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Gordon is most well known for her work in [[isospectral geometry]], for which [[hearing the shape of a drum]] is the prototypical example. 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 the spectrum of its [[Laplace–Beltrami operator]]). [[John Milnor]] observed that a theorem due to Witt implied the existence of a pair of 16-dimensional [[Torus|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 convex isospectral domains in the [[hyperbolic plane]]<ref name="Isospectral hyperbolic 1994">{{cite journal | last1=Gordon | first1=Carolyn S. | last2=Webb | first2=David L. | title=Isospectral convex domains in the hyperbolic plane | journal=Proceedings of the American Mathematical Society | publisher=American Mathematical Society (AMS) | volume=120 | issue=3 | year=1994 | issn=0002-9939 | doi=10.1090/s0002-9939-1994-1181165-0 | pages=981–983| s2cid=122328508 | doi-access=free }}</ref> and in [[Euclidean space]].<ref name=":1">[https://www.ams.org/mathscinet mathscinet]</ref>
Gordon is most well known for her work in [[isospectral geometry]], for which [[hearing the shape of a drum]] is the prototypical example. 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 the spectrum of its [[Laplace–Beltrami operator]]). [[John Milnor]] observed that a theorem due to Witt implied the existence of a pair of 16-dimensional [[Torus|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 convex isospectral domains in the [[hyperbolic plane]]<ref name="Isospectral hyperbolic 1994">{{cite journal | last1=Gordon | first1=Carolyn S. | last2=Webb | first2=David L. | title=Isospectral convex domains in the hyperbolic plane | journal=Proceedings of the American Mathematical Society | publisher=American Mathematical Society (AMS) | volume=120 | issue=3 | year=1994 | issn=0002-9939 | doi=10.1090/s0002-9939-1994-1181165-0 | pages=981–983| s2cid=122328508 | doi-access=free }}</ref> and in [[Euclidean space]].<ref name=":1">[https://www.ams.org/mathscinet mathscinet]</ref>


Gordon has written or coauthored over 30 articles on isospectral geometry including work on isospectral closed [[Riemannian manifold]]s 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," due to [[Toshikazu Sunada]]. 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.<ref name="not locally isometric ">{{cite journal | last1=Gordon | first1=Carolyn S. | last2=Wilson | first2=Edward N. | title=Continuous families of isospectral Riemannian metrics which are not locally isometric | journal=Journal of Differential Geometry | publisher=International Press of Boston | volume=47 | issue=3 | date=1997-01-01 | issn=0022-040X | doi=10.4310/jdg/1214460548 | page=| s2cid=17159822 }}</ref>
Gordon has written or coauthored over 30 articles on isospectral geometry including work on isospectral closed [[Riemannian manifold]]s 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," due to [[Toshikazu Sunada]]. 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.<ref name="not locally isometric ">{{cite journal | last1=Gordon | first1=Carolyn S. | last2=Wilson | first2=Edward N. | title=Continuous families of isospectral Riemannian metrics which are not locally isometric | journal=Journal of Differential Geometry | publisher=International Press of Boston | volume=47 | issue=3 | date=1997-01-01 | issn=0022-040X | doi=10.4310/jdg/1214460548 | page=| s2cid=17159822 | doi-access=free }}</ref>


Gordon has also worked on projects concerning the [[homology class]], length spectrum (the collection of lengths of all [[closed geodesic]]s, together with multiplicities) and [[geodesic flow]] on isospectral Riemannian manifolds.<ref name=":1" /><ref name="Gordon Mao ">{{cite arXiv | last1=Gordon | first1=Carolyn | last2=Mao | first2=Yiping | title=Geodesic Conjugacy in two-step nilmanifolds | year=1995 |eprint=math/9503220}}</ref>
Gordon has also worked on projects concerning the [[homology class]], length spectrum (the collection of lengths of all [[closed geodesic]]s, together with multiplicities) and [[geodesic flow]] on isospectral Riemannian manifolds.<ref name=":1" /><ref name="Gordon Mao ">{{cite arXiv | last1=Gordon | first1=Carolyn | last2=Mao | first2=Yiping | title=Geodesic Conjugacy in two-step nilmanifolds | year=1995 |eprint=math/9503220}}</ref>

Revision as of 07:13, 24 August 2023

Carolyn S. Gordon
Carolyn S. Gordon, Headshot, 2016
Gordon in 2016
Born (1950-12-26) December 26, 1950 (age 73)
Charleston, West Virginia
Alma materWashington University in St. Louis
Known forInverse spectral problems, homogeneous spaces
Awards
Scientific career
FieldsMathematics
InstitutionsDartmouth College
Doctoral advisorEdward Nathan Wilson

Carolyn S. Gordon (born 1950)[1] is a mathematician and Benjamin Cheney Professor of Mathematics at Dartmouth College. She is most well known for giving a negative answer to the question "Can you hear the shape of a drum?" in her work with David Webb and Scott A. Wolpert. She is a Chauvenet Prize winner and a 2010 Noether Lecturer.

