Louis Hirsch Kauffman (born February 3, 1945) 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, structure, space, models, and change.
History
On ...
,
topologist
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
, and professor of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
in the Department of Mathematics, Statistics, and Computer science at the
University of Illinois at Chicago
The University of Illinois Chicago (UIC) is a Public university, public research university in Chicago, Illinois. Its campus is in the Near West Side, Chicago, Near West Side community area, adjacent to the Chicago Loop. The second campus esta ...
. He is known for the introduction and development of the
bracket polynomial In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although it is not an invariant of knots or links (as it is not invariant under type I Reidemeister mov ...
and the
Kauffman polynomial
In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as
:F(K)(a,z)=a^L(K)\,,
where w(K) is the writhe of the link diagram and L(K) is a polynomial in ''a'' and ' ...
.
Biography
Kauffman was
valedictorian
Valedictorian is an academic title for the highest-performing student of a graduating class of an academic institution.
The valedictorian is commonly determined by a numerical formula, generally an academic institution's grade point average (GPA ...
of his graduating class at Norwood Norfolk Central High School in 1962. He received his
B.S.
A Bachelor of Science (BS, BSc, SB, or ScB; from the Latin ') is a bachelor's degree awarded for programs that generally last three to five years.
The first university to admit a student to the degree of Bachelor of Science was the University ...
at the
Massachusetts Institute of Technology
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...
in 1966 and his
Ph.D.
A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...
in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
from
Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
in 1972 (with
William Browder as thesis advisor).
Kauffman has worked at many places as a visiting professor and researcher, including the
University of Zaragoza
The University of Zaragoza, sometimes referred to as Saragossa University () is a public university with teaching campuses and research centres spread over the three provinces of Aragon, Spain. Founded in 1542, it is one of the oldest universiti ...
in Spain, the
University of Iowa
The University of Iowa (UI, U of I, UIowa, or simply Iowa) is a public university, public research university in Iowa City, Iowa, United States. Founded in 1847, it is the oldest and largest university in the state. The University of Iowa is org ...
in Iowa City, the
Institut des Hautes Études Scientifiques
The Institut des hautes études scientifiques (IHÉS; English: Institute of Advanced Scientific Studies) is a French research institute supporting advanced research in mathematics and theoretical physics. It is located in Bures-sur-Yvette, just ...
in Bures Sur Yevette, France, the
Institut Henri Poincaré
The Henri Poincaré Institute (or IHP for ''Institut Henri Poincaré'') is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS). It is located in the 5th arrond ...
in Paris, France, the
University of Bologna
The University of Bologna ( it, Alma Mater Studiorum – Università di Bologna, UNIBO) is a public research university in Bologna, Italy. Founded in 1088 by an organised guild of students (''studiorum''), it is the oldest university in continuo ...
, Italy, the
Federal University of Pernambuco
Federal University of Pernambuco ( pt, Universidade Federal de Pernambuco, UFPE) is a public university in Recife, Brazil, established in 1946. UFPE has 70 undergraduate courses and 175 postgraduate courses. , UFPE had 35,000 students and 2,000 ...
in Recife, Brazil, and the
Newton Institute in Cambridge England.
He is the founding editor and one of the managing editors of the ''
Journal of Knot Theory and Its Ramifications
The ''Journal of Knot Theory and Its Ramifications'' was established in 1992 by Louis Kauffman and was the first journal purely devoted to knot theory. It is an interdisciplinary journal covering developments in knot theory, with emphasis on creat ...
'', and editor of the ''World Scientific Book Series On Knots and Everything''. He writes a column entitled Virtual Logic for the journal ''Cybernetics and Human Knowing''
From 2005 to 2008 he was president of the
American Society for Cybernetics
The American Society for Cybernetics (ASC) is an American non-profit scholastic organization for the advancement of cybernetics as a science , a discipline, a meta-discipline and the promotion of cybernetics as basis for an interdisciplinary di ...
. He plays
clarinet in the ChickenFat Klezmer Orchestra in Chicago.
Work
Kauffman's research interests are in the fields of cybernetics, topology and foundations of mathematics and physics. His work is primarily in the topics of
knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
and connections with
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
,
quantum theory
Quantum theory may refer to:
Science
*Quantum mechanics, a major field of physics
*Old quantum theory, predating modern quantum mechanics
* Quantum field theory, an area of quantum mechanics that includes:
** Quantum electrodynamics
** Quantum ...
,
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
,
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
and foundations. In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
he introduced and developed the
bracket polynomial In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although it is not an invariant of knots or links (as it is not invariant under type I Reidemeister mov ...
and
Kauffman polynomial
In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as
:F(K)(a,z)=a^L(K)\,,
where w(K) is the writhe of the link diagram and L(K) is a polynomial in ''a'' and ' ...
.
Bracket polynomial
In the mathematical field of
knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
, the
bracket polynomial In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although it is not an invariant of knots or links (as it is not invariant under type I Reidemeister mov ...
, also known as the ''Kauffman bracket'', is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
invariant of
framed link
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of ...
s. Although it is not an invariant of knots or links (as it is not invariant under type I
Reidemeister move
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
Life
He was a brother of Marie Neurath.
Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Götting ...
s), a suitably "normalized" version yields the famous
knot invariant
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...
called the
Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynom ...
