Richard Evan Schwartz (born August 11, 1966) is an American
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
notable for his contributions
[ to ]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 group ...
and to an area of mathematics known as billiards
Cue sports are a wide variety of games of skill played with a cue, which is used to strike billiard balls and thereby cause them to move around a cloth-covered table bounded by elastic bumpers known as .
There are three major subdivisions ...
. Geometric group theory is a relatively new area of mathematics beginning around the late 1980s[M. Gromov, ''Hyperbolic Groups'', in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263.] 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
Cue sports are a wide variety of games of skill played with a cue, which is used to strike billiard balls and thereby cause them to move around a cloth-covered table bounded by elastic bumpers known as .
There are three major subdivisions ...
'', which are dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s based on a convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
shape in a plane. He has explored geometric iterations involving polygons
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...
,[ and he has been credited for developing the mathematical concept known as the ]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 n ...
. 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 2018 he is a professor of mathematics at Brown University.
Career
Schwartz was born in Los Angeles
Los Angeles ( ; es, Los Ángeles, link=no , ), often referred to by its initials L.A., is the largest city in the state of California and the second most populous city in the United States after New York City, as well as one of the wor ...
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.
The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the California St ...
in 1987, and then a Ph. D. in mathematics from Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...
in 1991 under the supervision of William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) 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.
Thursto ...
. He taught at the University of Maryland
The University of Maryland, College Park (University of Maryland, UMD, or simply Maryland) is a public university, public Land-grant university, land-grant research university in College Park, Maryland. Founded in 1856, UMD is the Flagship un ...
. He is currently the Chancellor's Professor of Mathematics at Brown University. He lives with his wife and two daughters in Barrington, Rhode Island
Barrington is a suburban, residential town in Bristol County, Rhode Island located approximately southeast of Providence. It was founded by Congregationalist separatists from Swansea, Massachusetts and incorporated in 1717.
Barrington was cede ...
.
Schwartz is credited by other mathematicians for introducing the concept of the 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 n ...
.
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
Sergei Tabachnikov, also spelled Serge, (in Russian: Сергей Львович Табачников; born in 1956) is a Russian mathematician who works in geometry and dynamical systems. He is currently a Professor of Mathematics at Pennsylvan ...
to show that the pentagram map is "completely integrable."
In his spare time he draws comic book
A comic book, also called comicbook, comic magazine or (in the United Kingdom and Ireland) simply comic, is a publication that consists of comics art in the form of sequential juxtaposed panel (comics), panels that represent individual scenes. ...
s,[ writes computer programs, listens to ]music
Music is generally defined as the The arts, art of arranging sound to create some combination of Musical form, form, harmony, melody, rhythm or otherwise Musical expression, expressive content. Exact definition of music, definitions of mu ...
and exercises. He admired the late Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eigh ...
n mathematician Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
and dedicated a paper to him.[ He played an ]April Fool's joke
April Fools' Day or All Fools' Day is an annual custom on 1 April consisting of practical jokes and hoaxes. Jokesters often expose their actions by shouting "April Fools!" at the recipient. Mass media can be involved in these pranks, which may b ...
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
''The Brown Daily Herald'' is the student newspaper of Brown University in Providence, Rhode Island.
Established in 1866 and published daily since 1891, The ''Herald'' is the second-oldest student newspaper among America's college dailies. It ...
''. 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 for young children published in 2010, entitled ''You Can Count on Monsters'', which became a bestseller.][ Each monster has a graphic which gives a mini-lesson about its properties, such as being a ]prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
or a lesson about factoring; for example, the graphic monster for the number five was a five-sided star or pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aro ...
.[ A year after publication, it was featured prominently on '']National Public Radio
National Public Radio (NPR, stylized in all lowercase) is an American privately and state funded nonprofit media organization headquartered in Washington, D.C., with its NPR West headquarters in Culver City, California. It differs from othe ...
'' in January 2011 and became a bestseller for a few days on the online bookstore Amazon
Amazon most often refers to:
* Amazons, a tribe of female warriors in Greek mythology
* Amazon rainforest, a rainforest covering most of the Amazon basin
* Amazon River, in South America
* Amazon (company), an American multinational technolog ...
[ as well as earning international acclaim.]
The ''Los Angeles Times
The ''Los Angeles Times'' (abbreviated as ''LA Times'') is a daily newspaper that started publishing in Los Angeles in 1881. Based in the LA-adjacent suburb of El Segundo since 2018, it is the sixth-largest newspaper by circulation in the ...
'' 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 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. ...
classification of rank one lattices: Any quasi-isometry of a hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
lattice is equivalent to a commensurator
In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer.
Commensurability in group th ...
.
*A proof of the 1989 Goldman Goldman is a Jewish surname. Notable people with the surname include:
* Alan J. Goldman (1932–2010), American expert in operations research
* Alan H. Goldman (born 1945), American philosopher
*Alan S. Goldman (born 1958), American chemist
* Alain ...
–Parker conjecture: This is a complete description of the moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such ...
of the complex hyperbolic ideal triangle groups.
*A proof that a triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colli ...
has a periodic billiard path provided all its angles are less than 100 degrees
*A solution of the 1960 Moser
Moser may refer to:
* Moser (surname)
* An individual who commits the act of Mesirah in Judaism
Places
* Moser Glacier, a glacier on the west coast of Graham Land, Antarctica
* Moser River, Nova Scotia, Canada
* Moser Bay Seaplane Base, a p ...
–Neumann
Neumann is German language, German and Yiddish language, Yiddish for "new man", and one of the List of the most common surnames in Europe#Germany, 20 most common German surnames.
People
* Von Neumann family, a Jewish Hungarian noble family
A� ...
problem: There exists an outer billiard
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 p ...
s system with an unbounded orbit.
*A solution of the 5-electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary partic ...
case of J. J. Thomson
Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered.
In 1897, Thomson showed that ...
's 1904 problem
Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
: The triangular bipyramid
In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.
As the name suggests, i ...
is the configuration of 5 electrons on the sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
that minimizes the Coulomb potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
.
*The introduction of the 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 n ...
and a later proof (with Sergei Tabachnikov and Valentin Ovsienko) of its complete integrability.
Corresponding articles
*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)
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''
American Mathematical Society, (2015)
''Mostly Surfaces''
American Mathematical Society, (2011)
''The Octagonal PETs''
American Mathematical Society, (2014)
''Really Big Numbers''
American Mathematical Society, (2014) Winner of the 201
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
The National Science Foundation (NSF) is an independent agency of the United States government that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National ...
Postdoctoral Fellow
*1996 Sloan Research Fellow
The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This program is one of the oldest of its kind in the United States.
...
*2002 Invited Speaker, International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be rename ...
, Beijing
*2003 Guggenheim Fellow
Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...
*2009 Clay Research Scholar
*2017 class of Fellow
A fellow is a concept whose exact meaning depends on context.
In learned or professional societies, it refers to a privileged member who is specially elected in recognition of their work and achievements.
Within the context of higher education ...
s of the American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
"for contributions to dynamics, geometry, and experimental mathematics and for exposition".2017 Class of the Fellows of the AMS
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
, retrieved 2016-11-06.
References
External links
Richard Evan Schwartz's Homepage
Richard Evan Schwartz's Author Page
Seminar talk by V Ovsienko on pentagram maps
{{DEFAULTSORT:Schwartz, Richard
Algebraic geometers
Additive combinatorialists
Mathematics educators
Brown University faculty
University of California, Berkeley alumni
University of California, Los Angeles alumni
American children's writers
1966 births
Living people
Fellows of the American Mathematical Society