Volume 11 (2011), Number 1. Abstracts

H. Andersen and M. Kaneda. Rigidity of Tilting Modules [PDF]

Let U_q denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that q is a complex root of unity of odd order and that U_q is obtained via Lusztig's q-divided powers construction. We prove that all regular projective (tilting) modules for U_q are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl modules for U_q. On the other hand, we give examples of non-rigid indecomposable tilting modules as well as non-rigid Weyl modules. These examples are for type B₂ and in this case as well as for type A₂ we calculate explicitly the Loewy structure for all regular Weyl modules.

We also demonstrate that these results carry over to the modular case when the highest weights in question are in the so-called Jantzen region. At the same time we show by examples that as soon as we leave this region non-rigid tilting modules do occur.

Keywords. Modular representations, modules for quantum groups, rigid modules, tilting modules

2000 Mathematics Subject Classification. 17B37, 20G05

Y. Genzmer and E. Paul. Normal Forms of Foliations and Curves Defined by a Function with a Generic Tangent Cone [PDF]

We first describe the local and global moduli spaces of germs of foliations defined by analytic functions in two variables with p transverse smooth branches, and with integral multiplicities (in the univalued holomorphic case) or complex multiplicities (in the multivalued “Darboux” case). We specify normal forms in each class. Then we study on these moduli space the distribution C induced by the following equivalence relation: two points are equivalent if and only if the corresponding foliations have the same analytic invariant curves up to analytical conjugacy. Therefore, the space of leaves of C is the moduli space of curves. We prove that C is rationally integrable. These rational integrals give a complete system of invariants for these generic plane curves, which extend the well-known cross-ratios between branches.

Keywords. Holomorphic foliation, moduli of curve, singularities

2000 Mathematics Subject Classification. 34M35, 32S65, 32G13

Y. Ihara and K. Matsumoto. On log L and L'/L for L-Functions and the Associated “M-Functions”: Connections in Optimal Cases [PDF]

Let L(s,χ) be either log L(s,χ) or L'/L(s,χ), associated with an (abelian) L-function L(s,χ) of a global field K. For any quasi-character ψ: ℂ→ℂ^× of the additive group of complex numbers, consider the average “Avg_fχ=f” of ψ(L(s,χ)) over all Dirichlet characters χ on K with a given prime conductor f. This paper contains (i) study of the limit as N(f)→∞ of this average, (ii) basic studies of the analytic function \tilde M_s(z₁,z₂) in 3 complex variables arising from (i) (here, (z₁,z₂)∈ℂ² is the natural parameter for ψ), and (iii) application to value-distribution theory for {L(s,χ)}_χ. Our base field K is either a function field over a finite field, or a special type of number field: the rational number field ℚ or an imaginary quadratic field. But in the number field case, the Generalized Riemann Hypothesis is assumed in (i) and (iii).

Keywords. L-function, Value distribution, Mean value theorem, Arithmetic Dirichlet series, function field over finite field

2000 Mathematics Subject Classification. Primary: 11R42; Secondary: 11M38, 11M41

R. Minlos. On Point-Like Interaction between n Fermions and Another Particle [PDF]

In this note the point-like interaction of n fermions with a particle of a different nature is considered in a framework of the theory of self-adjoint extensions of symmetric operators. It introduces the family of extensions of the original symmetric operator, most natural from the physical point of view (the so-called Ter-Martirosian–Skornyakov' extensions). Here we prove that for n≤4 and large enough values of the mass of the separate particle these extensions are self-adjoint and bounded from below.

Keywords. Symmetric operator, selfadjoint extension, boundedness below, Ter-Martirosian–Skornyakov extension

2000 Mathematics Subject Classification. 81Q10, 47S30

N. Moshchevitin. A Note on Badly Approximable Affine Forms and Winning Sets [PDF]

We prove a result on inhomogeneous Diophantine approximations related to the theory of (α,β)-games.

Keywords. Diophantine approximations, inhomogeneous linear forms, Schmidt's games

2000 Mathematics Subject Classification. 11J20

O. Musin. On Rigid Hirzebruch Genera [PDF]

The classical multiplicative (Hirzebruch) genera of manifolds have the wonderful property which is called rigidity. Rigidity of a genus h means that if a compact connected Lie group G acts on a manifold X, then the equivariant genus h^G(X) is independent on G, i.e., h^G(X)=h(X).

In this paper we are considering the rigidity problem for stably complex manifolds. In particular, we are proving that a genus is rigid if and only if it is a generalized Todd genus.

Keywords. Hirzebruch genus, rigid genus, complex bordism

2000 Mathematics Subject Classification. 55N22, 57R77

N. Nadirashvili. On Derivatives of Viscosity Solutions to Fully Nonlinear Elliptic Equations [PDF]

We prove that partial derivatives of viscosity solutions of elliptic fully nonlinear equations are viscosity solutions of linear elliptic equations.

Keywords. Fully nonlinear elliptic equation, elliptic equations with measurable coefficients

2000 Mathematics Subject Classification. 35J60

A. Shcherbakov. Metrics and Smooth Uniformisation of Leaves of Holomorphic Foliations [PDF]

We consider foliations of complex projective manifolds by analytic curves. In a generic case each leaf is hyperbolic and there exists unique Poincaré metric on the leaves. It is shown that in a generic case this metric smoothly depends on a leaf. The manifold of universal covering of the leaves passing through some transversal base has a natural complex structure. It is shown that this structure can be defined as a smooth almost complex structure on the product of the base and a fiber and there exists a natural pseudoconvex exhaustion.

Keywords. Foliations, Poincaré metric, almost complex structures

2000 Mathematics Subject Classification. 32Q30 (53C12)

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