Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

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.

     

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 .

     

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 .

     

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 .

     

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

     

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 .

     

Weighted estimates in for Laplace's equation on Lipschitz domains

Let , , be a bounded Lipschitz domain. For Laplace's equation in , we study the Dirichlet and Neumann problems with boundary data in the weighted space , where , is a fixed point on , and denotes the surface measure on . We prove that there exists such that the Dirichlet problem is uniquely solvable if , and the Neumann problem is uniquely solvable if . If is a domain, one may take . The regularity for the Dirichlet problem with data in the weighted Sobolev space is also considered. Finally we establish the weighted estimates with general weights for the Dirichlet and regularity problems.

     

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.

     

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 .

     

Simple corona -algebras

Let be a non-unital and -unital simple -algebra. We show that if is simple, then is purely infinite. We also show that is simple if and only if has a continuous scale provided that is not isomorphic to the compact operators.

     

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.

     

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.

     

A join theorem for the computably enumerable degrees

It is shown that for any computably enumerable (c.e.) degree , if , then there is a c.e. degree such that (so is lowand is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low c.e. degrees are not elementarily equivalent as partial orderings.

     

Maps between non-commutative spaces

Let be a graded ideal in a not necessarily commutative graded -algebra in which for all . We show that the map induces a closed immersion between the non-commutative projective spaces with homogeneous coordinate rings and . We also examine two other kinds of maps between non-commutative spaces. First, a homomorphism between not necessarily commutative -graded rings induces an affine map from a non-empty open subspace . Second, if is a right noetherian connected graded algebra (not necessarily generated in degree one), and is a Veronese subalgebra of , there is a map ; we identify open subspaces on which this map is an isomorphism. Applying these general results when is (a quotient of) a weighted polynomial ring produces a non-commutative resolution of (a closed subscheme of) a weighted projective space.

     

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.

     

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.

     

Dual Radon transforms on affine Grassmann manifolds

Fix , and let and denote the affine Grassmann manifolds of - and -planes in . We investigate the Radon transform associated with the inclusion incidence relation. For the generic case and n$">, we will show that the range of this transform is given by smooth functions on annihilated by a system of Pfaffian type differential operators. We also study aspects of the exceptional case .