Members in
this research group are:
Dennis Hejhal, professor
Andreas Juhl, professor
Andreas Strömbergsson, forskarassistent, docent
Anders Södergren, doktorand
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Seminar
The
DNA seminar (Dynamical Systems, Number theory, Analysis)
Research topics
Some subjects that are of interest to the people in this
group are:
Here is a poster (in Swedish) which gives a first introduction to some of these topics.
Quantum chaos is a subject, wherein one seeks to determine the precise level of randomness manifested by
quantum-mechanical "particles" in a variety of geometrically 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.
Computational spectral theory in this case is numerical computation of
the spectra of the hyperbolic Laplacian on hyperbolic surfaces defined
by Hecke congruence subgroups of the modular group.
Fredrik Strömberg is currently computing Maaas waveforms for Hecke congruence subgroups
of the modular group (both with trivial character
and in certain cases a Dirichlet character). See the pictures of waveforms to the right.
(For more pictures visit Gallery of Maass waveforms.)
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