Richard Schwartz (mathematician)
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[1] 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,[2] and he has been credited for developing the mathematical concept known as the pentagram map. In addition, he is author of a mathematics picture book for young children.[3] In 2018 he is a professor of mathematics at Brown University.
Career
[edit]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.[4] 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.[2] 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.[5] Schwartz observed these geometric patterns, partly by experimenting with computers.[6] He has collaborated with mathematicians Valentin Ovsienko[7] and Sergei Tabachnikov[8] to show that the pentagram map is "completely integrable."[9]
In his spare time he draws comic books,[10] writes computer programs, listens to music and exercises. He admired the late Russian mathematician Vladimir Arnold and dedicated a paper to him.[9] 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.[11] Colleagues such as mathematician Jeffrey Brock describe Schwartz as having a "very wry sense of humor."[11]
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 for young children published in 2010, entitled You Can Count on Monsters, which became a bestseller.[10] 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.[10] 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[10] as well as earning international acclaim.[12] The Los Angeles Times suggested that the book helped to "take the scariness out of arithmetic."[13] Mathematician Keith Devlin, on NPR, agreed, saying that Schwartz "very skillfully and subtly embeds mathematical ideas into the drawings." [14] [10]
Publications
[edit]Selected contributions
[edit]- 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.
Corresponding articles
[edit]- R. E. Schwartz, "The Quasi-Isometry Classification of Rank One Lattices Publ. Math. IHÉS (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)
Published books
[edit]- 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, American Mathematical Society, (2015)[10][15]
- 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
[edit]- 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".[16]
References
[edit]- ^ M. Gromov, Hyperbolic Groups, in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263.
- ^ a b Fedor Soloviev (June 27, 2011). "Integrability of the pentagram map". Duke Mathematical Journal. 162 (15). arXiv:1106.3950. doi:10.1215/00127094-2382228. S2CID 119586878.
The pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons
- ^ "Top 10/Top5/Editor's Picks/Editor's Note". Brown Daily Herald. February 3, 2011. Retrieved 2011-06-27.
- ^ "Richard Schwartz - the Mathematics Genealogy Project".
- ^ Max Glick (April 15, 2011). "The pentagram map and Y-patterns". arXiv:1005.0598 [math.CO].
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.
- ^ 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". Communications in Mathematical Physics. 299 (2): 409–446. arXiv:0810.5605. Bibcode:2010CMaPh.299..409O. doi:10.1007/s00220-010-1075-y. S2CID 2616239. Retrieved 2011-06-27.
- ^ a b Valentin Ovsienko; Richard Schwartz; Serge Tabachnikov (2011-06-27). "Discrete monodromy, pentagrams, and the method of condensation". Journal of Fixed Point Theory and Applications. 3 (2). Springerlink: 379–409. arXiv:0709.1264. doi:10.1007/s11784-008-0079-0. S2CID 17099073.
- ^ a b c d e f Ben Kutner (February 2, 2011). "Math and monsters add up in children's book". Brown Daily Herald. Retrieved 2011-06-27.
- ^ a b "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.
- ^ NPR Staff (January 22, 2011). "Math Isn't So Scary With Help From These Monsters". NPR. Retrieved 2011-06-27.
- ^ "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.);
- ^ 2017 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2016-11-06.
