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A complete classification of finitely generated involutive commutative two-valued groups is obtained. Three series of such two-valued groups are constructed: principal, unipotent and special, and it is shown that any finitely generated involutive commutative two-valued group is isomorphic to a two-valued group belonging to one of these series. A number of classification results are obtained for topological involutive commutative two-valued groups in the Hausdorff and locally compact cases. The classification of algebraic involutive two-valued groups in the one-dimensional case is also discussed., 67 pages; slightly amended English version of the paper published in Uspekhi Matematicheskikh Nauk (Russian Math Surveys)

A simple integral formula as an iterated residue is presented for the Baker-Akhiezer function related to $A_n$ type root system both in the rational and trigonometric cases. We present also a formula for the Baker-Akhiezer function as a Selberg-type integral and generalise it to the deformed $A_{n,1}$-case. These formulas can be interpreted as new cases of explicit evaluation of Selberg-type integrals., 13 pages, 2 figures

V. Fock [7] introduced an interesting function $��(x)$, $x \in {\mathbb R}$ related to Markov numbers. We explain its relation to Federer-Gromov's stable norm and Mather's $��$-function, and use this to study its properties. We prove that $��$ and its natural generalisations are differentiable at every irrational $x$ and non-differentiable otherwise, by exploiting the relation with length of closed geodesics on the punctured or one-hole tori with the hyperbolic metric and the results by Bangert [3] and McShane- Rivin [19]., 9 pages, 2 figures

We study Lam�� operators of the form $$L = -\frac{d^2}{dx^2} + m(m+1)��^2\wp(��x+z_0),$$ with $m\in\mathbb{N}$ and $��$ a half-period of $\wp(z)$. For rectangular period lattices, we can choose $��$ and $z_0$ such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lam�� operator has a band structure with not more than $m$ gaps. In the first part of the paper, we prove that the opened gaps are precisely the first $m$ ones. In the second part, we study the Lam�� spectrum for a generic period lattice when the potential is complex-valued. We concentrate on the $m=1$ case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the $m=2$ case, paying particular attention to the rhombic lattices.

We show that the orthogonal separation coordinates on the sphere $S^n$ are naturally parametrised by the real version of the Deligne-Mumford-Knudsen moduli space $\bar M_{0,n+2}(R)$ of stable curves of genus zero with $n+2$ marked points. We use the combinatorics of Stasheff polytopes tessellating $\bar M_{0,n+2}(R)$ to classify the different canonical forms of separation coordinates and deduce an explicit construction of separation coordinates and St\"ackel systems from the mosaic operad structure on $\bar M_{0,n+2}(R)$., Comment: Extended version, 20 pages, 4 figures

We introduce some discrete analogues of the Dirac magnetic monopole on a unit sphere S^2 and explain how to compute the corresponding spectrum using the representation theory of finite groups. The main examples are certain magnetic Laplacians on the regular polyhedral graphs, coming from induced representations of the binary polyhedral groups., Comment: 18 pages, 6 figures

The Matsuo-Cherednik correspondence is an isomorphism from solutions of Knizhnik-Zamolodchikov equations to eigenfunctions of generalized Calogero-Moser systems associated to Coxeter groups G and a multiplicity function m on their root systems. We apply this correspondence to the most degenerate case of zero spectral parameters. The space of eigenfunctions is then the space H_m of m-harmonic polynomials, recently introduced in math-ph/0105014. We compute the Poincare' polynomials for the space H_m and of its isotypical components corresponding to each irreducible representation of the group G. We also give an explicit formula for m-harmonic polynomials of lowest positive degree in the S_n case., Comment: 22 pages