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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.

Gaudin hamiltonians form families of r-dimensional abelian Lie subalgebras of the holonomy Lie algebra of the arrangement of reflection hyperplanes of a Coxeter group of rank r. We consider the set of principal Gaudin subalgebras, which is the closure in the appropriate Grassmannian of the set of spans of Gaudin hamiltonians. We show that principal Gaudin subalgebras form a smooth projective variety isomorphic to the De Concini-Procesi compactification of the projectivized complement of the arrangement of reflection hyperplanes., Comment: 13 pages, 2 figures; added detailed description of the B_2 and B_3 cases in the new version

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