Connections with prescribed first Pontrjagin form

Let be a principal bundle over a manifold of dimension . If , then we prove that every differential 4-form representing the first Pontrjagin class of is the Pontrjagin form of some connection on .     

Zero entropy, non-integrable geodesic flows and a non-commutative rotation vector

Let be a -step nilpotent Lie algebra; we say is non-integrable if, for a generic pair of points , the isotropy algebras do not commute: . Theorem: If is a simply-connected -step nilpotent Lie group, is non-integrable, is a cocompact subgroup, and is a left-invariant Riemannian metric, then the geodesic flow of on is neither Liouville nor non-commutatively integrable with first integrals. The proof uses a generalization of the rotation vector pioneered by Benardete and Mitchell.

     

Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let be a sequence of orthonormal polynomials and the restriction of to , where is the maximum zero of . Then and the composite are operator monotone on . Furthermore, for every polynomial with a positive leading coefficient there is a real number so that the inverse function of defined on is semi-operator monotone, that is, for matrices , implies

     

Anderson's double complex and gamma monomials for rational function fields

We investigate algebraic -monomials of Thakur's positive characteristic -function, by using Anderson and Das' double complex method of computing the sign cohomology of the universal ordinary distribution. We prove that the -monomial associated to an element of the second sign cohomology of the universal ordinary distribution of generates a Kummer extension of some Carlitz cyclotomic function field, which is also a Galois extension of the base field . These results are characteristic- analogues of those of Deligne on classical -monomials, proofs of which were given by Das using the double complex method. In this paper, we also obtain some results on -monomials of Carlitz's exponential function.

     

On the Iwasawa -invariants of real abelian fields

For a prime number and a number field , let denote the projective limit of the -parts of the ideal class groups of the intermediate fields of the cyclotomic -extension over . It is conjectured that is finite if is totally real. When is an odd prime and is a real abelian field, we give a criterion for the conjecture, which is a generalization of results of Ichimura and Sumida. Furthermore, in a special case where divides the degree of , we also obtain a rather simple criterion.

     

Contractive projections and operator spaces

Parallel to the study of finite-dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces , , generalizing the row and column Hilbert spaces and , and we show that an atomic subspace that is the range of a contractive projection on is isometrically completely contractive to an -sum of the and Cartan factors of types 1 to 4. In particular, for finite-dimensional , this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w-closed -triples without an infinite-dimensional rank 1 w-closed ideal.

     

A path-transformation for random walks and the Robinson-Schensted correspondence

The author and Marc Yor recently introduced a path-transformation with the property that, for belonging to a certain class of random walks on , the transformed walk has the same law as the original walk conditioned never to exit the Weyl chamber . In this paper, we show that is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of and . The corresponding results for the Brownian model are recovered by Donsker's theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformation and the Robinson-Schensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the Robinson-Schensted algorithm and, moreover, extends easily to a continuous setting.

     

On the Clifford algebra of a binary form

The Clifford algebra of a binary form of degree is the -algebra , where is the ideal generated by . has a natural homomorphic image that is a rank Azumaya algebra over its center. We prove that the center is isomorphic to the coordinate ring of the complement of an explicit -divisor in , where is the curve and is the genus of .

     

A classification and examples of rank one chain domains

A chain order of a skew field is a subring of so that implies Such a ring has rank one if , the Jacobson radical of is its only nonzero completely prime ideal. We show that a rank one chain order of is either invariant, in which case corresponds to a real-valued valuation of or is nearly simple, in which case and are the only ideals of or is exceptional in which case contains a prime ideal that is not completely prime. We use the group of divisorial of with the subgroup of principal to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index of in Using the covering group of and the result that the group ring is embeddable into a skew field for a skew field, examples of rank one chain orders are constructed for each possible exceptional case.

     

Poset block equivalence of integral matrices

Given square matrices and with a poset-indexed block structure (for which an block is zero unless ), when are there invertible matrices and with this required-zero-block structure such that ? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain . As one application, when is a field we classify such matrices up to similarity by matrices respecting the block structure. We also give complete invariants for equivalence under the additional requirement that the diagonal blocks of and have determinant . The invariants involve an associated diagram (the ``-web') of -module homomorphisms. The study is motivated by applications to symbolic dynamics and -algebras.

     

Self-intersection class for singularities and its application to fold maps

Let be a generic smooth map with corank one singularities between manifolds, and let be the singular point set of . We define the self-intersection class of using an incident class introduced by Rimányi but with twisted coefficients, and give a formula for in terms of characteristic classes of the manifolds. We then apply the formula to the existence problem of fold maps.

