Integrable Systems and Factorization Problems

The present lectures were prepared for the Faro International Summer School on Factorization and Integrable Systems in September 2000. They were intended for participants with the background in Analysis and Operator Theory but without special knowledge of Geometry and Lie Groups. In order to make the main ideas reasonably clear, I tried to use only matrix algebras such as $\frak{gl}(n)$ and its natural subalgebras; Lie groups used are either GL(n) and its subgroups, or loop groups consisting of matrix-valued functions on the circle (possibly admitting an extension to parts of the Riemann sphere). I hope this makes the environment sufficiently easy to live in for an analyst. The main goal is to explain how the factorization problems (typically, the matrix Riemann problem) generate the entire small world of Integrable Systems along with the geometry of the phase space, Hamiltonian structure, Lax representations, integrals of motion and explicit solutions. The key tool will be the \emph{% classical r-matrix} (an object whose other guise is the well-known Hilbert transform). I do not give technical details, unless they may be exposed in a few lines; on the other hand, all motivations are given in full scale whenever possible.Comment: LaTeX 2.09, 69 pages. Introductory lectures on Integrable systems, Classical r-matrices and Factorization problem

The positive equivariant symplectic homology as an invariant for some contact manifolds

We show that positive $S^1$-equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley-Zehnder indices of all good periodic Reeb orbits on the boundary of the Liouville domain is lacunary, the positive $S^1$-equivariant symplectic homology can be computed; it is generated by those orbits. We prove a "Viterbo functoriality" property: when one Liouville domain is embedded into an other one, there is a morphism (reversing arrows) between their positive $S^1$-equivariant symplectic homologies and morphisms compose nicely. These properties allow us to give a proof of Ustilovsky's result on the number of non isomorphic contact structures on the spheres $S^{4m+1}$. They also give a new proof of a Theorem by Ekeland and Lasry on the minimal number of periodic Reeb orbits on some hypersurfaces in $\mathbb{R}^{2n}$. We extend this result to some hypersurfaces in some negative line bundles.Comment: Correction in the computations of the action, no modifications of the result

Multiple Dirichlet Series for Affine Weyl Groups

Let $W$ be the Weyl group of a simply-laced affine Kac-Moody Lie group, excepting $\tilde{A}_n$ for $n$ even. We construct a multiple Dirichlet series $Z(x_1, \ldots x_{n+1})$, meromorphic in a half-space, satisfying a group $W$ of functional equations. This series is analogous to the multiple Dirichlet series for classical Weyl groups constructed by Brubaker-Bump-Friedberg, Chinta-Gunnells, and others. It is completely characterized by four natural axioms concerning its coefficients, axioms which come from the geometry of parameter spaces of hyperelliptic curves. The series constructed this way is optimal for computing moments of character sums and L-functions, including the fourth moment of quadratic L-functions at the central point via $\tilde{D}_4$ and the second moment weighted by the number of divisors of the conductor via $\tilde{A}_3$. We also give evidence to suggest that this series appears as a first Fourier-Whittaker coefficient in an Eisenstein series on the twofold metaplectic cover of the relevant Kac-Moody group. The construction is limited to the rational function field $\mathbb{F}_q(t)$, but it also describes the $p$-part of the multiple Dirichlet series over an arbitrary global field