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 addition, he is a bestselling author of a mathematics picture book for young children. His published work usually appears under the name Richard Evan Schwartz. In 2011 he is a professor of mathematics at Brown University.
Schwartz was born in Los Angeles on August 11, 1966. In his youth he played tennis and enjoyed video games. 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. 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, writes computer programs, listens to music and exercises. He admired the late Russian mathematician Vladimir Arnold and dedicated a paper to him. He played an April Fool's joke on fellow mathematics professors at Brown University by sending an email suggesting that students could be admitted randomly, along with references to bogus studies which purportedly suggested that there were benefits to having a certain population of the student body selected at random; the story was reported in the Brown Daily Herald. Colleagues such as mathematician Jeffrey Brock describe Schwartz as having a "very wry sense of humor."
In 2003, Schwartz was teaching one of his young daughters about number basics and developed a poster of the first 100 numbers using colorful monsters. This project gelled into a mathematics book published in 2010, entitled You Can Count on Monsters for young children which became a bestseller. Each monster has a graphic which gives a mini-lesson about its properties, such as being a prime number or a lesson about factoring; for example, the graphic monster for the number five was a five-sided star or pentagram. He estimated that he wrote the book during 1000 hours. A year after publication, it was featured prominently on National Public Radio in January 2011 and became a bestseller for a few days on the online bookstore Amazon as well as earning international acclaim. 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." He's been compared to a mathematical version of Dr. Seuss. 
- 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 Sergei Tabachnikov and Valentin Ovsienko) of its complete integrability.
- R.E. Schwartz, "The Quasi-Isometry Classification of Rank One Lattices Publ. Math. IHES (1995) 82 133–168
- R.E. Schwartz, "Ideal Triangle Groups, Dented Tori, and Numerical Analysis" Ann. of. Math (2001)
- R.E. Schwartz "Obtuse Triangular Billiards II: 100 Degrees worth of periodic billiard paths" Journal of Experimental Math (2008)
- R.E. Schwartz, "Unbounded orbits for Outer Billiards", Journal of Modern Dynamics (2007)
- R.E. Schwartz, "The 5-electron case of Thompson's Problem" preprint (2010).
- R.E. Schwartz "The Pentagram Map" Journal of Experimental Math (1992)
- V. Ovsienko, R.E. Schwartz, S.Tabachnikov, "The Pentagram Map: A Completely Integrable System", Communications in Mathematical Physics (2010)
- Spherical CR Geometry and Dehn Surgery, Annals of Mathematics Studies no. 165 (2007), Princeton University Press
- Outer Billiards on Kites, Annals of Mathematics Studies no. 171 (2009)
- You Can Count on Monsters A.K. Peters Ltd., (2010)
- Mostly Surfaces, American Math Society, (2011)
- Really Big Numbers, American Math Society, (2014)
- 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
- "Journal articles by Richard Evan Schwartz". SpringerLink. 1996–2010. Retrieved 2011-06-27. "Elementary Surprises in Projective Geometry -- Discrete monodromy, pentagrams, and the method of condensation -- The quasi-isometry classification of rank one lattices -- Degenerating the complex hyperbolic ideal triangle groups -- Quasi-isometric rigidity and diophantine approximation -- A Conformal Averaging Process on the Circle -- Desargues Theorem, Dynamics, and Hyperplane Arrangements -- The Density of Shapes in Three-Dimensional Barycentric Subdivision -- Real hyperbolic on the outside, complex hyperbolic on the inside -- Symmetric patterns of geodesics and automorphisms of surface groups"
- M. Gromov, Hyperbolic Groups, in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75-263.
- Fedor Soloviev (27 Jun 2011). "Integrability of the Pentagram Map". arXiv:1106.3950. "The pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons. ... arXiv:1106.3950v2 -- math.AG --".
- "Top 10/Top5/Editor’s Picks/Editor’s Note". Brown Daily Herald. 3 Feb 2011. Retrieved 2011-06-27. "BOOKS is braving the number jungle with the help of You Can Count on Monsters, a picture book by Brown prof Richard Schwartz that may just cure our childhood fear of factoring."
- Valentin Ovsienko, Richard Schwartz, and Sergei Tabachnikov (2011-06-27). "Quasiperiodic Motion for the Pentagram Map". Google User Content. Retrieved 2011-06-27. "Richard Evan Schwartz: Department of Mathematics, Brown University, Providence, RI 02912, USA,"
- Max Glick (15 Apr 2011). "The pentagram map and Y-patterns". arXiv:1005.0598. "The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its "shortest" diagonals, and output the smaller polygon which they cut out. We employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map.".
