I strongly believe in the absence of barriers between "pure" and "applied" mathematics and in the encouragement of links between them. Interdisciplinarity has led to the greatest achievements in Science: Isaac Newton and Henri Poincaré are only two examples, among the brightest. We are fortunately witnessing a return of these ideals which had been somewhat forgotten during part of the past century. Here is one of my favorite quotations (it is due to Bertrand Russell in "The Study of Mathematics"):
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."
My main areas of interest and research are:
- Harmonic analysis in symplectic spaces: Feichtinger's theory of modulation spaces, partial differential equations (including pseudo-differential calculus and microlocal analysis), applications to time-frequency analysis;
- Symplectic geometry and topology (symplectic and Lagrangian path intersection indices: Maslov index, Conley-Zehnder index, etc...) and their relations with the theory of the metaplectic group;
- Mathematical physics, especially quantum mechanics and its semi-classical formulations; "dequantization".
- The interplay between time-freequency analysis (also called Gabor analysis) and phase-space quantum mechanics;
- Foundational questions in quantum physics.
About symplectic geometry:
Symplectic geometry has deep roots in mathematics and physics going back to Huygens' study of optics and the Hamilton-Jacobi formulation of mechanics. It has grown to touch virtually all branches of mathematics, including many parts of dynamical systems, representation theory, combinatorics, packing problems, algebraic geometry, and differential topology.
About mathematical physics:
"Mathematical physics spans every subfield of physics. Its aim is to apply the most powerful mathematical techniques available to the formulation and solution of physical problems. Mathematics is the language of theoretical physics and, like other languages, it provides a means of organizing thought and expressing ideas in a precise consistent manner. Physicists who are articulate in the language of mathematics have made the greatest contributions to the modern formulations of physics. The list is long and includes such names as Newton, Maxwell, Einstein, Schrödinger, Heisenberg, Weyl, Wigner and Dirac." (David Rowe, author of Chapter 18 in the IUPAP Year 2000 book: http://www.iupap.org/). Here is a good description of mathematical physics (from Stack Exchange):
"Mathematical physics is a branch of mathematics. It explores relations between abstract concepts, proves certain results contingent upon certain hypotheses, and establishes an interlinked set of tools that can be used to study anything that happens to match the relations and hypotheses on hand. This branch is motivated by the theories used in physics. It may seek to prove certain truths that were simply assumed by physicists, or carefully delineate the conditions under which certain theories hold, or even provide generally applicable tools to physicists, who can, in turn, apply them to nature. Mathematical physicists are mathematicians who are intrigued/inspired by physics."
I enjoy writing and assembling mathematical material into a coherent package, so a lot of my effort has gone into writing books concurrently with research papers; it is fun to write books and you learn lots from doing it! See my Research page for details.
During the summer session 2003 I gave a special "FIRST FACULTY IN RESIDENCE" course on some of these topics at the University of Colorado at Boulder.