A new variational characterization of -dimensional space forms

A Riemannian manifold is associated with a Schouten -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with characterized the three-dimensional space forms.

     

On the adjunction mapping of very ample vector bundles of corank one

Let be a very ample vector bundle of rank over a smooth complex projective variety of dimension . The structure of being known when , we investigate the structure of the adjunction mapping when .

     

On orbital partitions and exceptionality of primitive permutation groups

Let and be transitive permutation groups on a set such that is a normal subgroup of . The overgroup induces a natural action on the set of non-trivial orbitals of on . In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples where fixes no elements of ; such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples where is a partition of such that is transitive on ; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple in a TOD is exceptional; conversely if an exceptional triple is such that is cyclic of prime-power order, then there exists a partition of such that is a TOD. This paper characterizes TODs such that is primitive and is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs.

     

Asymptotic relations among Fourier coefficients of automorphic eigenfunctions

A detailed stationary phase analysis is presented for noncompact parameter ranges of the family of elementary eigenfunctions on the hyperbolic plane , , the eigenvalue, and the Macdonald-Bessel function. The phase velocity of on is a double-valued vector field, the tangent field to the pencil of geodesics tangent to the horocycle . For a multi-term stationary phase expansion is presented in for uniform in parameters. An application is made to find an asymptotic relation for the Fourier coefficients of a family of automorphic eigenfunctions. In particular, for automorphic with coefficients and eigenvalue it is shown for the special range that is for large, improving by an order of magnitude for this special range upon the estimate from the general Hecke bound . An exposition of the WKB asymptotics of the Macdonald-Bessel functions is presented.

     

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 .

     

On Diophantine definability and decidability in some infinite totally real extensions of

Let be a number field, and a set of its non-Archimedean primes. Then let . Let be a finite set of prime numbers. Let be the field generated by all the -th roots of unity as and . Let be the largest totally real subfield of . Then for any 0$">, there exist a number field , and a set of non-Archimedean primes of such that has density greater than , and has a Diophantine definition over the integral closure of in .

     

Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc

Let be a holomorphic self-map of the unit disc . For every , there is a measure on (sometimes called Aleksandrov measure) defined by the Poisson representation . Its singular part measures in a natural way the ``affinity' of for the boundary value . The affinity for values inside is provided by the Nevanlinna counting function of . We introduce a natural measure-valued refinement of and establish that the measures are obtained as boundary values of the refined Nevanlinna counting function . More precisely, we prove that is the weak limit of whenever converges to non-tangentially outside a small exceptional set . We obtain a sharp estimate for the size of in the sense of capacity.

     

Spinors as automorphisms of the tangent bundle

We show that, on a -manifold endowed with a -structure induced by an almost-complex structure, a self-dual (positive) spinor field is the same as a bundle morphism acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of on tangent vectors, and that the squaring map acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kähler and symplectic structures.

     

How to make a triangulation of polytopal

We introduce a numerical isomorphism invariant for any triangulation of . Although its definition is purely topological (inspired by the bridge number of knots), reflects the geometric properties of . Specifically, if is polytopal or shellable, then is ``small' in the sense that we obtain a linear upper bound for in the number of tetrahedra of . Conversely, if is ``small', then is ``almost' polytopal, since we show how to transform into a polytopal triangulation by local subdivisions. The minimal number of local subdivisions needed to transform into a polytopal triangulation is at least . Using our previous results [The size of triangulations supporting a given link, Geometry & Topology 5 (2001), 369-398], we obtain a general upper bound for exponential in . We prove here by explicit constructions that there is no general subexponential upper bound for in . Thus, we obtain triangulations that are ``very far' from being polytopal. Our results yield a recognition algorithm for that is conceptually simpler, although somewhat slower, than the famous Rubinstein-Thompson algorithm.

     

Completely isometric representations of

Let be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier algebra , which is dual to the representation of the measure algebra , on . The image algebras of and in are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group , there is a natural completely isometric representation of on , which can be regarded as a duality result of Neufang's completely isometric representation theorem for .

     

The flat model structure on

Given a cotorsion pair in an abelian category with enough objects and enough objects, we define two cotorsion pairs in the category of unbounded chain complexes. We see that these two cotorsion pairs are related in a nice way when is hereditary. We then show that both of these induced cotorsion pairs are complete when is the ``flat' cotorsion pair of -modules. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new ``flat' model category structure on . In the last section we use the theory of model categories to show that we can define using a flat resolution of and a cotorsion coresolution of .

