On homeomorphic Bernoulli measures on the Cantor space

Let be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights and . It is a long-standing open problem to characterize those and such that and are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending to ). The (possibly) weaker property of and being continuously reducible to each other is equivalent to a property of and called binomial equivalence. In this paper we define an algebraic property called ``refinability' and show that, if and are refinable and binomially equivalent, then and are topologically equivalent. Next we show that refinability is equivalent to a fairly simple algebraic property. Finally, we give a class of examples of binomially equivalent and refinable numbers; in particular, the positive numbers and such that and are refinable, so the corresponding measures are topologically equivalent.

     

Surjectivity for Hamiltonian -spaces in -theory

Let be a compact connected Lie group, and a Hamiltonian -space with proper moment map . We give a surjectivity result which expresses the -theory of the symplectic quotient in terms of the equivariant -theory of the original manifold , under certain technical conditions on . This result is a natural -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian -spaces. We discuss this lemma in detail and highlight the differences between the -theory and rational cohomology versions of this lemma.

We also introduce a -theoretic version of equivariant formality and prove that when the fundamental group of is torsion-free, every compact Hamiltonian -space is equivariantly formal. Under these conditions, the forgetful map is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in admits an equivariant extension in .

     

The -stable pieces of the wonderful compactification

Let be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification of into finite many -stable pieces, which was introduced by Lusztig. In this paper, we will investigate the closure of any -stable piece in . We will show that the closure is a disjoint union of some -stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many -orbits.

     

The distance function from the boundary in a Minkowski space

Let the space be endowed with a Minkowski structure (that is, is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class ), and let be the (asymmetric) distance associated to . Given an open domain of class , let be the Minkowski distance of a point from the boundary of . We prove that a suitable extension of to (which plays the rôle of a signed Minkowski distance to ) is of class in a tubular neighborhood of , and that is of class outside the cut locus of (that is, the closure of the set of points of nondifferentiability of in ). In addition, we prove that the cut locus of has Lebesgue measure zero, and that can be decomposed, up to this set of vanishing measure, into geodesics starting from and going into along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point outside the cut locus the pair , where denotes the (unique) projection of on , and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

     

Vertex operator algebras, extended diagram, and McKay's observation on the Monster simple group

We study McKay's observation on the Monster simple group, which relates the -involutions of the Monster simple group to the extended diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices of the lattice obtained by removing one node from the extended diagram at each time. We then construct a certain coset (or commutant) subalgebra associated with in the lattice VOA . There are two natural conformal vectors of central charge in such that their inner product is exactly the value predicted by Conway (1985). The Griess algebra of coincides with the algebra described in his Table 3. There is a canonical automorphism of of order . Such an automorphism can be extended to the Leech lattice VOA , and it is in fact a product of two Miyamoto involutions. In the sequel (2005) to this article, the properties of will be discussed in detail. It is expected that if is actually contained in the Moonshine VOA , the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.

     

Projectivity and freeness over comodule algebras

Let be a Hopf algebra and an -simple right -comodule algebra. It is shown that under certain hypotheses every -Hopf module is either projective or free as an -module and is either a quasi-Frobenius or a semisimple ring. As an application it is proved that every weakly finite (in particular, every finite dimensional) Hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras, and the latter are Frobenius algebras. Similar results are obtained for -simple -module algebras.

     

Torsion on elliptic curves in isogeny classes

Let be an elliptic curve over a number field and its -isogeny class. We are interested in determining the orders and the types of torsion groups in . For a prime , we give the range of possible types of -primary parts of when runs over . One of our results immediately gives a simple proof of a theorem of Katz on the order of maximal -primary torsion in .

