Volume 10 (2010), Number 3. Abstracts

A. Bucur and A. Diaconu. Moments of Quadratic Dirichlet L-Functions over Rational Function Fields [PDF]

We establish the meromorphic continuation of a multiple Dirichlet series associated to the fourth moment of quadratic Dirichlet L-functions, over the rational function field Fq(T) with q odd, up to its natural boundary. This is the first such result in which the group of functional equations is infinite; in such cases, it is expected that the series cannot be continued everywhere but can at least be extended to a large enough region to deduce asymptotics at the central point. In this case, these asymptotics coincide with existing predictions for the fourth moment of the symplectic family of quadratic Dirichlet L-functions. The construction uses the Weyl group action of a particular Kac–Moody algebra; this suggests an approach to higher moments using appropriate non-affine Kac–Moody algebras.

Keywords. Moments of quadratic Dirichlet L-functions, multiple Dirichlet series, finite field, rational function field, Coxeter group, roots.

2000 Mathematics Subject Classification. Primary: 14G10, 14G15, 20F55, 11F68, 11M32; Secondary: 11M26, 11T24.

M. Garay and D. van Straten. Classical and Quantum Integrability [PDF]

It is a well-known problem to decide if a classical hamiltonian system that is integrable, in the Liouville sense, can be quantised to a quantum integrable system. We identify the obstructions to do so, and show that the obstructions vanish under certain conditions.

Keywords. Micro-local analysis, non-commutative geometry.

2000 Mathematics Subject Classification. 81S10.

A. Glutsyuk. On Density of Horospheres in Dynamical Laminations [PDF]

Sullivan's dictionary relates two domains of complex dynamics: Kleinian groups and rational iterations on the Riemann sphere. In 1997 M. Lyubich and Y. Minsky have extended the Sullivan's dictionary by constructing an analogue of the hyperbolic manifold of a Kleinian group: the so-called quotient hyperbolic lamination associated to a rational function. This is an abstract topological space constructed from the space of backward orbits of the rational function that carries a “foliation” (more precisely, lamination) by hyperbolic 3-manifolds (that may be singular). The hyperbolic leaves are dense, may be after deleting at most finite number of isolated leaves. Each hyperbolic leaf is foliated by horospheres, which form the unstable foliation (horospheric lamination) for the leafwise vertical geodesic flow. We consider the total laminated space with isolated hyperbolic leaves deleted. We prove that the horospheric lamination is topologically transitive (and there are a lot of dense horospheres), if and only if the corresponding rational function does not belong to the following list of exceptions: powers, Chebyshev polynomials, Lattès examples. We show that the horospheric lamination is minimal, if the corresponding function does not belong to the same list of exceptions and is critically nonrecurrent without parabolics.

Keywords. Rational function, natural extension, repelling periodic orbit, affine lamination, hyperbolic lamination, horosphere, minimality.

2000 Mathematics Subject Classification. 58F23, 57M50.

S. Gusein-Zade, I. Luengo, and A. Melle-Hernández. On Generating Series of Classes of Equivariant Hilbert Schemes of Fat Points [PDF]

We discuss different definitions of equivariant (with respect to an action of a finite group on a manifold) Hilbert schemes of zero-dimensional subschemes and compute generating series of classes of equivariant Hilbert schemes for actions of cyclic groups on the plane in some cases.

Keywords. Hilbert schemes of zero-dimensional subschemes, group actions, generating series.

2000 Mathematics Subject Classification. 14C05, 14G10.

G. Lusztig. Parabolic Character Sheaves, III [PDF]

The main purpose of this paper is to define a class of simple perverse sheaves (called character sheaves) on certain ind-varieties associated to a loop group. This has applications to a geometric construction of certain affine Hecke algebras with unequal parameters, as will be shown elsewhere.

Keywords. Character sheaf, parahoric subgroup, ind-variety, affine Hecke algebra.

2000 Mathematics Subject Classification. 20G99.

V. Meshkov, A. Omelchenko, M. Petrov, and E. Tropp. Dyck and Motzkin Triangles with Multiplicities [PDF]

Exponential generating functions for the Dyck and Motzkin triangles are constructed for various assignments of multiplicities to the arrows of these triangles. The possibility to build such a function provided that the generating function for paths that end on the axis is a priori unknown is analyzed. Asymptotic estimates for the number of paths are obtained for large values of the path length.

Keywords. Dyck and Motzkin triangles, Dyck and Motzkin paths, paths with multiplicities, exact enumeration of paths, generating function, asymptotic enumeration.

2000 Mathematics Subject Classification. 05A16, 05A15.

A. Ramadoss. The Mukai Pairing and Integral Transforms in Hochschild Homology [PDF]

Let X be a separated smooth proper scheme over a field of characteristic 0. Following Shklyarov, we construct a (non-degenerate) pairing on the Hochschild homology of perf(X), and hence, on the Hochschild homology of X. On the other hand the Hochschild homology of X also has the Mukai pairing (see papers by Căldăraru). If X is Calabi–Yau, this pairing arises from the action of the class of a genus 0 Riemann-surface with two incoming closed boundaries and no outgoing boundary in H0(M0(2,0)) on the algebra of closed states of a version of the B-Model on X. We show that these pairings almost coincide. This is done via a different view of the construction of integral transforms in Hochschild homology that originally appeared in Căldăraru's work. This is used to prove that the “more natural” construction of integral transforms in Hochschild homology by Shklyarov coincides with that of Căldăraru. These results give rise to a Hirzebruch Riemann–Roch theorem for the sheafification of the Dennis trace map.

Keywords. Hochschild homology, integral transforms, Mukai pairing, Dennis trace map, Hirzebruch–Riemann–Roch.

2000 Mathematics Subject Classification. 19L10, 14F05, 19D55, 14C40.

W. Veech. Decoding Rauzy Induction: Bufetov's Question [PDF]

Answering a question posed by Alexander Bufetov, it is proved that if a pair of (i.d.o.c.) interval exchanges on [0, 1) have identical sequences of visitation matrices with respect to Rauzy induction, then the exchanges are homeomorphically conjugate and, in particular, are governed by the same permutation.

Keywords. Interval exchange, Rauzy induction.

2000 Mathematics Subject Classification. 37E05, 37B10.

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