- Richard Evan Schwartz, Serge Tabachnikov (2010). "The Pentagram Integrals on Inscribed Polygons". Mendeley. Retrieved 2011-06-27. "The pentagram map is a natural iteration on projective equivalence classes of (twisted) n-gons in the projective plane. It was recently proved (OST) that the pentagram map is completely integrable, with the complete set of Poisson commuting integrals given by the polynomials O1,...,On/2,On and E1,...,En/2,En, previously constructed in S3. These polynomials are somewhat reminiscent of the symmetric polynomials. It was observed in computer experiments that if a polygon is inscribed into a conic then Oi=Ei for all i. The goal of the paper is to prove this theorem. The proof is combinatorial, and it was also suggested by computer experimentation."
- V Ovsienko (2011-06-27). "The Pentagram map: a discrete integrable system". University of Cambridge. Retrieved 2011-06-27. "(academic lecture by mathematician V Ovsienko on the pentagram map subject)"
- Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov (2010). "The Pentagram Map: A Discrete Integrable System". Microsoft Academic Search. pp. 409–446. doi:10.1007/s00220-010-1075-y. Retrieved 2011-06-27. "The pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons (a twisted polygon is a map from Z into the projective plane that is periodic modulo a projective transformation). We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. Journal: Communications in Mathematical Physics - COMMUN MATH PHYS , vol. 299, no. 2, pp. 409-446, 2010 doi:10.1007/s00220-010-1075-y"
- Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov (2011-06-27). "Journal of Fixed Point Theory and Applications". Springerlink. Retrieved 2011-06-27. "This paper studies the pentagram map, a projectively natural iteration on the space of polygons. Inspired by a method from the theory of ordinary differential equations, the paper constructs roughly n algebraically independent invariants for the map, when it is defined on the space of n-gons. These invariants strongly suggest that the pentagram map is a discrete completely integrable system. The paper also relates the pentagram map to Dodgson’s method of condensation for computing determinants, also known as the octahedral recurrence. Journal of Fixed Point Theory and Applications, Volume 3, Number 2, 379-409, doi:10.1007/s11784-008-0079-0"
- Ben Kutner (February 2, 2011). "Math and monsters add up in children’s book". Brown Daily Herald. Retrieved 2011-06-27.
- "Merit blind admissions fool math profs on April 1st.". Brown Daily Herald. April 17, 2008. Retrieved 2011-06-27.
- PRNewsWire News Releases (March 21, 2011). "You Can Count on Monsters Proclaimed a Self-Learning Tool That Makes Math Fun". Boston Globe. Retrieved 2011-06-27. "You Can Count on Monsters, a creatively educational children's book that illustrates prime and composite numbers through colorful monsters-themed geometrical designs, has earned international acclaim and stellar sales since its January debut on NPR's Weekend Edition."
- "Summer reading: Children's books". Los Angeles Times. May 22, 2011. Retrieved 2011-06-27. "Richard Evan Schwartz -- CRC Press: $24.95, ages 4-8 -- Math is more fun when monsters are along! This colorful illustrated journey through the factor trees of 1-100 features a different creature for each prime number to help take the scariness out of arithmetic."
- NPR Staff (January 22, 2011). "Math Isn't So Scary With Help From These Monsters". NPR. Retrieved 2011-06-27. "Richard Evan Schwartz, a math professor at Brown University, has written and illustrated a children's book called You Can Count On Monsters. Mathematician Keith Devlin talks with NPR's Scott Simon about how the book makes finding prime numbers fun. "This is one of the most amazing math books for kids I have ever seen…," Devlin says. "Great colors, it's wonderful, and yet because [Schwartz] knows the mathematics, he very skillfully and subtly embeds mathematical ideas into the drawings." What Schwartz does is draw monsters to represent different prime and composite numbers."
- "BOOKS CALENDAR". Providence Journal. May 11, 2010. Retrieved 2011-06-27. "Meet children’s book authors: Mary Jane Begin, author of "Willow Buds" and Liz Goulet Dubois, author of "What Kind of Rabbit Are You?" (10 a.m.-noon); Karen Dugan, author of "Ms. April & Ms. Mae" and Richard Evan Schwartz, author of "You Can Count on Monsters" (Noon-2 p.m.);" </ref name=twsZA15> Ben Kutner (February 2, 2011). "Math and monsters add up in children’s book". Brown Daily Herald. Retrieved 2011-06-27.