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.

     

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.

     

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 .

     

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.

     

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.

     

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.

     

A sharp weak type inequality for martingale transforms and other subordinate martingales

If is a martingale difference sequence, a sequence of numbers in , and a positive integer, then

Here denotes the best constant. If , then as was shown by Burkholder. We show here that for the case 2$">, and that is also the best constant in the analogous inequality for two martingales and indexed by , right continuous with limits from the left, adapted to the same filtration, and such that is nonnegative and nondecreasing in . In Section 7, we prove a similar inequality for harmonic functions.

     

Threefolds with vanishing Hodge cohomology

We consider algebraic manifolds of dimension 3 over with for all and 0$">. Let be a smooth completion of with , an effective divisor on with normal crossings. If the -dimension of is not zero, then is a fibre space over a smooth affine curve (i.e., we have a surjective morphism from to such that the general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of is and the -dimension of is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of .

     

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 .

     

Dimension of families of determinantal schemes

A scheme of codimension is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous matrix and is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers and we denote by (resp. ) the locus of good (resp. standard) determinantal schemes of codimension defined by the maximal minors of a matrix where is a homogeneous polynomial of degree .

In this paper we address the following three fundamental problems: To determine (1) the dimension of (resp. ) in terms of and , (2) whether the closure of is an irreducible component of , and (3) when is generically smooth along . Concerning question (1) we give an upper bound for the dimension of (resp. ) which works for all integers and , and we conjecture that this bound is sharp. The conjecture is proved for , and for under some restriction on and . For questions (2) and (3) we have an affirmative answer for and , and for under certain numerical assumptions.

     

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 .

     

On the -torsion of elliptic curves and elliptic surfaces in characteristic

We study the extension generated by the -coordinates of the -torsion points of an elliptic curve over a function field of characteristic . If is a non-isotrivial elliptic surface in characteristic with a -torsion section, then for 11$"> our results imply restrictions on the genus, the gonality, and the -rank of the base curve , whereas for such a surface can be constructed over any base curve . We also describe explicitly all occurring in the cases where the surface is rational or or the base curve is rational, elliptic or hyperelliptic.

     

Tangent algebraic subvarieties of vector fields

An algebraic commutative group is associated to any vector field on a complete algebraic variety . The group acts on and its orbits are the minimal subvarieties of which are tangent to . This group is computed in the case of a vector field on .

     

Stable and finite Morse index solutions on or on bounded domains with small diffusion

In this paper, we study bounded solutions of on (where and sometimes ) and show that, for most 's, the weakly stable and finite Morse index solutions are quite simple. We then use this to obtain a very good understanding of the stable and bounded Morse index solutions of on with Dirichlet or Neumann boundary conditions for small .

     

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 .

     

Plane Cremona maps, exceptional curves and roots

We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection and , respectively) on a smooth complex projective surface which admits a birational morphism to . The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of is in the -orbit of the class of the total transform of some point blown up by if is exceptional, or in the -orbit of a simple root if is root, where is the Weyl group acting on ; we give an arithmetical criterion, which adapts an analogous criterion suggested by Hudson for homaloidal divisors, and a geometrical one. Secondly, we prove that the irreducibility of the exceptional or root divisor is a necessary and sufficient condition in order that could be transformed into a line by some plane Cremona map, and in most cases for its contractibility. Finally, we provide irreducibility criteria for proper homaloidal, exceptional and effective root divisors.

     

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 , .

     

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

     

Compactness of isospectral potentials

The Schrödinger operator , of a compact Riemannian manifold , has pure point spectrum. Suppose that is a smooth reference potential. Various criteria are given which guarantee the compactness of all satisfying . In particular, compactness is proved assuming an a priori bound on the norm of , where n/2-2$"> and . This improves earlier work of Brüning. An example involving singular potentials suggests that the condition n/2-2$"> is appropriate. Compactness is also proved for non-negative isospectral potentials in dimensions .