Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds

We translate the problem of finding Anosov diffeomorphisms on a nilmanifold which is covered by a free nilpotent Lie group into a problem of constructing matrices in whose eigenvalues satisfy certain conditions. Afterwards, we show how this translation can then be solved in some specific situations. The paper starts with a section on polynomial permutations of , a subject which is of interest on its own.

     

A Calabi-Yau threefold with Brauer group

We compute the Brauer group of a Calabi-Yau threefold discovered by the first author and Sorin Popescu, and find it is , the largest known Brauer group of a non-singular Calabi-Yau threefold.

     

A counterexample to the maximality of toric varieties

We present a counterexample to the conjecture of Bihan, Franz, McCrory, and van Hamel concerning the maximality of toric varieties. There exists a six dimensional projective toric variety with the sum of the Betti numbers of strictly less than the sum of the Betti numbers of .

     

Minkowski's conjecture, well-rounded lattices and topological dimension

Let be the diagonal subgroup, and identify with the space of unimodular lattices in . In this paper we show that the closure of any bounded orbit

meets the set of well-rounded lattices. This assertion implies Minkowski's conjecture for and yields bounds for the density of algebraic integers in totally real sextic fields.

The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of and .

     

Alternating forms and the Brauer group of a geometric field

We compute the theory of for any proabelian group , using a natural isomorphism with the group of continuous alternating forms. We use this to establish a sort of generic behavioral ideal, or role model, for the Brauer group Br of a geometric field of characteristic zero. We show this ideal is attained in several interesting cases.

     

Modular differential equations of second order with regular singularities at elliptic points for

We give a definition of the modular differential equations of weight for a discrete subgroup for ; in this paper we set . We solve such equations admitting regular singularities at elliptic points for in terms of the Eisenstein series and the Gauss hypergeometric series. Furthermore, we give a series of such modular differential equations parametrized by an even integer , and discuss some properties of solution spaces. We find several equations which are solved by a modular form of weight .

     

A categorification of integral Specht modules

We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories of category for the Lie algebra .

     

boundedness of discrete singular Radon transforms

We prove that if is a Calderón-Zygmund kernel and is a polynomial of degree with real coefficients, then the discrete singular Radon transform operator

extends to a bounded operator on , . This gives a positive answer to an earlier conjecture of E. M. Stein and S. Wainger.

     

Hilbert modules over a class of semicrossed products

Given the disk algebra and an automorphism , there is associated a non-self-adjoint operator algebra called the semicrossed product of with . Buske and Peters showed that there is a one-to-one correspondence between the contractive Hilbert modules over and pairs of contractions and on satisfying . In this paper, we show that the orthogonally projective and Shilov Hilbert modules over correspond to pairs of isometries on satisfying . The problem of commutant lifting for is left open, but some related results are presented.     

Cohomology operations for Lie algebras

If is a Lie algebra over and its centre, the natural inclusion extends to a representation of the exterior algebra of in the cohomology of . We begin a study of this representation by examining its Poincaré duality properties, its associated higher cohomology operations and its relevance to the toral rank conjecture. In particular, by using harmonic forms we show that the higher operations presented by Goresky, Kottwitz and MacPherson (1998) form a subalgebra of , and that they can be assembled to yield an explicit Hirsch-Brown model for the Borel construction associated to .

     

Testing polycyclicity of finitely generated rational matrix groups

We describe algorithms for testing polycyclicity and nilpotency for finitely generated subgroups of and thus we show that these properties are decidable. Variations of our algorithm can be used for testing virtual polycyclicity and virtual nilpotency for finitely generated subgroups of .

     

Spectra of subdivision operators

Let be a sequence of complex numbers and except for finitely many . The subdivision operator associated with is the bi-infinite matrix . This operator plays an important role in wavelet analysis and subdivision algorithms. As the adjoint it is closely related to the well-known transfer operators (also called Ruelle operator).

In this paper we show that for any , the spectrum of in is always a closed disc centered at the origin. Moreover, except for finitely many points, all the points in the open disc of the spectrum lie in the residual spectrum.

     

Complex Brjuno functions

The Brjuno function arises naturally in the study of analytic small divisors problems in one dimension. It belongs to and it is stable under Hölder perturbations. It is related to the size of Siegel disks by various rigorous and conjectural results.

In this work we show how to extend the Brjuno function to a holomorphic function on , the complex Brjuno function. This has an explicit expression in terms of a series of transformed dilogarithms under the action of the modular group. The extension is obtained using a complex analogue of the continued fraction expansion of a real number. Since our method is based on the use of hyperfunctions, it applies to less regular functions than the Brjuno function and it is quite general.

We prove that the harmonic conjugate of the Brjuno function is bounded. Its trace on is continuous at all irrational points and has a jump of at each rational point .

     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

     

On mod 3 Galois representations with conductor 4

The non-existence is proved of 2-dimensional mod 3 irreducible representations of of Artin conductor dividing 4.

     

On the genus of elliptic fibrations

A simply connected topological space is called elliptic if both and are finite-dimensional -vector spaces. In this paper, we consider fibrations for which the fibre is elliptic and is evenly graded. We show that in the generic cases, the genus of such a fibration is completely determined by generalized Chern classes of the fibration.

     

The s-elementary frame wavelets are path connected

An s-elementary frame wavelet is a function which is a frame wavelet and is defined by a Lebesgue measurable set such that . In this paper we prove that the family of s-elementary frame wavelets is a path-connected set in the -norm. This result also holds for s-elementary -dilation frame wavelets in in general. On the other hand, we prove that the path-connectedness of s-elementary frame wavelets cannot be strengthened to uniform path-connectedness. In fact, the sets of normalized tight frame wavelets and frame wavelets are not uniformly path-connected either.

     

planes in

We establish a homeomorphism between the moduli space of ordered -tuples of 2-dimensional linear subspaces (mod ) and the quotient by simultaneous conjugation of a certain open subset . For , this leads to an explicit computation of the moduli space of central 2-arrangements in mod and its subspace of those classes that contain a complex hyperplane arrangement.

     

Morita equivalence for quantum Heisenberg manifolds

We discuss Morita equivalence within the family 0, \mu,\nu\in\mathbb{R}\}$"> of quantum Heisenberg manifolds. Morita equivalence classes are described in terms of the parameters , and the rank of the free abelian group associated to the -algebra .

     

Average size of -Selmer groups of elliptic curves, I

In this paper, we study a class of elliptic curves over with -torsion group , and prove that the average order of the -Selmer groups is bounded.