Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula

We consider the problem of explicitly computing dimensions of spaces of automorphic or modular forms in level one, for a split classical group $\mathbf{G}$ over $\mathbb{Q}$ such that $\mathbf{G}(\R)$ has discrete series. Our main contribution is an algorithm calculating orbital integrals for the characteristic function of $\mathbf{G}(\mathbb{Z}_p)$ at torsion elements of $\mathbf{G}(\mathbb{Q}_p)$. We apply it to compute the geometric side in Arthur's specialisation of his invariant trace formula involving stable discrete series pseudo-coefficients for $\mathbf{G}(\mathbb{R})$. Therefore we explicitly compute the Euler-Poincar\'e characteristic of the level one discrete automorphic spectrum of $\mathbf{G}$ with respect to a finite-dimensional representation of $\mathbf{G}(\mathbb{R})$. For such a group $\mathbf{G}$, Arthur's endoscopic classification of the discrete spectrum allows to analyse precisely this Euler-Poincar\'e characteristic. For example one can deduce the number of everywhere unramified automorphic representations $\pi$ of $\mathbf{G}$ such that $\pi_{\infty}$ is isomorphic to a given discrete series representation of $\mathbf{G}(\mathbb{R})$. Dimension formulae for the spaces of vector-valued Siegel modular forms are easily derived.Comment: 89 pages, 28 tables, comments welcome. Much more data available at http://www.math.ens.fr/~taibi/dimtrace

Heisenberg model in pseudo-Euclidean spaces

We construct analogues of the classical Heisenberg spin chain model (or the discrete Neumann system) on pseudo-spheres and light-like cones in the pseudo-Euclidean spaces and show their complete Hamiltonian integrability. Further, we prove that the Heisenberg model on a light--like cone leads to a new example of integrable discrete contact system.Comment: 6 page

Axiomatizations of signed discrete Choquet integrals

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lov\'asz extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory

Discrete orbits, recurrence and solvable subgroups of Diff(C^2,0)

We discuss the local dynamics of a subgroup of Diff(C^2,0) possessing locally discrete orbits as well as the structure of the recurrent set for more general groups. It is proved, in particular, that a subgroup of Diff(C^2,0) possessing locally discrete orbits must be virtually solvable. These results are of considerable interest in problems concerning integrable systems.Comment: The first version of this paper and "A note on integrability and finite orbits for subgroups of Diff(C^n,0)" are an expanded version of our paper "Discrete orbits and special subgroups of Diff(C^n,0)". An intermediate version re-submitted to the journal on March 2015 is available at http://www.fep.up.pt/docentes/hreis/publications.htm where there is also a comparison between these 3 version

Contact complete integrability

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical $\R\ltimes \R^{n-1}$ structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of $n$ contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures. We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories

Motivic integration and the Grothendieck group of pseudo-finite fields

We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.Comment: 11 page