Early life and education

Gordon received her Bachelor of Science degree from the Purdue University. She entered graduate studies at the Washington University in St. Louis, earning her Doctor of Philosophy in mathematics in 1979. Her doctoral advisor was Edward Nathan Wilson and her thesis was on isometry groups of homogeneous manifolds. She completed a postdoc at Technion Israel Institute of Technology and held positions at Lehigh University and Washington University in St. Louis.

Career

This is the Gordon–Webb–Wolpert example of two flat surfaces with the same spectrum. Notice that both polygons have the same area and perimeter.

Gordon is most well known for her work in isospectral geometry, for which hearing the shape of a drum is the prototypical example. 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 the spectrum of its Laplace–Beltrami operator). 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 convex isospectral domains in the hyperbolic plane[2] and in Euclidean space.[3]

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," due to Toshikazu Sunada. 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.[4]

Gordon has also worked on projects concerning the homology class, length spectrum (the collection of lengths of all closed geodesics, together with multiplicities) and geodesic flow on isospectral Riemannian manifolds.[3][5]

Selected awards and honors

In 2001 Gordon and Webb were awarded the Mathematical Association of America Chauvenet Prize for their 1996 American Scientist paper, "You can't hear the shape of a drum". In 1990 she was awarded an AMS Centennial Fellowship by the American Mathematical Society for outstanding early career research. In 1999 Gordon presented an AMS-MAA joint invited address. In 2010 she was selected as a Noether Lecturer.[6] In 2012 she became a fellow of the American Mathematical Society[7] and of the American Association for the Advancement of Science.[8] She was also an AMS Council member at large from 2005 to 2007.[9] In 2017 she was selected as a fellow of the Association for Women in Mathematics in the inaugural class.[10] Gordon was featured in the Women's History Month tribute in the March 2018 edition of the AMS Notices.[11]

Selected articles

  • Gordon, Carolyn (2001), "Isospectral Deformations of Metrics on Spheres", Inventiones Mathematicae, 145 (2): 317–331, arXiv:math/0005156, Bibcode:2001InMat.145..317G, doi:10.1007/s002220100150, S2CID 14078869
  • Gordon, Carolyn; Webb, David (1996), "You can't hear the shape of a drum", American Scientist, 84 (January–February): 46–55, Bibcode:1996AmSci..84...46G
  • Gordon, Carolyn (1994), Isospectral Closed Riemannian Manifolds which are not Locally Isometric II, Contemporary Mathematics, vol. 173, American Mathematical Society, pp. 121–131, doi:10.1090/conm/173/01821, ISBN 9780821851852
  • Gordon, Carolyn; Webb, David; Wolpert, Scott (1992), "Isospectral plane domains and surfaces via Riemannian orbifolds", Inventiones Mathematicae, 110: 1–22, Bibcode:1992InMat.110....1G, doi:10.1007/BF01231320, S2CID 122258115
  • Gordon, Carolyn; Webb, David L.; Wolpert, Scott (1992), "One Cannot Hear the Shape of a Drum", Bulletin of the American Mathematical Society, 27: 134–138, arXiv:math/9207215, doi:10.1090/S0273-0979-1992-00289-6, S2CID 15950835
  • Gordon, Carolyn; Wilson, Edward (1984), "Isospectral deformations of compact solvmanifolds", Journal of Differential Geometry, 19: 241–256, doi:10.4310/jdg/1214438431

Personal life

Gordon is married to David Webb. She cites raising her daughter, Annalisa, as her greatest joy in life.[11]

References

  1. ^ Birth year from ISNI authority control file, accessed 2018-11-27.
  2. ^ Gordon, Carolyn S.; Webb, David L. (1994). "Isospectral convex domains in the hyperbolic plane". Proceedings of the American Mathematical Society. 120 (3). American Mathematical Society (AMS): 981–983. doi:10.1090/s0002-9939-1994-1181165-0. ISSN 0002-9939. S2CID 122328508.
  3. ^ a b mathscinet
  4. ^ Gordon, Carolyn S.; Wilson, Edward N. (1997-01-01). "Continuous families of isospectral Riemannian metrics which are not locally isometric". Journal of Differential Geometry. 47 (3). International Press of Boston. doi:10.4310/jdg/1214460548. ISSN 0022-040X. S2CID 17159822.
  5. ^ Gordon, Carolyn; Mao, Yiping (1995). "Geodesic Conjugacy in two-step nilmanifolds". arXiv:math/9503220.
  6. ^ "Carolyn Gordon Named 2010 Noether Lecturer". Association for Women in Mathematics. Retrieved 7 April 2019.
  7. ^ List of Fellows of the American Mathematical Society, retrieved 2013-01-19.
  8. ^ Elected fellows, AAAS, retrieved 2017-10-30.
  9. ^ "AMS Committees". American Mathematical Society. Retrieved 2023-03-29.
  10. ^ "AWM Fellows Program". awm-math.org/awards/awm-fellows/. Association for Women in Mathematics. Retrieved 9 January 2021.
  11. ^ a b "Notices of the AMS" (PDF). p. 265. Retrieved 10 September 2018.
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