Mathematics III

The program was headed by Professor Yngve Domar until his retirement on 30 June 1993. Dennis Hejhal, full professor at the University of Minnesota since 1978, was appointed to be his successor, and Hejhal's professorship at Uppsala began on 1 July 1994. Hejhal's areas of interest center around complex analysis, analytic number theory, quantum chaos, and high-performance computing. His work in these areas is of international stature, and was supported by National Science Foundation in the USA for over 20 years. During the next few years, Hejhal will be seeking to form an active research group in his areas at Uppsala.

Certainly, within complex variable (and, more generally, classical analysis), there is already a strong tradition existent at Uppsala. Hejhal's interest in Riemann surface theory and discontinuous groups (particularly computational aspects of the same) can be seen as a natural continuation of this tradition. The situation for number theory and quantum chaos is entirely different, however. It is not too far wrong to say that, in Uppsala, number theory has lain essentially dormant since the 50's and the time of Nagell. (Even then, the thrust was primarily algebraic as opposed to analytic: Hejhal is thus making a "new start"..) Quantum chaos is a much newer subject, wherein one seeks to determine the precise level of randomness manifested by quantum-mechanical "particles" in a variety of gemetrically simple classical systems. It is a subject that, for all practical purposes, is new to Swedish mathematics (though some very preliminary aspects, involving eigenfunctions of the Laplacian, can be seen in early work of Carleman and Pleijel). The exciting thing for many researchers (including Hejhal) is how number theory offers a natural inroad into this area.

At least on Lobachevsky space, quantum chaos features a blend of theoretical physics, number theory, discontinuous groups, trace formulae (a la Selberg), ergodic theory, dynamical systems, and (experiments using) high- performance computers. Seeking to provide rigorous underpinnings for what one "sees" experimentally brings one face-to-face with a whole series of deep open problems (e.g. Riemann Hypothesis and the Sato-Tate conjecture for Fourier coefficients of modular forms). Arithmetic surfaces in Lobachevsky space are one of the main categories of classically chaotic systems (in terms of the geodesic flow), so the appearance of problems of this type is not wholly unexpected given that the quantum-mechanical "particles" are simply automorphic eigenfunctions of the Laplacian.

Sten Kaijser and his former student Fan Ming work in functional analysis. Their main field is interpolation theory for Banach spaces. Fan Ming wrote a thesis about a natural generalization of the usual complex method of interpolation, where the value of a Banach space valued function at a point is replaced by a derivative. This work has been continued and extended. The main part of the research is devoted to a study of interpolation methods. Fan Ming is mainly working with real methods whereas Kaijser considers the relation between real and complex methods. Besides work in interpolation theory Kaijser has also recently studied problems in the theory of convex sets and also in general Banach algebras.

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