     

The -theory of is undecidable

The three quantifier theory of , the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of that lies between the two and three quantifier theories with but includes function symbols.

Theorem. The two quantifier theory of , the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on ) is undecidable.

The same result holds for various lattices of ideals of which are natural extensions of preserving join and infimum when it exits.

     

Complete hyperelliptic integrals of the first kind and their non-oscillation

Let be a real polynomial of degree , and be an oval contained in the level set . We study complete Abelian integrals of the form

where are real and is a maximal open interval on which a continuous family of ovals exists. We show that the -dimensional real vector space of these integrals is not Chebyshev in general: for any 1$">, there are hyperelliptic Hamiltonians and continuous families of ovals , , such that the Abelian integral can have at least zeros in . Our main result is Theorem 1 in which we show that when , exceptional families of ovals exist, such that the corresponding vector space is still Chebyshev.

     

Koszul homology and extremal properties of Gin and Lex

For every homogeneous ideal in a polynomial ring and for every we consider the Koszul homology with respect to a sequence of of generic linear forms. The Koszul-Betti number is, by definition, the dimension of the degree part of . In characteristic , we show that the Koszul-Betti numbers of any ideal are bounded above by those of the gin-revlex of and also by those of the Lex-segment of . We show that iff is componentwise linear and that and iff is Gotzmann. We also investigate the set of all the gin of and show that the Koszul-Betti numbers of any ideal in are bounded below by those of the gin-revlex of . On the other hand, we present examples showing that in general there is no is such that the Koszul-Betti numbers of any ideal in are bounded above by those of .

     

Chern numbers of ample vector bundles on toric surfaces

This article shows a number of strong inequalities that hold for the Chern numbers , of any ample vector bundle of rank on a smooth toric projective surface, , whose topological Euler characteristic is . One general lower bound for proven in this article has leading term . Using Bogomolov instability, strong lower bounds for are also given. Using the new inequalities, the exceptions to the lower bounds 4e(S)$"> and e(S)$"> are classified.

     

Fundamental solutions for non-divergence form operators on stratified groups

We construct the fundamental solutions and for the non-divergence form operators and , where the 's are Hörmander vector fields generating a stratified group and is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of allow us to construct using both -saturation and approximation arguments.

     

The periodic isoperimetric problem

Given a discrete group of isometries of , we study the -isoperimetric problem, which consists of minimizing area (modulo ) among surfaces in which enclose a -invariant region with a prescribed volume fraction. If is a line group, we prove that solutions are either families of round spheres or right cylinders. In the doubly periodic case we prove that for most rank two lattices, solutions must be spheres, cylinders or planes. For the remaining rank two lattices we show, among other results, an isoperimetric inequality in terms of the topology of the isoperimetric surfaces. Finally, we study the case where (the group of symmetries of the integer rank three lattice ) and other crystallographic groups of cubic type. We prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than , and we give an isoperimetric inequality for -invariant regions that, for instance, implies that the area (modulo ) of a surface dividing the three space in two -invariant regions with equal volume fractions, is at least (the conjectured solution is the classical Schwarz triply periodic minimal surface whose area is ). Another consequence of this isoperimetric inequality is that -symmetric surfaces (other than families of spheres) cannot be isoperimetric for the lattice group .

     

On the representation of integers as linear combinations of consecutive values of a polynomial

Let be a cyclotomic field with ring of integers and let be a polynomial whose values on belong to . If the ideal of generated by the values of on is itself, then every algebraic integer of may be written in the following form:

for some integer , where the 's are roots of unity of . Moreover, there are two effective constants and such that the least integer (for a fixed ) is less than , where

     

A Capelli Harish-Chandra homomorphism

For a real reductive dual pair the Capelli identities define a homomorphism from the center of the universal enveloping algebra of the larger group to the center of the universal enveloping algebra of the smaller group. In terms of the Harish-Chandra isomorphism, this map involves a -shift. We view a dual pair as a Lie supergroup and offer a construction of the homomorphism based solely on the Harish-Chandra's radial component maps. Thus we provide a geometric interpretation of the -shift.