Research: Most of my research is in the area of spectral geometry. This is a subfield of geometric analysis, and as a graduate student many years ago I was trained in classical analysis at UCLA by S-Y Alice Chang and John Garnett. In fact it is not only analysts, who investigate questions in spectral geometry. There are results in spectral geometry by analysts, various kinds of geometers, number theorists, applied mathematicians, mathematical physicists, and others.
Spectral geometry most usually means the study of how the shape or " geometry" of an object is related to the values of the natural frequencies of the object. Roughly speaking, these natural frequencies are the frequencies at which the object can vibrate. A vibrating object often produces a sound, and the frequencies can be heard as the dominant tone and the overtones of the sound. The well known question highlighting what spectral geometry is all about is the question "Can one hear the shape of a drum?". In mathematical terms, the natural frequencies of an object are the eigenvalues of a partial differential operator called the Laplacian. This Laplacian acts on real valued functions defined on the object, and roughly what is does is compute a second order derivative of the function. The eigenvalues of the Laplacian form an infinite sequence of numbers tending to infinity. In spectral geometry we study how this infinite sequence of numbers depends on the shape of the object. Mathematicians working in spectral geometry also study the eigenvalues of other operators of mathematical interest, as well as the eigenvalues of the Laplacian. These eigenvalues might or might not correspond to physical quantities that you can measure. I usually study spectral geometry for nice smooth objects known as smooth manifolds, but some people work on rough objects and even discrete objects like graphs.
For people who like to know the full story, I should mention that many spectral geometers (including me) who work on the Laplacian on smooth manifolds study the whole sequence of eigenvalues of the Laplacian. Now the low eigenvalues of the Laplacian often give accurate values for the frequencies at which a real life object vibrates, but the very high eigenvalues do not correspond to genuine physical vibrations of the object because of molecular forces and damping. These effects are not included in the linear model where the vibration is driven by the Laplacian alone. This means that my research is rather different from that of an engineer who wishes to model precisely the vibrations of a real life object. In actual fact the questions I work on are more closely related to mathematics arising in quantum physics and string theory. I mention that although the Laplacian is a linear differential operator, the study of the spectrum of the Laplacian often involves analyzing non-linear partial differential equations.
To get down to details, given a smooth compact manifold of dimension n, one can use the eigenvalues of the Laplacian to form a function Z(s) of the complex variable s. This function Z(s) is known as a spectral zeta function. It is similar to the Riemann zeta function except that instead of summing up the counting numbers raised to the power -s, you sum up the non-zero eigenvalues of the Laplacian raised to the power -s. This series converges when the real part of s is greater than n/2, but it turns out you can analytically continue it to all complex values of s, and you get a meromorphic function of s. Recently I have been studying how this spectral zeta function is related to the geometry of the manifold. Surprisingly this is sometimes easier than studying individual eigenvalues of the Laplacian. Two quantities occurring in geometry and physics which can be defined using the spectral zeta function are the zeta-regularized determinant, given by exp(-Z'(0)), and the ADM mass. My recent work studies such quantities. I have previously also worked on the spectral geometry of Toeplitz operators on manifolds, and my early work was in geometric measure theory. (Updated 08/02/07.)
Online Publications
A negative mass theorem for the 2-torus
Extremals for Logarithmic HLS inequalities on compact manifolds
Hessian of the zeta function for the Laplacian on forms
Hessians of Spectral Zeta Functions
Critical metrics for the determinant of the Laplacian in odd dimensions
Address:
Department of Mathematics
University of California, San Diego,
9500 Gilman Drive
La Jolla,
California 92093-0112 USA
Mailcode: 0112
Phone: (858) 534-2772
Fax: (858) 534-5273
Office: APM
7426
Email: okikiolu@euclid.ucsd.edu
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