Valence of complex-valued planar harmonic functions

The valence of a function at a point is the number of distinct, finite solutions to . Let be a complex-valued harmonic function in an open set . Let denote the critical set of and the global cluster set of . We show that partitions the complex plane into regions of constant valence. We give some conditions such that has empty interior. We also show that a component is an -fold covering of some component . If is simply connected, then is univalent on . We explore conditions for combining adjacent components to form a larger region of univalence. Those results which hold for functions on open sets in are first stated in that form and then applied to the case of planar harmonic functions. If is a light, harmonic function in the complex plane, we apply a structure theorem of Lyzzaik to gain information about the difference in valence between components of sharing a common boundary arc in .

     

Asymptotic properties of convolution operators and limits of triangular arrays on locally compact groups

We consider a locally compact group and a limiting measure of a commutative infinitesimal triangular system (c.i.t.s.) of probability measures on . We show, under some restrictions on , or , that belongs to a continuous one-parameter convolution semigroup. In particular, this result is valid for symmetric c.i.t.s. on any locally compact group . It is also valid for a limiting measure which has `full' support on a Zariski connected -algebraic group , where is a local field, and any one of the following conditions is satisfied: (1) is a compact extension of a closed solvable normal subgroup, in particular, is amenable, (2) has finite one-moment or (3) has density and in case the characteristic of is positive, the radical of is -defined. We also discuss the spectral radius of the convolution operator of a probability measure on a locally compact group , we show that it is always positive for any probability measure on , and it is also multiplicative in case of symmetric commuting measures.

     

Prescribing analytic singularities for solutions of a class of vector fields on the torus

We consider the operator acting on distributions on the two-torus where and are real-valued, real analytic functions defined on the unit circle We prove, among other things, that when changes sign, given any subset of the set of the local extrema of the local primitives of there exists a singular solution of such that the projection of its analytic singular support is furthermore, for any and any closed subset of there exists such that and We also provide a microlocal result concerning the trace of at

     

On Ore's conjecture and its developments

The -component of the index of a number field , , depends only on the completions of at the primes over . More precisely, equals the index of the -algebra . If is normal, then for some normal over and some , and we write for its index. In this paper we describe an effective procedure to compute for all and all normal and tamely ramified extensions of , hence to determine for all Galois number fields that are tamely ramified at . Using our procedure, we are able to exhibit a counterexample to a conjecture of Nart (1985) on the behaviour of .

     

Geometry and ergodic theory of non-hyperbolic exponential maps

We deal with all the maps from the exponential family such that the orbit of zero escapes to infinity sufficiently fast. In particular all the parameters are included. We introduce as our main technical devices the projection of the map to the infinite cylinder and an appropriate conformal measure . We prove that , essentially the set of points in returning infinitely often to a compact region of disjoint from the orbit of , has the Hausdorff dimension , that the -dimensional Hausdorff measure of is positive and finite, and that the -dimensional packing measure is locally infinite at each point of . We also prove the existence and uniqueness of a Borel probability -invariant ergodic measure equivalent to the conformal measure . As a byproduct of the main course of our considerations, we reprove the result obtained independently by Lyubich and Rees that the -limit set (under ) of Lebesgue almost every point in , coincides with the orbit of zero under the map . Finally we show that the the function , , is continuous.

     

Spike-layered solutions for an elliptic system with Neumann boundary conditions

We prove the existence of nonconstant positive solutions for a system of the form , in , with Neumann boundary conditions on , where is a smooth bounded domain and , are power-type nonlinearities having superlinear and subcritical growth at infinity. For small values of , the corresponding solutions and admit a unique maximum point which is located at the boundary of .

     

Boundary Hölder and estimates for local solutions of the tangential Cauchy-Riemann equation

We study the local solvability of the tangential Cauchy-Riemann equation on an open neighborhood of a point when is a generic -concave manifold of real codimension in , where . Our method is to first derive a homotopy formula for in when is the intersection of with a strongly pseudoconvex domain. The homotopy formula gives a local solution operator for any -closed form on without shrinking. We obtain Hölder and estimates up to the boundary for the solution operator. RÉSUMÉ. Nous étudions la résolubilité locale de l'opérateur de Cauchy- Riemann tangentiel sur un voisinage d'un point d'une sous-variété générique -concave de codimension quelconque de . Nous construisons une formule d'homotopie pour le sur , lorsque est l'intersection de et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un opérateur de résolution pour toute forme -fermée sur . Nous en déduisons des estimations et des estimations hölderiennes jusqu'au bord pour la solution de l'équation de Cauchy-Riemann tangentielle sur .

     

Nonabelian cohomology with coefficients in Lie groups

In this paper we prove some properties of the nonabelian cohomology of a group with coefficients in a connected Lie group . When is finite, we show that for every -submodule of which is a maximal compact subgroup of , the canonical map is bijective. In this case we also show that is always finite. When and is compact, we show that for every maximal torus of the identity component of the group of invariants , is surjective if and only if the -action on is -semisimple, which is also equivalent to the fact that all fibers of are finite. When , we show that is always surjective, where is a maximal compact torus of the identity component of . When is cyclic, we also interpret some properties of in terms of twisted conjugate actions of .

