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.

     

Lattice invariants and the center of the generic division ring

Let be a finite group, let be a -lattice, and let be a field of characteristic zero containing primitive roots of 1. Let be the quotient field of the group algebra of the abelian group . It is well known that if is quasi-permutation and -faithful, then is stably equivalent to . Let be the center of the division ring of generic matrices over . Let be the symmetric group on symbols. Let be a prime. We show that there exist a split group extension of by a -elementary group, a -faithful quasi-permutation -lattice , and a one-cocycle in such that is stably isomorphic to . This represents a reduction of the problem since we have a quasi-permutation action; however, the twist introduces a new level of complexity. The second result, which is a consequence of the first, is that, if is algebraically closed, there is a group extension of by an abelian -group such that is stably equivalent to the invariants of the Noether setting .

     

Morse index and uniqueness for positive solutions of radial -Laplace equations

We study the positive radial solutions of the Dirichlet problem in , 0$"> in , on , where , 1$">, is the -Laplace operator, is the unit ball in centered at the origin and is a function. We are able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive from this information on the Morse index. In particular, we show that positive radial solutions of Mountain Pass type have Morse index 1 in the subspace of radial functions of . We use this to prove uniqueness and nondegeneracy of positive radial solutions when is of the type and .

     

regularity of Fourier integral operators with caustics

The caustics of Fourier integral operators are defined as caustics of the corresponding Schwartz kernels (Lagrangian distributions on ). The caustic set of the canonical relation is characterized as the set of points where the rank of the projection is smaller than its maximal value, . We derive the estimates on Fourier integral operators with caustics of corank (such as caustics of type , ). For the values of and outside of a certain neighborhood of the line of duality, , the estimates are proved to be caustics-insensitive.

We apply our results to the analysis of the blow-up of the estimates on the half-wave operator just before the geodesic flow forms caustics.

     

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.

     

Radon transforms on affine Grassmannians

We develop an analytic approach to the Radon transform , where is a function on the affine Grassmann manifold of -dimensional planes in , and is a -dimensional plane in the similar manifold k$">. For , we prove that this transform is finite almost everywhere on if and only if , and obtain explicit inversion formulas. We establish correspondence between Radon transforms on affine Grassmann manifolds and similar transforms on standard Grassmann manifolds of linear subspaces of . It is proved that the dual Radon transform can be explicitly inverted for , and interpreted as a direct, ``quasi-orthogonal" Radon transform for another pair of affine Grassmannians. As a consequence we obtain that the Radon transform and the dual Radon transform are injective simultaneously if and only if . The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.

     

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 .

     

Spectral properties and dynamics of quantized Henon maps

We study a generalization of the Airy function, and use its properties to investigate the dynamics and spectral properties of the unitary operators on of the form , where is a real polynomial of odd degree, is a real number, and is the Fourier transform. We show that is a quantization of the classical Henon map

and show that for 0$"> sufficiently large, has purely continuous spectrum. This fact has implications for the dynamics of , which are shown to correspond when the condition is satisfied to the dynamics of its classical counterpart on .

     

Ideals in a perfect closure, linear growth of primary decompositions, and tight closure

This paper is concerned with tight closure in a commutative Noetherian ring of prime characteristic , and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal of has linear growth of primary decompositions, then tight closure (of ) `commutes with localization at the powers of a single element'. It is shown in this paper that, provided has a weak test element, linear growth of primary decompositions for other sequences of ideals of that approximate, in a certain sense, the sequence of Frobenius powers of would not only be just as good in this context, but, in the presence of a certain additional finiteness property, would actually imply that tight closure (of ) commutes with localization at an arbitrary multiplicatively closed subset of .

Work of M. Katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the Frobenius powers of has only finitely many maximal members. This paper develops, through a careful analysis of the ideal theory of the perfect closure of , strategies for showing that tight closure (of a specified ideal of ) commutes with localization at an arbitrary multiplicatively closed subset of and for showing that the union of the associated primes of the tight closures of the Frobenius powers of is actually a finite set. Several applications of the strategies are presented; in most of them it was already known that tight closure commutes with localization, but the resulting affirmative answers to Katzman's question in the various situations considered are believed to be new.

     

Value groups, residue fields, and bad places of rational function fields

We classify all possible extensions of a valuation from a ground field to a rational function field in one or several variables over . We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field in one variable, we consider the relative algebraic closure of in the henselization of with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of . In the ``tame case', we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the -adics.

     

A bracket power characterization of analytic spread one ideals

The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let be regular nonunits in a local (Noetherian) ring and assume that , the integral closure of , where . Then the main result shows that for all but finitely many units in that are non-congruent modulo and for all large integers and it holds that for and not divisible by , where is the -th bracket power of . And, conversely, if there exist positive integers , , and such that has a basis such that , then has analytic spread one.

     

Poincaré's closed geodesic on a convex surface

We present a new proof for the existence of a simple closed geodesic on a convex surface . This result is due originally to Poincaré. The proof uses the -dimensional Riemannian manifold of piecewise geodesic closed curves on with a fixed number of corners, chosen sufficiently large. In we consider a submanifold formed by those elements of which are simple regular and divide into two parts of equal total curvature . The main burden of the proof is to show that the energy integral , restricted to , assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on .

     

C*-algebras associated with interval maps

For each piecewise monotonic map of , we associate a pair of C*-algebras and and calculate their K-groups. The algebra is an AI-algebra. We characterize when and are simple. In those cases, has a unique trace, and is purely infinite with a unique KMS state. In the case that is Markov, these algebras include the Cuntz-Krieger algebras , and the associated AF-algebras . Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and -transformations. For the case of interval exchange maps and of -transformations, the C*-algebra coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.

     

Flat holomorphic connections on principal bundles over a projective manifold

Let be a connected complex linear algebraic group and its unipotent radical. A principal -bundle over a projective manifold will be called polystable if the associated principal -bundle is so. A -bundle over is polystable with vanishing characteristic classes of degrees one and two if and only if admits a flat holomorphic connection with the property that the image in of the monodromy of the connection is contained in a maximal compact subgroup of .

     

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

     

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.

     

On peak-interpolation manifolds for

Let be a bounded, weakly convex domain in , , having real-analytic boundary. is the algebra of all functions holomorphic in and continuous up to the boundary. A submanifold is said to be complex-tangential if lies in the maximal complex subspace of for each . We show that for real-analytic submanifolds , if is complex-tangential, then every compact subset of is a peak-interpolation set for .

     

Oppenheim conjecture for pairs consisting of a linear form and a quadratic form

Let be a nondegenerate quadratic form and a nonzero linear form of dimension 3$">. As a generalization of the Oppenheim conjecture, we prove that the set is dense in provided that and satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.

     

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 .