Richard Schwartz

Richard Evan Schwartz (born August 11, 1966) is an American mathematician notable for his contributions to geometric group theory and to an area of mathematics known as billiards. Geometric group theory is a relatively new area of mathematics beginning around the late 1980s which explores finitely generated groups, and seeks connections between their algebraic properties and the geometric spaces on which these groups act. He has worked on what mathematicians refer to as billiards, which are dynamical systems based on a convex shape in a plane. He has explored geometric iterations involving polygons, and he has been credited for developing the mathematical concept known as the pentagram map. In 2018 he is a professor of mathematics at Brown University.

Career Schwartz was born in Los Angeles on August 11, 1966. He attended John F. Kennedy High School in Los Angeles from 1981 to 1984, then earned a B. S. in mathematics from U.C.L.A. in 1987, and then a Ph. D. in mathematics from Princeton University in 1991 under the supervision of William Thurston. He taught at the University of Maryland. He is currently the Chancellor's Professor of Mathematics at Brown University. He lives with his wife and two daughters in Barrington, Rhode Island.

Schwartz is credited by other mathematicians for introducing the concept of the pentagram map. According to Schwartz's conception, a convex polygon would be inscribed with diagonal lines inside it, by drawing a line from one point to the next point—that is, by skipping over the immediate point on the polygon. The intersection points of the diagonals would form an inner polygon, and the process could be repeated. Schwartz observed these geometric patterns, partly by experimenting with computers. He has collaborated with mathematicians Valentin Ovsienko and Sergei Tabachnikov to show that the pentagram map is "completely integrable."

In his spare time he draws comic books, Colleagues such as mathematician Jeffrey Brock describe Schwartz as having a "very wry sense of humor." The Los Angeles Times suggested that the book helped to "take the scariness out of arithmetic." Mathematician Keith Devlin, on NPR, agreed, saying that Schwartz "very skillfully and subtly embeds mathematical ideas into the drawings."

Publications

Selected contributions The quasi-isometry classification of rank one lattices: Any quasi-isometry of a hyperbolic lattice is equivalent to a commensurator. A proof of the 1989 Goldman–Parker conjecture: This is a complete description of the moduli space of the complex hyperbolic ideal triangle groups. A proof that a triangle has a periodic billiard path provided all its angles are less than 100 degrees. A solution of the 1960 Moser–Neumann problem: There exists an outer billiards system with an unbounded orbit. A solution of the 5-electron case of J. J. Thomson's 1904 problem: The triangular bipyramid is the configuration of 5 electrons on the sphere that minimizes the Coulomb potential. The introduction of the pentagram map and a later proof (with Valentin Ovsienko and Sergei Tabachnikov) of its complete integrability. Mostly Surfaces, American Mathematical Society, (2011) The Octagonal PETs, American Mathematical Society, (2014) Really Big Numbers, American Mathematical Society, (2014), Winner of the 2015 MSRI Mathical Books for Kids from Tots to Teens Award Gallery of the Infinite, American Mathematical Society, (2016) The Projective Heat Map, American Mathematical Society, (2017)

Selected awards 1993 National Science Foundation Postdoctoral Fellow 1996 Sloan Research Fellow 2002 Invited Speaker, International Congress of Mathematicians, Beijing 2003 Guggenheim Fellow 2009 Clay Research Scholar 2017 class of Fellows of the American Mathematical Society "for contributions to dynamics, geometry, and experimental mathematics and for exposition". In June 2026, the conference "Geometry, Dynamics, and Computer-Assisted Proofs" is held in his honor at Heidelberg University.