Quasicrystals are nonperiodic configurations in space that have many properties of crystals (i.e., periodic configurations). An example is the famous Penrose's tiling discovered in 1973. These objects attracted attention of mathematicians and physicists. In 1984, physicists obtained quasicrystallic metallic alloys with 5th order symmetry which is forbidden for crystals. Very soon mathematicians discovered remarkable connections between quasicrystals and harmonic analysis, mathematical physics, number theory, mathematical logic, dynamical systems. Some aspects of this theory is the subject of the lecture.
The talk presents the results of the joint work with A.Vershik and M.Yor. We define a sigma-finite measure on the space of discrete measures on a measurable space which is equivalent to the law of the classical gamma process and is invariant under an infinite-dimensional group of multiplicators. This measure was first discovered in the works by Gelfand--Graev--Vershik2. O.V. Demchenko. Formal groups and the Hilbert symbol.
on the representation theory of current groups, but we construct it explicitly using some properties of the gamma process. The above invariance property is a natural generalization of the corresponding property of the Lebesgue measure in Rn, and this allows us to call the constructed measure the infinite-dimensional Lebesgue measure. It enjoys many distinguished properties, some of them will be considered in the talk.
The following special functions are presented in the system:
1) hypergeometric-type functions, 2) Heun functions,
3) Painleve transcendents, 4) orthogonal polynomials.
For these functions basic differential equations with
emphasize to singularities, integral relations, series expansions, Riemann's
P-symbols and other important characteristics are given. Output in TEX
format is available. The numerical values of functions can be computed
by Jaffe-Lay algorithm.