Emmy Noether Lectures

2010 Lecturer: Carolyn S. Gordon

You Can’t Hear the Shape of a Manifold

Inverse spectral problems ask how much information about an object is encoded in spectral data. For example, Mark Kac’s question “Can you hear the shape of a drum?” asks whether a plane domain, viewed as a vibrating membrane, is determined by the Dirichlet eigenvalue spectrum of the associated Laplacian, equivalently, by the characteristic frequencies of vibration. The lecture will focus on Kac’s question and its generalization to Riemannian manifolds. We will consider methods for constructing manifolds with the same spectral data and compare examples of such “sound-alike” manifolds. We will also refer to related constructions on discrete and quantum graphs.
Brief Biography:
Carolyn S. Gordon is the Benjamin Cheney Professor of Mathematics at Dartmouth College and was selected as the 31st Noether Lecturer because of her fundamental contributions to inverse spectral problems.
Gordon received her B.S. and M.S. in Mathematics from Purdue University and her Ph.D. from Washington University. She began her career as the Lady Davis Postdoctoral Fellow at Technion Israel Institute of Technology, followed by positions at Lehigh University and Washington University before joining the Dartmouth faculty in 1992.
Gordon’s papers have appeared in diverse settings – from research journals to popular journals such as the Intelligencer. She was awarded a Centennial Fellowship by the American Mathematical Society in1990.
She and David Webb received the Chauvenet Prize from the Mathematical Association of America in 2001 for their 1996 American Scientist paper, “You can’t hear the shape of a drum.” Gordon has given numerous seminars and colloquia at universities throughout the world. She was the principal speaker at the Conference Board on Mathematical Sciences conference “Advances in Inverse Spectral Geometry” in 1996. She has been an AMS Invited Speaker at the Joint Mathematics Meetings and an AMS-MAA Invited Speaker at MathFest. She is a member of the editorial board of the Journal of Geometric Analysis and the Korean Mathematics Journal.
Gordon is a Past President of the Association for Women in Mathematics and continues to be a very active member. Many mathematicians will know her as the organizer of the AWM January workshops, a role she held for a number of years. She is currently a member of the AWM Policy and Advocacy Committee. Gordon is a former member of the Executive Council of the Conference Board on Mathematical Sciences and has held elected positions on the Editorial Boards Committee and the Council of the American Mathematical Society. She has served on many AMS committees including the Committee on the Profession, and the Committee on Committees.
Gordon’s research interests are in Riemannian geometry with emphasis on inverse spectral problems and on the geometry of Lie groups. Mark Kac’s question “Can one hear the shape of a drum?” asks whether the eigenvalue spectrum of the Laplacian on a plane domain determines the domain up to congruence. Gordon is particularly well-known for her work on this question and its analog for more general Riemannian manifolds.  Among her constructions are the first examples of domains with the same eigenvalue spectrum (joint work with David Webb and Scott Wolpert) and continuous families of isospectral Riemannian metrics on spheres.