     

Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles

Given a bounded domain in with smooth boundary, the cut locus is the closure of the set of nondifferentiability points of the distance from the boundary of . The normal distance to the cut locus, , is the map which measures the length of the line segment joining to the cut locus along the normal direction , whenever . Recent results show that this map, restricted to boundary points, is Lipschitz continuous, as long as the boundary of is of class . Our main result is the global Hölder regularity of in the case of a domain with analytic boundary. We will also show that the regularity obtained is optimal, as soon as the set of the so-called regular conjugate points is nonempty. In all the other cases, Lipschitz continuity can be extended to the whole domain . The above regularity result for is also applied to derive the Hölder continuity of the solution of a system of partial differential equations that arises in granular matter theory and optimal mass transfer.

     

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.

     

Analyticity on translates of a Jordan curve

Let be a domain in which is symmetric with respect to the real axis and whose boundary is a real analytic simple closed curve. Translate vertically to get where is such that . We prove that if is a continuous function on such that for each , the function has a continuous extension to which is holomorphic on , then is holomorphic on .

     

Torsion freeness of symmetric powers of ideals

Let be an ideal in a Noetherian commutative ring with unit, let be an integer, and let be the canonical surjective -module homomorphism from the th symmetric power of to the th power of . When or when is a perfect Gorenstein ideal of grade , we provide a necessary and sufficient condition for to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of . When is a maximal ideal of we show that is an isomorphism if and only if is a regular local ring. In all three cases for our results yield that if is an isomorphism, then is also an isomorphism for each .

     

Iwasawa theory for -local spectra

The Iwasawa algebra is a power series ring in one variable over the -adic integers. It has long been studied by number theorists in the context of -extensions of number fields. It also arises, however, as a ring of operations in -adic topological -theory. In this paper we study -local stable homotopy theory using the structure theory of modules over the Iwasawa algebra. In particular, for odd we classify -local spectra up to pseudo-equivalence (the analogue of pseudo-isomorphism for -modules) and give an Iwasawa-theoretic classification of the thick subcategories of the weakly dualizable spectra.

     

Curves of genus 2 with group of automorphisms isomorphic to or

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.

     

The odd primary -structure of low rank Lie groups and its application to exponents

A compact, connected, simple Lie group localized at an odd prime is shown to be homotopy equivalent to a product of homotopy associative, homotopy commutative spaces, provided the rank of is low. This holds for , for example, if . The homotopy equivalence is usually just as spaces, not multiplicative spaces. Nevertheless, the strong multiplicative features of the factors can be used to prove useful properties, which after looping can be transferred multiplicatively to . This is applied to prove useful information about the torsion in the homotopy groups of , including an upper bound on its exponent.

     

Compatible valuations and generalized Milnor -theory

Given a field and a subgroup of there is a minimal group for which there exists an -compatible valuation whose units are contained in . Assuming that has finite index in and contains for prime, we describe in computable -theoretic terms.

     

The Cech filtration and monodromy in log crystalline cohomology

For a strictly semistable log scheme over a perfect field of characteristic we investigate the canonical Cech spectral sequence abutting the Hyodo-Kato (log crystalline) cohomology of and beginning with the log convergent cohomology of its various component intersections . We compare the filtration on arising from with the monodromy operator on . We also express through residue maps and study relations with singular cohomology. If lifts to a proper strictly semistable (formal) scheme over a finite totally ramified extension of , with generic fibre , we obtain results on how the simplicial structure of (as analytic space) is reflected in .

     

Equivalence of domains arising from duality of orbits on flag manifolds III

In Gindikin and Matsuki 2003, we defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of nonholomorphic type. This conjecture was proved for closed in Wolf and Zierau 2000 and 2003, Fels and Huckleberry 2005, and Matsuki 2006 and for open in Matsuki 2006. It was proved for the other orbits in Matsuki 2006, when is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed -orbit when is of Hermitian type. Thus the conjecture is completely solved affirmatively.

     

Disklikeness of planar self-affine tiles

We consider the disklikeness of the planar self-affine tile generated by an integral expanding matrix and a consecutive collinear digit set . Let be the characteristic polynomial of . We show that the tile is disklike if and only if . Moreover, is a hexagonal tile for all the cases except when , in which case is a square tile. The proof depends on certain special devices to count the numbers of nodal points and neighbors of and a criterion of Bandt and Wang (2001) on disklikeness.