Volume 9 (2009), Number 4. Abstracts

M. Brunella. Nonuniformisable Foliations on Compact Complex Surfaces [PDF]

We give a complete classification of holomorphic foliations on compact complex surfaces which are not uniformisable, i.e., for which universal coverings of the leaves do not glue together in a Hausdorff way. This leads to complex analogs of the Reeb component defined on certain Hopf surfaces and certain Kato surfaces.

Keywords. Holomorphic foliations, Reeb component, uniformisation, nonkahlerian compact complex surfaces.

2000 Mathematics Subject Classification. 32J15, 37F75, 57R30.

P. Forrester and E. Nordenstam. The Anti-Symmetric GUE Minor Process [PDF]

Our study is initiated by a multi-component particle system underlying the tiling of a half hexagon by three species of rhombi. In this particle system species j consists of ⌊j/2⌋ particles which are interlaced with neigbouring species. The joint probability density function (PDF) for this particle system is obtained, and is shown in a suitable scaling limit to coincide with the joint eigenvalue PDF for the process formed by the successive minors of anti-symmetric GUE matrices, which in turn we compute from first principles. The correlations for this process are determinantal and we give an explicit formula for the corresponding correlation kernel in terms of Hermite polynomials. Scaling limits of the latter are computed, giving rise to the Airy kernel, extended Airy kernel and bead kernel at the soft edge and in the bulk, as well as a new kernel at the hard edge.

Keywords. Random matrices, tilings, point processes.

2000 Mathematics Subject Classification. Primary: 15A52; Secondary: 60G57.

D. Grigoriev. Analogue of Newton–Puiseux Series for Non-Holonomic D-Modules and Factoring [PDF]

We introduce a concept of a fractional derivatives series and prove that any linear partial differential equation in two independent variables has a fractional derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to D-modules having infinite-dimensional space of solutions (i.e., non-holonomic D-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors.

Keywords. Newton–Puiseux series for D-modules, fractional derivatives, factoring linear partial differential operators.

2000 Mathematics Subject Classification. 35C10, 35D05, 68W30.

V. Grines, F. Laudenbach, and O. Pochinka. Self-indexing Energy Function for Morse–Smale Diffeomorphisms on 3-Manifolds [PDF]

The paper is devoted to finding conditions for the existence of a self-indexing energy function for Morse–Smale diffeomorphisms on a 3-manifold M3. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of M3 of a special type with respect to the considered diffeomorphism.

Keywords. Morse–Smale diffeomorphism, Morse–Lyapunov function, Heegaard splitting.

2000 Mathematics Subject Classification. 37B25, 37D15, 57M30.

A. Kirillov and R. Sakamoto. Paths and Kostka–Macdonald Polynomials [PDF]

We give several equivalent combinatorial descriptions of the space of states for the box-ball systems, and connect certain partition functions for these models with the q-weight multiplicities of the tensor product of the fundamental representations of the Lie algebra gl(n). As an application, we give an elementary proof of the special case t=1 of the Haglund–Haiman–Loehr formula. Also, we propose a new class of combinatorial statistics that naturally generalize the so-called energy statistics.

Keywords. Crystals, paths, energy and tau functions, box-ball systems, Kostka–Macdonald polynomials.

2000 Mathematics Subject Classification. 05E10, 20C35.

H. Movasati and E. Vieira. Projective Limit Cycles [PDF]

In this article we study projective cycles in P2R. Our inspiring example is the Jouanolou foliation of odd degree which has a hyperbolic projective limit cycle. We prove that only odd degree foliations may have projective cycles and that foliations with exactly one real simple singularity have a projective cycle. We also prove that after a perturbation of a generic Hamiltonian foliation with a projective cycle, we have a projective limit cycle if and only if the perturbation is not Hamiltonian.

Keywords. Holomorphic foliations, holonomy, vanishing cycle.

2000 Mathematics Subject Classification. 34C07.

D. Panyushev. Properties of Weight Posets for Weight Multiplicity Free Representations [PDF]

We study weight posets of weight multiplicity free (WMF) representations of reductive Lie algebras. Specifically, we are interested in relations between dim R and the number of edges in the Hasse diagram of the corresponding weight poset #E(R). We compute the number of edges and upper covering polynomials for the weight posets of all WMF-representations. We also point out non-trivial isomorphisms between weight posets of different irreducible WMF-representations.

Our main results concern WMF-representations associated with periodic gradings or Z-gradings of simple Lie algebras. For Z-gradings, we prove that 0 < 2 dim R − #E(Rh, where h is the Coxeter number of g. For periodic gradings, we prove that 0 ≤ 2 dim R − #E(R).

Keywords. Hasse diagram, weight poset, root order, grading of a Lie algebra.

2000 Mathematics Subject Classification. Primary: 05E15; Secondary: 06A07, 17B20.

C. Sabbah. Fourier–Laplace Transform of Irreducible Regular Differential Systems on the Riemann Sphere, II [PDF]

This article is devoted to the complete proof of the main theorem in the author's paper of 2004 showing that the Fourier–Laplace transform of an irreducible regular differential system on the Riemann sphere underlies, at finite distance, a polarizable regular twistor D-module.

Keywords. Flat bundle, harmonic metric, twistor D-module, Fourier–Laplace transform.

2000 Mathematics Subject Classification. Primary: 32S40; Secondary: 14C30, 34Mxx.

M. Troyanov. On the Hodge Decomposition in Rn [PDF]

We prove a version of Lp-Hodge decomposition for differential forms in Euclidean space and a generalization to the class of Lizorkin currents. Using these tools, we also compute Lq,p-cohomology of Rn.

Keywords. Hodge decomposition, temperate currents, Lizorkin currents.

2000 Mathematics Subject Classification. 58A10, 42B, 42B20, 58A14.

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