Translation equivalence in free groups

Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when two elements in a free group have the property that for every free isometric action of on an -tree the translation lengths of and on are equal.

     

Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

In this paper we study the space of effective -cycles in with the homology class equal to an integral multiple of the homology class of Schubert variety of type . When is a proper linear subspace of a linear space in , we know that is already complicated. We will show that for a smooth Schubert variety in a Hermitian symmetric space, any irreducible subvariety with the homology class , , is again a Schubert variety of type , unless is a non-maximal linear space. In particular, any local deformation of such a smooth Schubert variety in Hermitian symmetric space is obtained by the action of the Lie group .

     

Rotation topological factors of minimal -actions on the Cantor set

In this paper we study conditions under which a free minimal -action on the Cantor set is a topological extension of the action of rotations, either on the product of -tori or on a single -torus . We extend the notion of linearly recurrent systems defined for -actions on the Cantor set to -actions, and we derive in this more general setting a necessary and sufficient condition, which involves a natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one of these two types.

     

Values of Gauss maps of complete minimal surfaces in on annular ends

Let be a complete minimal surface in and let be an annular end of which is conformal to , where is the conformal coordinate. Let be the generalized Gauss map of . We show that must intersect every hyperplane in , with the possible exception of hyperplanes in general position.

     

The Cech filtration and monodromy in log crystalline cohomology

For a strictly semistable log scheme over a perfect field of characteristic we investigate the canonical Cech spectral sequence abutting the Hyodo-Kato (log crystalline) cohomology of and beginning with the log convergent cohomology of its various component intersections . We compare the filtration on arising from with the monodromy operator on . We also express through residue maps and study relations with singular cohomology. If lifts to a proper strictly semistable (formal) scheme over a finite totally ramified extension of , with generic fibre , we obtain results on how the simplicial structure of (as analytic space) is reflected in .

     

-continuity properties of the symmetric -stable process

Let be a domain of finite Lebesgue measure in and let be the symmetric -stable process killed upon exiting . Each element of the set of eigenvalues associated to , regarded as a function of , is right continuous. In addition, if is Lipschitz and bounded, then each is continuous in and the set of associated eigenfunctions is precompact.

     

Postnikov pieces and -homotopy theory

We present a constructive method to compute the cellularization with respect to for any integer of a large class of -spaces, namely all those which have a finite number of non-trivial -homotopy groups (the pointed mapping space is a Postnikov piece). We prove in particular that the -cellularization of an -space having a finite number of -homotopy groups is a -torsion Postnikov piece. Along the way, we characterize the -cellular classifying spaces of nilpotent groups.

     

Composition operators on uniform algebras, essential norms, and hyperbolically bounded sets

Let be a uniform algebra, and let be a self-map of the spectrum of that induces a composition operator on . The object of this paper is to relate the notion of ``hyperbolic boundedness' introduced by the authors in 2004 to the essential spectrum of . It is shown that the essential spectral radius of is strictly less than if and only if the image of under some iterate of is hyperbolically bounded. The set of composition operators is partitioned into ``hyperbolic vicinities" that are clopen with respect to the essential operator norm. This partition is related to the analogous partition with respect to the uniform operator norm.

     

A homotopy principle for maps with prescribed Thom-Boardman singularities

Let and be smooth manifolds of dimensions and ( ) respectively. Let denote an open subspace of which consists of all Boardman submanifolds of symbols with . An -regular map refers to a smooth map such that . We will prove what is called the homotopy principle for -regular maps on the existence level. Namely, a continuous section of over has an -regular map such that and are homotopic as sections.

     

Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles

Given a bounded domain in with smooth boundary, the cut locus is the closure of the set of nondifferentiability points of the distance from the boundary of . The normal distance to the cut locus, , is the map which measures the length of the line segment joining to the cut locus along the normal direction , whenever . Recent results show that this map, restricted to boundary points, is Lipschitz continuous, as long as the boundary of is of class . Our main result is the global Hölder regularity of in the case of a domain with analytic boundary. We will also show that the regularity obtained is optimal, as soon as the set of the so-called regular conjugate points is nonempty. In all the other cases, Lipschitz continuity can be extended to the whole domain . The above regularity result for is also applied to derive the Hölder continuity of the solution of a system of partial differential equations that arises in granular matter theory and optimal mass transfer.

