Carolyn S. Gordon
Encyclopedia
Carolyn S. Gordon is a mathematician
and professor
of mathematics
at Dartmouth College
since 1992. She received her Bachelor of Science
degree from the Purdue University
, then studied at the Washington University, earning her Doctor of Philosophy
in mathematics in 1979. Her doctoral advisor was Edward Nathan Wilson and her
thesis was on isometry group
s of homogeneous manifolds. She completed a postdoc at Technion Israel Institute of Technology and held positions at Lehigh University
and Washington University. In 1990 she was awarded
an AMS Centennial Fellowship by the American Mathematical Society
.
Gordon is most well known for her work in isospectral geometry which concerns hearing the shape of a drum
. In 1966 Mark Kac
asked whether the shape of a drum could be determined by the sound it makes (whether a Riemannian manifold
is determined by its Laplace spectrum). John Milnor
observed that a theorem due to Witt implied the existence of a pair of 16-dimensional tori that have the same spectrum but different shapes. However, the problem in two dimensions remained open until 1992, when Gordon, with coauthors Webb and Wolpert, constructed a pair of regions in the Euclidean plane that have different shapes but identical eigenvalues (see figure on right). In further work, Gordon and Webb produced isospectral domains in hyperbolic space
and convex isospectral domains in the Euclidean plane.
Gordon has written or coauthored over 30 articles on isospectral geometry including work on isospectral closed Riemannian manifolds with a common Riemannian covering. These isospectral Riemannian manifolds have the same local geometry but different topology. They can be found using the Sunada method. In 1993 she found isospectral Riemannian manifolds which are not locally isometric and, since that time, has worked with coauthors to produce a number of other such examples.
Gordon has also worked on projects concerning the homology class, length spectrum and geodesic flow on isospectral Riemannian manifolds.
Chauvenet Prize
for their 1996 American Scientist
paper, "You can't hear the shape of a drum".
In 1999 Gordon presented an AMS-MAA joint invited address. In 2010 she was selected as a Noether Lecturer.
Ruth Gornet Washington University 1993
Elizabeth Stanhope Dartmouth College
2002
Emily Proctor Dartmouth College
2003
Emily Dryden Dartmouth College
2004
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....
and professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
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...
at Dartmouth College
Dartmouth College
Dartmouth College is a private, Ivy League university in Hanover, New Hampshire, United States. The institution comprises a liberal arts college, Dartmouth Medical School, Thayer School of Engineering, and the Tuck School of Business, as well as 19 graduate programs in the arts and sciences...
since 1992. She received her Bachelor of Science
Bachelor of Science
A Bachelor of Science is an undergraduate academic degree awarded for completed courses that generally last three to five years .-Australia:In Australia, the BSc is a 3 year degree, offered from 1st year on...
degree from the Purdue University
Purdue University
Purdue University, located in West Lafayette, Indiana, U.S., is the flagship university of the six-campus Purdue University system. Purdue was founded on May 6, 1869, as a land-grant university when the Indiana General Assembly, taking advantage of the Morrill Act, accepted a donation of land and...
, then studied at the Washington University, earning her Doctor of Philosophy
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
in mathematics in 1979. Her doctoral advisor was Edward Nathan Wilson and her
thesis was on isometry group
Isometry group
In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...
s of homogeneous manifolds. She completed a postdoc at Technion Israel Institute of Technology and held positions at Lehigh University
Lehigh University
Lehigh University is a private, co-educational university located in Bethlehem, Pennsylvania, in the Lehigh Valley region of the United States. It was established in 1865 by Asa Packer as a four-year technical school, but has grown to include studies in a wide variety of disciplines...
and Washington University. In 1990 she was awarded
an AMS Centennial Fellowship by the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
.
Hearing the shape of a drum
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of basic harmonics, via the use of mathematical theory...
. In 1966 Mark Kac
Mark Kac
Mark Kac was a Polish mathematician. His main interest was probability theory. His question, "Can one hear the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry. Kac completed his Ph.D...
asked whether the shape of a drum could be determined by the sound it makes (whether a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
is determined by its Laplace spectrum). John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
observed that a theorem due to Witt implied the existence of a pair of 16-dimensional tori that have the same spectrum but different shapes. However, the problem in two dimensions remained open until 1992, when Gordon, with coauthors Webb and Wolpert, constructed a pair of regions in the Euclidean plane that have different shapes but identical eigenvalues (see figure on right). In further work, Gordon and Webb produced isospectral domains in hyperbolic space
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...
and convex isospectral domains in the Euclidean plane.
Gordon has written or coauthored over 30 articles on isospectral geometry including work on isospectral closed Riemannian manifolds with a common Riemannian covering. These isospectral Riemannian manifolds have the same local geometry but different topology. They can be found using the Sunada method. In 1993 she found isospectral Riemannian manifolds which are not locally isometric and, since that time, has worked with coauthors to produce a number of other such examples.
Gordon has also worked on projects concerning the homology class, length spectrum and geodesic flow on isospectral Riemannian manifolds.
Awards and honors
In 2001 Gordon and Webb were awarded the Mathematical Association of AmericaMathematical Association of America
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
Chauvenet Prize
Chauvenet Prize
The Chauvenet Prize is the highest award for mathematical expository writing. It consists of a prize of $1,000 and a certificate, and is awarded yearly by the Mathematical Association of America in recognition of an outstanding expository article on a mathematical topic. The prize is named in...
for their 1996 American Scientist
American Scientist
American Scientist is the bimonthly science and technology magazine published since 1913 by Sigma Xi. Each issue includes four to five feature articles written by scientists and engineers. These authors review research in all fields of science...
paper, "You can't hear the shape of a drum".
In 1999 Gordon presented an AMS-MAA joint invited address. In 2010 she was selected as a Noether Lecturer.
Doctoral Students
He Ouyang Washington University 1991Ruth Gornet Washington University 1993
Elizabeth Stanhope Dartmouth College
Dartmouth College
Dartmouth College is a private, Ivy League university in Hanover, New Hampshire, United States. The institution comprises a liberal arts college, Dartmouth Medical School, Thayer School of Engineering, and the Tuck School of Business, as well as 19 graduate programs in the arts and sciences...
2002
Emily Proctor Dartmouth College
Dartmouth College
Dartmouth College is a private, Ivy League university in Hanover, New Hampshire, United States. The institution comprises a liberal arts college, Dartmouth Medical School, Thayer School of Engineering, and the Tuck School of Business, as well as 19 graduate programs in the arts and sciences...
2003
Emily Dryden Dartmouth College
Dartmouth College
Dartmouth College is a private, Ivy League university in Hanover, New Hampshire, United States. The institution comprises a liberal arts college, Dartmouth Medical School, Thayer School of Engineering, and the Tuck School of Business, as well as 19 graduate programs in the arts and sciences...
2004