Riemannian geometry.
    This recently appeared in ``Symmetry in Geometry and Analysis, Volume 1,'' Birkhauser Progress in Mathematics, volume 357.
  • Here is a recent preprint discussing how the deformation space of complete affine 2-manifolds relates to a twisted cubic cone in 4-space. 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 appeared in the recent Contemporary Mathematics volume celebrating Ravi Kulkarni's 80th birthday.
  • 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.

    • Dynamics on moduli spaces

    The classification of geometric structures on manifolds naturally leads to dynamical systems involving actions of mapping class groups on character varieties. The above-mentioned paper proving Baues's theorem is a good example of how chaotic dynamics is involved in the classification problem.

  • My first paper with my colleague Giovanni Forni, Mixing Flows on moduli spaces of flat bundles over sruraces outlines a program to apply Teichmueller dynamics to mapping class group actions on character varieties.
    This paper 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 introduces 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.
  • A third paper, with Forni,Lawton and Matheus, is in preparation.
  • A ZblMath review of Michael Magee's article "Random Unitary Representations of Surface Groups I:Asymptotic Expansions."

    • Current and past experimental projects

    Visualization geometric structures and their related dynamical led to experimental projects.

    • I am currently investigating a remarkable family of real affine cubic surfaces which arise as relative SL(2)-character varieties for a one-holed torus. See if you can run the interactive Mathematica notebook I wrote with Ajeet Gary when he was an undergraduate at Maryland. This work is based on my 2003 Geometry & Topology paper
    • Computational Aspects of Discrete Subgroups of Lie groups, ICERM, 14-18 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 then-undergraduate Robert Benedetto, when we discovered compact components of the SL(2,R)-character variety of a four-holed sphere.
    • Co-founder, with Richard Schwartz, of Lab for Experimental Mathematics at MAryland (LEMMA), 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 co-organized the Geometry Labs United Conference which took place at ICERM, 16-17 July 2020.
      The experimental lab at Maryland was directly inspired by the former Geometry Center at the University of Minnesota, where I served on the Board of Governors until it shut down in 1998.
    • One of my first visualization projects was in the summer of 1992 was with Sergey Brin (who later cofounded Google) in the Undergraduate Apprecenticeship in Research and Scholarship program at the University of Maryland.
      Brin taught me how visualize Pappus's Theorem in Projective Geometry in Objective-C on a Next computer.

    Recent Administrative Activities

  • Brin Mathematics Research Center at the University of Maryland Department of Mathematics:
    • Member, Scientific Advisory Board.
    • I am co-organizing a workshop on Lorentzian, Affine and Hyperbolic Differential Geometry -- in Memory of Todd Drumm.
    • Last year I co-organized (with Steve Bradlow and Richard Wentworth) a workshop ``Advances in Higgs bundles." there.
  • I recently completed my second 3-year term serving on the Council of the American Mathematical Society and the Committee on Publications.
    My first term on the Council was 2006--2009, when I served on the Committee on Education.

  • Former member (with Patrick Brosnan and Abba Gumel), Department Colloquium Committee.
  • Administrative activities before 2015
  • Network Executive Committee (original co-PI, and director of Maryland hub), GEAR Network, an NSF Research Network in the Mathematical Sciences, 2011-2022; with Steve Bradlow, Steve Kerkchff, Richard Wentworth and Anna Wienhard.
  • Scientific Advisory Board (2020-2022), Institute for Computational and Experimental Research in Mathematics (ICERM), Brown University.

    • Recent lectures

  • 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.

    • Last updated: Monnday 24 March 2025.