     

Hyperpolygon spaces and their cores

Given an -tuple of positive real numbers , Konno (2000) defines the hyperpolygon space , a hyperkähler analogue of the Kähler variety parametrizing polygons in with edge lengths . The polygon space can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural -action, and the union of the precompact orbits is called the core. We study the components of the core of , interpreting each one as a moduli space of pairs of polygons in with certain properties. Konno gives a presentation of the cohomology ring of ; we extend this result by computing the -equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.

     

Cremer fixed points and small cycles

Let , , and let denote the sequence of convergents to the regular continued fraction of . Let be a function holomorphic at the origin, with a power series of the form . We assume that for infinitely many we simultaneously have (i) , (ii) the coefficients stay outside two small disks, and (iii) the series is lacunary, with for . We then prove that has infinitely many periodic orbits in every neighborhood of the origin.

     

Minimal invariant tori in the resonant regions for nearly integrable Hamiltonian systems

Consider a real analytical Hamiltonian system of KAM type that has degrees of freedom (2$">) and is positive definite in . Let . In this paper we show that for most rotation vectors in , in the sense of ()-dimensional Lebesgue measure, there is at least one ()-dimensional invariant torus. These tori are the support of corresponding minimal measures. The Lebesgue measure estimate on this set is uniformly valid for any perturbation.

     

The Farrell-Jones Isomorphism Conjecture for finite covolume hyperbolic actions and the algebraic -theory of Bianchi groups

We prove the Farrell-Jones Isomorphism Conjecture for groups acting properly discontinuously via isometries on (real) hyperbolic -space with finite volume orbit space. We then apply this result to show that, for any Bianchi group , , , and vanish for .

     

Ramsey families of subtrees of the dyadic tree

We show that for every rooted, finitely branching, pruned tree of height there exists a family which consists of order isomorphic to subtrees of the dyadic tree with the following properties: (i) the family is a subset of ; (ii) every perfect subtree of contains a member of ; (iii) if is an analytic subset of , then for every perfect subtree of there exists a perfect subtree of such that the set either is contained in or is disjoint from .

     

Estimates of the derivatives for parabolic operators with unbounded coefficients

We consider a class of second-order uniformly elliptic operators with unbounded coefficients in . Using a Bernstein approach we provide several uniform estimates for the semigroup generated by the realization of the operator in the space of all bounded and continuous or Hölder continuous functions in . As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation (0$">) and the nonhomogeneous Dirichlet Cauchy problem . Then, we prove two different kinds of pointwise estimates of that can be used to prove a Liouville-type theorem. Finally, we provide sharp estimates of the semigroup in weighted -spaces related to the invariant measure associated with the semigroup.

     

Toric residue and combinatorial degree

Consider an -dimensional projective toric variety defined by a convex lattice polytope . David Cox introduced the toric residue map given by a collection of divisors on . In the case when the are -invariant divisors whose sum is , the toric residue map is the multiplication by an integer number. We show that this number is the degree of a certain map from the boundary of the polytope to the boundary of a simplex. This degree can be computed combinatorially. We also study radical monomial ideals of the homogeneous coordinate ring of . We give a necessary and sufficient condition for a homogeneous polynomial of semiample degree to belong to in terms of geometry of toric varieties and combinatorics of fans. Both results have applications to the problem of constructing an element of residue one for semiample degrees.

     

Dynamical forcing of circular groups

In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties.

As an application, we show that the set consisting of rotation numbers which can be forced by finitely presented groups is an infinitely generated -module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number is forced by a pair , where is a finitely presented group and is some element, if the set of rotation numbers of as varies over is precisely the set .

We show that the set of subsets of which are of the form

where varies over countable groups, are exactly the set of closed subsets which contain and are invariant under . Moreover, we show that every such subset can be approximated from above by for finitely presented .

As another application, we construct a finitely generated group which acts faithfully on the circle, but which does not admit any faithful action, thus answering in the negative a question of John Franks.

     

Hausdorff measures, dimensions and mutual singularity

Let be a metric space. For a probability measure on a subset of and a Vitali cover of , we introduce the notion of a -Vitali subcover , and compare the Hausdorff measures of with respect to these two collections. As an application, we consider graph directed self-similar measures and in satisfying the open set condition. Using the notion of pointwise local dimension of with respect to , we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen's multifractal Hausdorff measures are mutually singular.

     

Hardy space of exact forms on

We show that the Hardy space of divergence-free vector fields on has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of . Using the duality result we prove a ``div-curl" type theorem: for in , is equivalent to a -type norm of , where the supremum is taken over all with This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on , study their atomic decompositions and dual spaces, and establish ``div-curl" type theorems on .

     

Good measures on Cantor space

While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure is the countable dense subset is clopen of the unit interval. It is a topological invariant for the measure. For the class of good measures it is a complete invariant. A full, nonatomic measure is good if whenever are clopen sets with , there exists a clopen subset of such that . These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense, conjugacy class.

     

Maximal holonomy of infra-nilmanifolds with -dimensional quaternionic Heisenberg geometry

Let be the quaternionic Heisenberg group of real dimension and let denote the maximal order of the holonomy groups of all infra-nilmanifolds with -geometry. We prove that . As an application, by applying Kim and Parker's result, we obtain that the minimum volume of a -dimensional quaternionic hyperbolic manifold with cusps is at least