Richard Schwartz (mathematician)

Last updated

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.

Contents

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. [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

Selected contributions

Corresponding articles

Published books

Selected awards

Related Research Articles

<span class="mw-page-title-main">Vladimir Arnold</span> Russian mathematician (1937–2010)

Vladimir Igorevich Arnold was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to several areas, including geometrical theory of dynamical systems theory, algebra, catastrophe theory, topology, real algebraic geometry, symplectic geometry, symplectic topology, differential equations, classical mechanics, differential geometric approach to hydrodynamics, geometric analysis and singularity theory, including posing the ADE classification problem.

<span class="mw-page-title-main">William Thurston</span> American mathematician (1946–2012)

William Paul Thurston was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.

<span class="mw-page-title-main">Kite (geometry)</span> Quadrilateral symmetric across a diagonal

In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.

In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

<span class="mw-page-title-main">Hyperbolic geometry</span> Non-Euclidean geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

<span class="mw-page-title-main">Tessellation</span> Tiling of a plane in mathematics

A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.

<span class="mw-page-title-main">Peter Lax</span> Hungarian-born American mathematician

Peter David Lax is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics.

<span class="mw-page-title-main">Mikhael Gromov (mathematician)</span> Russian-French mathematician

Mikhael Leonidovich Gromov is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of Institut des Hautes Études Scientifiques in France and a professor of mathematics at New York University.

<span class="mw-page-title-main">William Goldman (mathematician)</span> American mathematician

William Mark Goldman is a professor of mathematics at the University of Maryland, College Park. He received a B.A. in mathematics from Princeton University in 1977, and a Ph.D. in mathematics from the University of California, Berkeley in 1980.

In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work initiated by Jakob Nielsen.

Outer billiards is a dynamical system based on a convex shape in the plane. Classically, this system is defined for the Euclidean plane but one can also consider the system in the hyperbolic plane or in other spaces that suitably generalize the plane. Outer billiards differs from a usual dynamical billiard in that it deals with a discrete sequence of moves outside the shape rather than inside of it.

<span class="mw-page-title-main">Quasi-isometry</span> Function between two metric spaces that only respects their large-scale geometry

In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.

Brian Hayward Bowditch is a British mathematician known for his contributions to geometry and topology, particularly in the areas of geometric group theory and low-dimensional topology. He is also known for solving the angel problem. Bowditch holds a chaired Professor appointment in Mathematics at the University of Warwick.

In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane. The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections. Richard Schwartz introduced the pentagram map for a general polygon in a 1992 paper though it seems that the special case, in which the map is defined for pentagons only, goes back to an 1871 paper of Alfred Clebsch and a 1945 paper of Theodore Motzkin. The pentagram map is similar in spirit to the constructions underlying Desargues' theorem and Poncelet's porism. It echoes the rationale and construction underlying a conjecture of Branko Grünbaum concerning the diagonals of a polygon.

<span class="mw-page-title-main">Finite subdivision rule</span> Way to divide polygon into smaller parts

In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule.

<span class="mw-page-title-main">Hee Oh</span> South Korean American mathematician

Hee Oh is a South Korean mathematician who works in dynamical systems. She has made contributions to dynamics and its connections to number theory. She is a student of homogeneous dynamics and has worked extensively on counting and equidistribution for Apollonian circle packings, Sierpinski carpets and Schottky dances. She is currently the Abraham Robinson Professor of Mathematics at Yale University.

<span class="mw-page-title-main">Sergei Tabachnikov</span> American mathematician

Sergei Tabachnikov, also spelled Serge, is an American mathematician who works in geometry and dynamical systems. He is currently a Professor of Mathematics at Pennsylvania State University.

In the mathematical subject of geometric group theory, the Švarc–Milnor lemma is a statement which says that a group , equipped with a "nice" discrete isometric action on a metric space , is quasi-isometric to .

<span class="mw-page-title-main">Jeffrey Brock</span> American mathematician

Jeffrey Farlowe Brock is an American mathematician, working in low-dimensional geometry and topology. He is known for his contributions to the understanding of hyperbolic 3-manifolds and the geometry of Teichmüller spaces.

<span class="mw-page-title-main">Michael Kapovich</span> Russian-American mathematician

Michael Kapovich is a Russian-American mathematician.

References

  1. M. Gromov, Hyperbolic Groups, in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263.
  2. 1 2 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
  3. "Top 10/Top5/Editor's Picks/Editor's Note". Brown Daily Herald. February 3, 2011. Retrieved 2011-06-27.
  4. "Richard Schwartz - the Mathematics Genealogy Project".
  5. 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.
  6. Richard Evan Schwartz; Serge Tabachnikov (2010). "The Pentagram Integrals on Inscribed Polygons". Mendeley. Retrieved 2011-06-27.
  7. 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)
  8. 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.
  9. 1 2 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.
  10. 1 2 3 4 5 6 Ben Kutner (February 2, 2011). "Math and monsters add up in children's book". Brown Daily Herald. Retrieved 2011-06-27.
  11. 1 2 "Merit blind admissions fool math profs on April 1st". Brown Daily Herald. April 17, 2008. Retrieved 2011-06-27.
  12. 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.
  13. "Summer reading: Children's books". Los Angeles Times. May 22, 2011. Retrieved 2011-06-27.
  14. NPR Staff (January 22, 2011). "Math Isn't So Scary With Help From These Monsters". NPR. Retrieved 2011-06-27.
  15. "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.);
  16. 2017 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2016-11-06.