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.

     

Finiteness theorems for positive definite -regular quadratic forms

An integral quadratic form of variables is said to be -regular if globally represents all quadratic forms of variables that are represented by the genus of . For any , it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of variables that are -regular. We also investigate similar finiteness results for almost -regular and spinor -regular quadratic forms. It is shown that for any , there are only finitely many equivalence classes of primitive positive definite spinor or almost -regular quadratic forms of variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).

     

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.

     

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.

     

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.

     

Fixed points of commuting holomorphic mappings other than the Wolff point

Let be the unit disc of and let be such that . For 1$">, let . We study the behavior of on . In particular, we prove that . As a consequence, besides conditions for , we prove a conjecture of C. Cowen in case and are univalent mappings.

     

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.

     

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.

     

Regularity of weak solutions to the Monge-Ampère equation

We study the properties of generalized solutions to the Monge-Ampère equation , where the Borel measure satisfies a condition, introduced by Jerison, that is weaker than the doubling property. When , this condition, which we call , admits the possibility of vanishing or becoming infinite. Our analysis extends the regularity theory (due to Caffarelli) available when , which implies that is doubling. The main difference between the case and the case when is bounded between two positive constants is the need to use a variant of the Aleksandrov maximum principle (due to Jerison) and some tools from convex geometry, in particular the Hausdorff metric.

     

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.

     

The periodic Euler-Bernoulli equation

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

where the functions and are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by and . Here we develop a theory analogous to the theory of the Hill operator .

We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or -spectrum.

Our new analysis begins with a detailed study of the zeros of the function , for any given ``quasimomentum' , where is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to is , where is the discriminant and the period of ). We show that the multiplicity of any zero of can be one or two and (for some ) if and only if is also a zero of another entire function , independent of . Furthermore, we show that has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each -gap. If is a double zero of , it may happen that there is only one Floquet solution with quasimomentum ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.

Next we show that if is an open -gap of the pseudospectrum (i.e., ), then the Floquet matrix has a specific Jordan anomaly at and .

We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by the eigenvalues of this multipoint problem and show that is also characterized as the set of values of for which there is a proper Floquet solution such that .

We also show (Theorem 7) that each gap of the -spectrum contains exactly one and each -gap of the pseudospectrum contains exactly two 's, counting multiplicities. Here when we say ``gap' or ``-gap' we also include the endpoints (so that when two consecutive bands or -bands touch, the in-between collapsed gap, or -gap, is a point). We believe that can be used to formulate the associated inverse spectral problem.

As an application of Theorem 7, we show that if is a collapsed (``closed') -gap, then the Floquet matrix is diagonalizable.

Some of the above results were conjectured in our previous works. However, our conjecture that if all the -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.

     

On the inversion of the convolution and Laplace transform

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform of continuous or generalized functions . The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if is continuous, it is in if , and converges in an appropriate norm or Fréchet topology for generalized functions . As a corollary we obtain a new constructive inversion procedure for the convolution transform ; i.e., for given and we construct a sequence of continuous functions such that .

     

Induction theorems of surgery obstruction groups

Let be a finite group. It is well known that a Mackey functor is a module over the Burnside ring functor , where ranges over the set of all subgroups of . For a fixed homomorphism , the Wall group functor is not a Mackey functor if is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside ring functor as well as over the Grothendieck-Witt ring functor . In fact, we prove a more general result, that the functor assigning the equivariant surgery obstruction group on manifolds with middle-dimensional singular sets to each subgroup of is a module over the Burnside ring functor as well as over the special Grothendieck-Witt ring functor. As an application, we obtain a computable property of the functor described with an element in the Burnside ring.

     

Examples for the mod motivic cohomology of classifying spaces

Let be the classifying space of a compact Lie group . Some examples of computations of the motivic cohomology are given, by comparing with , and .

     

Construction of -structures and equivalences of derived categories

We associate a -structure to a family of objects in , the derived category of a Grothendieck category . Using general results on -structures, we give a new proof of Rickard's theorem on equivalence of bounded derived categories of modules. Also, we extend this result to bounded derived categories of quasi-coherent sheaves on separated divisorial schemes obtaining, in particular, Belinson's equivalences.

     

Twisted sums with spaces

If is a separable Banach space, we consider the existence of non-trivial twisted sums , where or For the case we show that there exists a twisted sum whose quotient map is strictly singular if and only if contains no copy of . If we prove an analogue of a theorem of Johnson and Zippin (for ) by showing that all such twisted sums are trivial if is the dual of a space with summable Szlenk index (e.g., could be Tsirelson's space); a converse is established under the assumption that has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with with strictly singular quotient map.

     

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

     

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.

     

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 .