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 .

     

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 .

     

On structurally stable diffeomorphisms with codimension one expanding attractors

We show that if a closed -manifold admits a structurally stable diffeomorphism with an orientable expanding attractor of codimension one, then is homotopy equivalent to the -torus and is homeomorphic to for . Moreover, there are no nontrivial basic sets of different from . This allows us to classify, up to conjugacy, structurally stable diffeomorphisms having codimension one orientable expanding attractors and contracting repellers on , .

     

On a refinement of the generalized Catalan numbers for Weyl groups

Let be an irreducible crystallographic root system with Weyl group , coroot lattice and Coxeter number , spanning a Euclidean space , and let be a positive integer. It is known that the set of regions into which the fundamental chamber of is dissected by the hyperplanes in of the form for and is equinumerous to the set of orbits of the action of on the quotient . A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of , are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case and .

     

Cut numbers of -manifolds

We investigate the relations between the cut number, and the first Betti number, of -manifolds We prove that the cut number of a ``generic' -manifold is at most This is a rather unexpected result since specific examples of -manifolds with large and are hard to construct. We also prove that for any complex semisimple Lie algebra there exists a -manifold with and Such manifolds can be explicitly constructed.

     

One-dimensional dynamical systems and Benford's law

Near a stable fixed point at 0 or , many real-valued dynamical systems follow Benford's law: under iteration of a map the proportion of values in with mantissa (base ) less than tends to for all in as , for all integer bases 1$">. In particular, the orbits under most power, exponential, and rational functions (or any successive combination thereof), follow Benford's law for almost all sufficiently large initial values. For linearly-dominated systems, convergence to Benford's distribution occurs for every , but for essentially nonlinear systems, exceptional sets may exist. Extensions to nonautonomous dynamical systems are given, and the results are applied to show that many differential equations such as , where is with F'(0)$">, also follow Benford's law. Besides generalizing many well-known results for sequences such as or the Fibonacci numbers, these findings supplement recent observations in physical experiments and numerical simulations of dynamical systems.

     

Gröbner bases of associative algebras and the Hochschild cohomology

We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Gröbner base. It enables us to compute the Hochschild (co)homology of the algebra. Let be a finitely generated algebra over a commutative ring with a (possibly infinite) Gröbner base on a free algebra , that is, is the quotient with the ideal of generated by . Given a Gröbner base for an -subbimodule of the free -bimodule generated by a set , we have a morphism of -bimodules from the free -bimodule generated by to sending the generator to the element . We construct a Gröbner base on for the -subbimodule Ker() of , and with this we have the free -bimodule generated by and an exact sequence . Applying this construction inductively to the -bimodule itself, we have a free -bimodule resolution of .

     

Fixed point index in symmetric products

Let be an open subset of a locally compact metric ANR and let be a continuous map. In this paper we study the fixed point index of the map that induces in the -symmetric product of , . This index can detect the existence of periodic orbits of period of , and it can be used to obtain the Euler characteristic of the -symmetric product of a manifold , . We compute for all orientable compact surfaces without boundary.

     

A quadratic approximation to the Sendov radius near the unit circle

Define to be the set of complex polynomials of degree with all roots in the unit disk and at least one root at . For a polynomial , define to be the distance between and the closest root of the derivative . Finally, define . In this notation, a conjecture of Bl. Sendov claims that .

In this paper we investigate Sendov's conjecture near the unit circle, by computing constants and (depending only on ) such that for near . We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov's conjecture.

     

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.

     

Weighted estimates in for Laplace's equation on Lipschitz domains

Let , , be a bounded Lipschitz domain. For Laplace's equation in , we study the Dirichlet and Neumann problems with boundary data in the weighted space , where , is a fixed point on , and denotes the surface measure on . We prove that there exists such that the Dirichlet problem is uniquely solvable if , and the Neumann problem is uniquely solvable if . If is a domain, one may take . The regularity for the Dirichlet problem with data in the weighted Sobolev space is also considered. Finally we establish the weighted estimates with general weights for the Dirichlet and regularity problems.

     

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.

     

Depth and cohomological connectivity in modular invariant theory

Let be a finite group acting linearly on a finite-dimensional vector space over a field of characteristic . Assume that  divides the order of so that is a modular representation and let be a Sylow -subgroup for . Define the cohomological connectivity of the symmetric algebra to be the smallest positive integer such that . We show that is a lower bound for the depth of . We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if is -nilpotent and is cyclic, then, for any modular representation, the depth of is .

     

Convergence of double Fourier series and -classes

The double Fourier series of functions of the generalized bounded variation class are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class defined here, have quadrant limits at every point and, for there exist at most countable sets and such that, for and is continuous at . It is shown that the previously studied class contains essentially discontinuous functions unless the sequence satisfies a strong condition.

     

Extension-orthogonal components of preprojective varieties

Let be a Dynkin quiver, and let be the corresponding preprojective algebra. Let be a set of pairwise different indecomposable irreducible components of varieties of -modules such that generically there are no extensions between and for all . We show that the number of elements in is at most the number of positive roots of . Furthermore, we give a module-theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.

     

Anosov automorphisms on compact nilmanifolds associated with graphs

We associate with each graph a -step simply connected nilpotent Lie group and a lattice in . We determine the group of Lie automorphisms of and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every there exist a -dimensional -step simply connected nilpotent Lie group which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice in such that admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups of a given dimension, satisfying the condition. Necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms.

     

Operators on spaces preserving copies of Schreier spaces

It is proved that an operator , compact metrizable, a separable Banach space, for which the -Szlenk index of is greater than or equal to , , is an isomorphism on a subspace of isomorphic to , the Schreier space of order . As a corollary, one obtains that a complemented subspace of with Szlenk index equal to contains a subspace isomorphic to .

     

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.

     

Harmonic maps and 2-cycles, realizing the Thurston norm

Let be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, , of a harmonic map with Morse-type singularities delivers the Thurston norm of its homology class .

In particular, for a map with connected fibers and any well-positioned oriented surface in the homology class of a fiber, we show that the Thurston number satisfies an inequality

Here the variation is can be expressed in terms of the -invariants of the fiber components, and the twist measures the complexity of the intersection of with a particular set of ``bad" fiber components. This complexity is tightly linked with the optimal ``-height" of , being lifted to the -induced cyclic cover .

Based on these invariants, for any Morse map , we introduce the notion of its twist . We prove that, for a harmonic , if and only if .

     

Characteristic subsurfaces and Dehn filling

Let be a simple knot manifold. Using the characteristic submanifold theory and the combinatorics of graphs in surfaces, we develop a method for bounding the distance between the boundary slope of an essential surface in which is not a fiber or a semi-fiber, and the boundary slope of a certain type of singular surface. Applications include bounds on the distances between exceptional Dehn surgery slopes. It is shown that if the fundamental group of has no non-abelian free subgroup, and if is a reducible manifold which is not homeomorphic to or , then . Under the same condition on , it is shown that if is Seifert fibered, then . Moreover, in the latter situation, character variety techniques are used to characterize the topological types of and in case the bound of is attained.