     

The free entropy dimension of hyperfinite von Neumann algebras

Suppose is a hyperfinite von Neumann algebra with a normal, tracial state and is a set of selfadjoint generators for . We calculate , the modified free entropy dimension of . Moreover, we show that depends only on and . Consequently, is independent of the choice of generators for . In the course of the argument we show that if is a set of selfadjoint generators for a von Neumann algebra with a normal, tracial state and has finite-dimensional approximants, then for any hyperfinite von Neumann subalgebra of Combined with a result by Voiculescu, this implies that if has a regular diffuse hyperfinite von Neumann subalgebra, then .

     

Explicit Lower bounds for residues at of Dedekind zeta functions and relative class numbers of CM-fields

Let be a given set of positive rational primes. Assume that the value of the Dedekind zeta function of a number field is less than or equal to zero at some real point in the range . We give explicit lower bounds on the residue at of this Dedekind zeta function which depend on , the absolute value of the discriminant of and the behavior in of the rational primes . Now, let be a real abelian number field and let be any real zero of the zeta function of . We give an upper bound on the residue at of which depends on , and the behavior in of the rational primes . By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields which depend on the behavior in of the rational primes . We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.

     

West's problem on equivariant hyperspaces and Banach-Mazur compacta

Let be a compact Lie group, a metric -space, and the hyperspace of all nonempty compact subsets of endowed with the Hausdorff metric topology and with the induced action of . We prove that the following three assertions are equivalent: (a) is locally continuum-connected (resp., connected and locally continuum-connected); (b) is a -ANR (resp., a -AR); (c) is an ANR (resp., an AR). This is applied to show that is an ANR (resp., an AR) for each compact (resp., connected) Lie group . If is a finite group, then is a Hilbert cube whenever is a nondegenerate Peano continuum. Let be the hyperspace of all centrally symmetric, compact, convex bodies , , for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing , and let be the complement of the unique -fixed point in . We prove that: (1) for each closed subgroup , is a Hilbert cube manifold; (2) for each closed subgroup acting non-transitively on , the -orbit space and the -fixed point set are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta and prove that and have the same -homotopy type.

     

Heat kernels on metric measure spaces and an application to semilinear elliptic equations

We consider a metric measure space and a heat kernel on satisfying certain upper and lower estimates, which depend on two parameters and . We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space . Namely, is the Hausdorff dimension of this space, whereas , called the walk dimension, is determined via the properties of the family of Besov spaces on . Moreover, the parameters and are related by the inequalities .

We prove also the embedding theorems for the space , and use them to obtain the existence results for weak solutions to semilinear elliptic equations on of the form

where is the generator of the semigroup associated with .

The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpinski carpet in .

     

Stratified transversality by isotopy

For an abstract stratified set or a -regular stratification, hence for any -, - or -regular stratification, we prove that after stratified isotopy of , a stratified subspace of , or a stratified map , can be made transverse to a fixed stratified map .

     

Spines and topology of thin Riemannian manifolds

Consider Riemannian manifolds for which the sectional curvature of and second fundamental form of the boundary are bounded above by one in absolute value. Previously we proved that if has sufficiently small inradius (i.e. all points are sufficiently close to the boundary), then the cut locus of exhibits canonical branching behavior of arbitrarily low branching number. In particular, if is thin in the sense that its inradius is less than a certain universal constant (known to lie between and ), then collapses to a triply branched simple polyhedral spine.

We use a graphical representation of the stratification structure of such a collapse, and relate numerical invariants of the graph to topological invariants of when is simply connected. In particular, the number of connected strata of the cut locus is a topological invariant. When is -dimensional and compact, has complexity in the sense of Matveev, and is a connected sum of copies of the real projective space , copies chosen from the lens spaces , and handles chosen from or , with 3-balls removed, where . Moreover, we construct a thin metric for every graph, and hence for every homeomorphism type on the list.

     

Hyperbolic mean growth of bounded holomorphic functions in the ball

We consider the hyperbolic Hardy class , . It consists of holomorphic in the unit complex ball for which and

where denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type -function and the area function are defined in terms of the invariant gradient of , and membership of is expressed by the property of the functions. As an application, we can characterize the boundedness and the compactness of the composition operator , defined by , from the Bloch space into the Hardy space .

     

The structure of equicontinuous maps

Let be a metric space, and be a continuous map. In this paper we prove that if is compact, and for all , then is equicontinuous if and only if there exist a pointwise recurrent isometric homeomorphism and a non-expanding map that is pointwise convergent to a fixed point such that is uniformly conjugate to a subsystem of the product map . In addition, we give some still simpler necessary and sufficient conditions of equicontinuous graph maps.

     

Stationary sets for the wave equation in crystallographic domains

Let be a crystallographic group in generated by reflections and let be the fundamental domain of We characterize stationary sets for the wave equation in when the initial data is supported in the interior of The stationary sets are the sets of time-invariant zeros of nontrivial solutions that are identically zero at .

We show that, for these initial data, the -dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group , This part comes from a corresponding odd symmetry of the initial data.

In physical language, the result is that if the initial source is localized strictly inside of the crystalline , then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one.