     

Deformed preprojective algebras of generalized Dynkin type

We introduce the class of deformed preprojective algebras of generalized Dynkin graphs (), (), , , and () and prove that it coincides with the class of all basic connected finite-dimensional self-injective algebras for which the inverse Nakayama shift of every non-projective simple module is isomorphic to its third syzygy .

     

Equivalence of domains arising from duality of orbits on flag manifolds III

In Gindikin and Matsuki 2003, we defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of nonholomorphic type. This conjecture was proved for closed in Wolf and Zierau 2000 and 2003, Fels and Huckleberry 2005, and Matsuki 2006 and for open in Matsuki 2006. It was proved for the other orbits in Matsuki 2006, when is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed -orbit when is of Hermitian type. Thus the conjecture is completely solved affirmatively.

     

Quadratic maps and Bockstein closed group extensions

Let be a central extension of the form where and are elementary abelian -groups. Associated to there is a quadratic map , given by the -power map, which uniquely determines the extension. This quadratic map also determines the extension class of the extension in and an ideal in which is generated by the components of . We say that is Bockstein closed if is an ideal closed under the Bockstein operator.

We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map given by yield Bockstein closed extensions.

On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension for some -lattice . In this situation, one may write for a ``binding matrix' with entries in . We find a direct way to calculate the module structure of in terms of . Using this, we study extensions where the lattice is diagonalizable/triangulable and find interesting equivalent conditions to these properties.

     

Generalized Ahlfors functions

Let be a bordered Riemann surface with genus and boundary components. Let be a smooth family of smooth Jordan curves in which all contain the point 0 in their interior. Let and let be the family of all bounded holomorphic functions on such that and for almost every . Then there exists a smooth up to the boundary holomorphic function with at most zeros on so that for every and such that for every . If, in addition, all the curves are strictly convex, then is unique among all the functions from the family .

     

Extension and separation properties of positive definite functions on locally compact groups

Continuing earlier work, we investigate two related aspects of the set of continuous positive definite functions on a locally compact group . The first one is the problem of when, for a closed subgroup of , every function in extends to some function in . The second one is the question whether elements in can be separated from by functions in which are identically one on .

     

The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends

Given an integer , let be the space of complete embedded singly periodic minimal surfaces in , which in the quotient have genus zero and Scherk-type ends. Surfaces in can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that consists of the -parameter family of singly periodic Scherk minimal surfaces. We prove that for each , there exists a natural one-to-one correspondence between and the space of convex unitary nonspecial polygons through the map which assigns to each the polygon whose edges are the flux vectors at the ends of (a special polygon is a parallelogram with two sides of length and two sides of length ). As consequence, reduces to the saddle towers constructed by Karcher.

     

Lusternik-Schnirelmann category of

First we give an upper bound of , the L-S category of a principal -bundle for a connected compact group with a characteristic map . Assume that there is a cone-decomposition of in the sense of Ganea that is compatible with multiplication. Then we have for , if is compressible into with trivial higher Hopf invariant . Second, we introduce a new computable lower bound, for . The two new estimates imply , where is a category weight due to Rudyak and Strom.

     

Scrollar syzygies of general canonical curves with genus

We prove that for a general canonical curve of genus , the space of th (last) scrollar syzygies is isomorphic to the Brill-Noether locus . Schreyer has conjectured that these scrollar syzygies span the space of all th (last) syzygies of . Using Mukai varieties we prove this conjecture for genus , and .

     

On generalizations of Lavrentieff's theorem for Polish group actions

It is shown that for every Polish group that is not locally compact there is a continuous action of on a -complete subset of a Polish space such that cannot be extended to any superset of in . This answers a question posed by Becker and Kechris and shows that an earlier theorem of them is optimal in several aspects.

     

Curves of genus 2 with group of automorphisms isomorphic to or

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.