. The bracket polynomial plays an important role in unifying the Jones polynomial with other
quantum invariant
In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.
List of invariants
*Finite type invariant
* Kon ...
s. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to state sum invariants of
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
s. Recently the bracket polynomial formed the basis for Mikhail Khovanov's construction of a homology for knots and links, creating
a stronger invariant than the Jones polynomial and such that the graded Euler characteristic of the
Khovanov homology is equal to the original
Jones polynomial. The generators for the chain complex of the Khovanov homology are states of the bracket polynomial decorated with elements
of a
Frobenius algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice dualit ...
.
Kauffman polynomial
The
Kauffman polynomial
In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as
:F(K)(a,z)=a^L(K)\,,
where w(K) is the writhe of the link diagram and L(K) is a polynomial in ''a'' and ' ...
is a 2-variable
knot polynomial
In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.
History
The first knot polynomial, the Alexander polynomial, was introdu ...
due to Louis Kauffman. It is defined as
:
where
is the
writhe
In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amou ...
and
is a
regular isotopy
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrume ...
invariant which generalizes the bracket polynomial.
Discrete ordered calculus
In 1994, Kauffman and Tom Etter wrote a draft proposal for a non-commutative ''discrete ordered calculus'' (DOC), which they presented in revised form in 1996. In the meantime, the theory was presented in a modified form by Kauffman and
H. Pierre Noyes
H. Pierre Noyes (December 10, 1923 – September 30, 2016) was an American theoretical physicist. He became a member of the faculty at the SLAC National Accelerator Laboratory at Stanford University in 1962. Noyes specialized in several areas o ...
together with a presentation of a derivation of free space
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
on this basis.
Awards and honors
He won a
Lester R. Ford Award
Lester is an ancient Anglo-Saxon surname and given name. Notable people and characters with the name include:
People
Given name
* Lester Bangs (1948–1982), American music critic
* Lester W. Bentley (1908–1972), American artist from Wisc ...
(with
Thomas Banchoff
Thomas Francis Banchoff (born April 7, 1938) is an American mathematician
specializing in geometry. He is a professor at Brown University, where he has taught since 1967. He is known for his research in differential geometry in three and four dim ...
) in 1978. Kauffman is the 1993 recipient of th
Warren McCullochaward of the
American Society for Cybernetics
The American Society for Cybernetics (ASC) is an American non-profit scholastic organization for the advancement of cybernetics as a science , a discipline, a meta-discipline and the promotion of cybernetics as basis for an interdisciplinary di ...
and the 1996 award of the Alternative Natural Philosophy Association for his work in discrete physics. He is the 2014 recipient of th
Norbert Wieneraward of the
American Society for Cybernetics
The American Society for Cybernetics (ASC) is an American non-profit scholastic organization for the advancement of cybernetics as a science , a discipline, a meta-discipline and the promotion of cybernetics as basis for an interdisciplinary di ...
.
In 2012 he became a fellow 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, ...
.
List of Fellows of the American Mathematical Society
retrieved 2013-01-27.
Publications
Louis H. Kauffman is author of several monographs on knot theory and mathematical physics. His publication list numbers over 170. Books:
* 1987
''On Knots''
Princeton University Press 498 pp.
* 1993, ''Quantum Topology (Series on Knots & Everything)'', with Randy A. Baadhio, World Scientific Pub Co Inc, 394 pp.
* 1994, ''Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds'', with Sostenes Lins, Princeton University Press, 312 pp.
* 1995, ''Knots and Applications (Series on Knots and Everything, Vol 6)''
* 1995, ''The Interface of Knots and Physics: American Mathematical Society Short Course January 2–3, 1995 San Francisco, California (Proceedings of Symposia in Applied Mathematics)'', with the American Mathematical Society.
* 1998, ''Knots at Hellas 98: Proceedings of the International Conference on Knot Theory and Its Ramifications'', with Cameron McA. Gordon, Vaughan F. R. Jones and Sofia Lambropoulou,
* 1999, ''Ideal Knots'', with Andrzej Stasiak and Vsevolod Katritch, World Scientific Publishing Company, 414 pp.
* 2002, ''Hypercomplex Iterations: Distance Estimation and Higher Dimensional Fractals (Series on Knots and Everything , Vol 17)'', with Yumei Dang and Daniel Sandin.
* 2006
''Formal Knot Theory''
Dover Publications, 272 pp.
* 2007, ''Intelligence of Low Dimensional Topology 2006'', with J. Scott Carter and Seiichi Kamada.
* 2012, ''Knots and Physics'' (4th ed.), World Scientific Publishing Company,
Articles and papers, a selection:
* 2001
The Mathematics of Charles Sanders Peirce
in: ''Cybernetics & Human Knowing'', Vol.8, no.1–2, 2001, pp. 79–110
References
External links
Louis Kauffman
homepage at uic.edu
Hypercomplex Fractals
*
ChickenFat Klezmer Orchestra
{{DEFAULTSORT:Kauffman, Louis
1945 births
Cyberneticists
Living people
20th-century American mathematicians
21st-century American mathematicians
Topologists
Princeton University alumni
University of Illinois Chicago faculty
Fellows of the American Mathematical Society