Louis Kauffman

Last updated
Louis H. Kauffman
Louis H. Kauffman, Oct 2014.jpg
Louis Kauffman in October 2014
Born (1945-02-03) February 3, 1945 (age 76)
Nationality American
Alma mater Princeton University
Massachusetts Institute of Technology
Scientific career
Fields Mathematics
Institutions University of Illinois at Chicago
Doctoral advisor William Browder

Louis Hirsch Kauffman (born February 3, 1945) is an American mathematician, topologist, and professor of Mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He is known for the introduction and development of the bracket polynomial and the Kauffman polynomial.

Contents

Biography

Kauffman was valedictorian of his graduating class at Norwood Norfolk Central High School in 1962. He received his B.S. at the Massachusetts Institute of Technology in 1966 and his Ph.D. in mathematics from Princeton University 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 in Spain, the University of Iowa in Iowa City, the Institut des Hautes Études Scientifiques in Bures Sur Yevette, France, the Institut Henri Poincaré in Paris, France, the University of Bologna, Italy, the Federal University of Pernambuco in Recife, Brazil, and the Newton Institute in Cambridge England. [1]

He is the founding editor and one of the managing editors of the Journal of Knot Theory and Its Ramifications , 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. 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 and connections with statistical mechanics, quantum theory, algebra, combinatorics and foundations. [2] In topology he introduced and developed the bracket polynomial and Kauffman polynomial.

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 moves), a suitably "normalized" version yields the famous knot invariant called the Jones polynomial. The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariants. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to state sum invariants of 3-manifolds. 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.

Kauffman polynomial

The Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is defined as

where is the writhe and is a regular isotopy 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. [3] In the meantime, the theory was presented in a modified form by Kauffman and H. Pierre Noyes together with a presentation of a derivation of free space Maxwell's equations on this basis. [4]

Awards and honors

He won a Lester R. Ford Award (with Thomas Banchoff) in 1978. [5] Kauffman is the 1993 recipient of the Warren McCulloch award of the American Society for Cybernetics and the 1996 award of the Alternative Natural Philosophy Association for his work in discrete physics. He is the 2014 recipient of the Norbert Wiener award of the American Society for Cybernetics. [6]

In 2012 he became a fellow of the American Mathematical Society. [7]

Publications

Louis H. Kauffman is author of several monographs on knot theory and mathematical physics. His publication list numbers over 170. [1] Books:

Articles and papers, a selection:

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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.

In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l.

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 polynomial in the variable with integer coefficients.

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In the mathematical field of knot theory, the bracket polynomial is a polynomial invariant of framed links. Although it is not an invariant of knots or links, a suitably "normalized" version yields the famous knot invariant called the Jones polynomial. The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariants. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to invariants of 3-manifolds.

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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

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Hernando Burgos Soto is a Canadian writer and mathematician, professor of mathematics at George Brown College. He is the author of several math papers in which he introduced some mathematics concepts and extended to tangles some celebrated results of knot theory about the Khovanov homology and the Jones polynomial. During his career as a mathematician, his interests have included Mathematical Statistics, Knot Theory, Algebraic Topology and more recently Mathematical Finance. He is comfortable writing in English and Spanish. When writing in Spanish, he works in the area of prose fiction writing short stories. Some of his short stories were published at the website Cuentos y Cuentos.

References

  1. 1 2 http://www.math.uic.edu/~kauffman/569.html
  2. "Presentation". Archived from the original on 2008-09-17. Retrieved 2007-09-26.
  3. T. Etter, L.H. Kauffman, ANPA West Journal, vol. 6, no. 1, pp. 3–5
  4. Louis H. Kauffman, H. Pierre Noyes, Discrete physics and the derivation of electromagnetism from the formalism of quantum mechanics, Proceedings of the Royal Society London A (1996), vol. 452, pp. 81–95
  5. Kauffman, Louis; Banchoff, Thomas (1977). "Immersions and Mod-2 quadratic forms". Amer. Math. Monthly. 84: 168–185. doi:10.2307/2319486. JSTOR   2319486.
  6. About SSC: Awards, retrieved 2014-11-02.
  7. List of Fellows of the American Mathematical Society, retrieved 2013-01-27.