Richard Schwartz

# Richard Schwartz

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Encyclopedia
Richard Evan Schwartz is an American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

notable for his contributions to geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...

and to an area of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

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 system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s based on a convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

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
Pentagram map
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...

. 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
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...

.

## Career

Schwartz was born in Los Angeles
Los Angeles, California
Los Angeles , with a population at the 2010 United States Census of 3,792,621, is the most populous city in California, USA and the second most populous in the United States, after New York City. It has an area of , and is located in Southern California...

on August 11, 1966. In his youth he played tennis
Tennis
Tennis is a sport usually played between two players or between two teams of two players each . Each player uses a racket that is strung to strike a hollow rubber ball covered with felt over a net into the opponent's court. Tennis is an Olympic sport and is played at all levels of society at all...

and enjoyed video games. He attended John F. Kennedy High School
John F. Kennedy High School (Los Angeles)
John F. Kennedy High School is located in Granada Hills, Los Angeles California, USA. It is in District 1 of the Los Angeles Unified School District....

in Los Angeles from 1981 to 1984, then earned a B. S.
Bachelor's degree
A bachelor's degree is usually an academic degree awarded for an undergraduate course or major that generally lasts for three or four years, but can range anywhere from two to six years depending on the region of the world...

in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

from U.C.L.A.
University of California, Los Angeles
The University of California, Los Angeles is a public research university located in the Westwood neighborhood of Los Angeles, California, USA. It was founded in 1919 as the "Southern Branch" of the University of California and is the second oldest of the ten campuses...

in 1987, and then a Ph. D.
Doctorate
A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder to teach in a specific field, A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder...

in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

from Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....

in 1991. He taught at the University of Maryland
University of Maryland
When the term "University of Maryland" is used without any qualification, it generally refers to the University of Maryland, College Park.University of Maryland may refer to the following:...

. He is currently the Chancellor's Professor of Mathematics at Brown University
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...

. He lives with his wife and two daughters in Barrington, Rhode Island
Barrington, Rhode Island
Barrington is a town in Bristol County, Rhode Island, United States. The population was 16,310 at the 2010 census.In July 2005, CNN/Money and Money magazine ranked Barrington sixth on its list of the 100 best places to live in the United States.-History:...

.

Schwartz is credited by other mathematicians for introducing the concept of the pentagram map
Pentagram map
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...

.
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 book
Comic book
A comic book or comicbook is a magazine made up of comics, narrative artwork in the form of separate panels that represent individual scenes, often accompanied by dialog as well as including...

s, writes computer programs, listens to music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

and exercises. He admired the late Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...

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 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
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

Factoring
Factoring can refer to the following:* A form of commercial finance - see factoring ; structured settlement factoring transaction* Factorization, a mathematical concept* Factoring a design, as in code refactoring...

; for example, the graphic monster for the number five was a five-sided star or pentagram
Pentagram
A pentagram is the shape of a five-pointed star drawn with five straight strokes...

. 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
Amazon.com
Amazon.com, Inc. is a multinational electronic commerce company headquartered in Seattle, Washington, United States. It is the world's largest online retailer. Amazon has separate websites for the following countries: United States, Canada, United Kingdom, Germany, France, Italy, Spain, Japan, and...

as well as earning international acclaim. The Los Angeles Times
Los Angeles Times
The Los Angeles Times is a daily newspaper published in Los Angeles, California, since 1881. It was the second-largest metropolitan newspaper in circulation in the United States in 2008 and the fourth most widely distributed newspaper in the country....

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
Dr. Seuss
Theodor Seuss Geisel was an American writer, poet, and cartoonist most widely known for his children's books written under the pen names Dr. Seuss, Theo LeSieg and, in one case, Rosetta Stone....

.

### Selected contributions

• The quasi-isometry
Quasi-isometry
In mathematics, a quasi-isometry is a means to compare the large-scale structure of metric spaces. The concept is especially important in Gromov's geometric group theory.-Definition:...

classification of rank one lattices: Any quasi-isometry of a hyperbolic
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

lattice is equivalent to a commensurator
Commensurator
In group theory, a branch of abstract algebra, the commensurator of a subgroup H of a group G is a specific subgroup of G.-Definition:...

.
• A proof of the 1989 Goldman-Parker conjecture: This is a complete description of the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...

of the complex hyperbolic ideal triangle groups.
• A proof that a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

has a periodic billiard
Dynamical billiards
A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed...

path provided all its angles are less than 100 degrees
• A solution of the 1960 Moser
Jürgen Moser
Jürgen Kurt Moser or Juergen Kurt Moser was a German-American mathematician.-Professional biography:...

-Neumann problem: There exists an outer billiards system with an unbounded orbit.
• A solution of the 5-electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

case of J. J. Thomson
J. J. Thomson
Sir Joseph John "J. J." Thomson, OM, FRS was a British physicist and Nobel laureate. He is credited for the discovery of the electron and of isotopes, and the invention of the mass spectrometer...

's 1904 problem: The triangular bipyramid is the configuration of 5 electrons on the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

that minimizes the Coulomb potential.
• The introduction of the pentagram map
Pentagram map
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...

and a later proof (with Sergei Tabachnikov and Valentin Ovsienko) of its complete integrability
Integrable system
In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical...

.

### Corresponding articles

• R.E. Schwartz, "The Quasi-Isometry Classification of Rank One Lattices Publ. Math. IHES
Publications Mathématiques de l'IHÉS
Publications Mathématiques de l'IHÉS is a mathematical journal. It is published by Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recherche Scientifique....

(1995) 82 133–168
• R.E. Schwartz, "Ideal Triangle Groups, Dented Tori, and Numerical Analysis" Ann. of. Math
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...

(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

• 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)

## Selected awards

• 1993 National Science Foundation
National Science Foundation
The National Science Foundation is a United States government agency that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National Institutes of Health...

Postdoctoral Fellow
• 1996 Sloan Research Fellow
• 2002 Invited Speaker, International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....

, Beijing
• 2003 Guggenheim Fellow
Guggenheim Fellowship
Guggenheim Fellowships are American grants that have been awarded annually since 1925 by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the arts." Each year, the foundation makes...

• 2009 Clay Research Scholar
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...