Quantifier elimination for algebraic -groups

We prove that if is an algebraic -group (in the sense of Buium over a differentially closed field of characteristic , then the first order structure consisting of together with the algebraic -subvarieties of , has quantifier-elimination. In other words, the projection on of a -constructible subset of is -constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.

     

-manifolds with planar presentations and the width of satellite knots

We consider compact -manifolds having a submersion to in which each generic point inverse is a planar surface. The standard height function on a submanifold of is a motivating example. To we associate a connectivity graph . For , is a tree if and only if there is a Fox reimbedding of which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of is a tree, then there is a level-preserving reimbedding of so that is a connected sum of handlebodies.

Corollary.

The width of a satellite knot is no less than the width of its pattern knot and so

.

     

Polar sets on metric spaces

We show that if is a proper metric measure space equipped with a doubling measure supporting a Poincaré inequality, then subsets of with zero -capacity are precisely the -polar sets; that is, a relatively compact subset of a domain in is of zero -capacity if and only if there exists a -superharmonic function whose set of singularities contains the given set. In addition, we prove that if is a -hyperbolic metric space, then the -superharmonic function can be required to be -superharmonic on the entire space . We also study the the following question: If a set is of zero -capacity, does there exist a -superharmonic function whose set of singularities is precisely the given set?

     

Resolutions for metrizable compacta in extension theory

We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, , , for , and (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map such that:

(a) is -acyclic,

(b) , and

(c) .

This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension when . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes .

If in addition , then (a) can be replaced by the stronger statement,

(aa) is -acyclic.

To say that a map is -acyclic means that for each , every map of the fiber to is nullhomotopic.

     

Algebra of dimension theory

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory, this algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes of graded groups . There are two geometric interpretations of these equivalence classes: 1) For pointed CW complexes and , if and only if the infinite symmetric products and are of the same extension type (i.e., iff for all compact ). 2) For pointed compact spaces and , if and only if and are of the same dimension type (i.e., for all Abelian groups ).

Dranishnikov's version of the Hurewicz Theorem in extension theory becomes for all simply connected .

The concept of cohomological dimension of a pointed compact space with respect to a graded group is introduced. It turns out iff for all . If and are two positive graded groups, then if and only if for all compact .

     

On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo and generalizations

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic

where denotes the number of odd parts of the partition and is the conjugate of . In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition congruence mod :

where () denotes the number of partitions of with and is the number of unrestricted partitions of . Andrews asked for a partition statistic that would divide the partitions enumerated by () into five equinumerous classes.

In this paper we discuss three such statistics: the ST-crank, the -quotient-rank and the -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan's congruence mod . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo . Finally, we discuss some new formulas for partitions that are -cores and discuss an intriguing relation between -cores and the Andrews-Garvan crank.

     

Filtrations in semisimple Lie algebras, I

In this paper, we study the maximal bounded -filtrations of a complex semisimple Lie algebra . Specifically, we show that if is simple of classical type , , or , then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain partial, but not definitive, results in this direction for the remaining exceptional algebras. Maximal bounded filtrations were first introduced in the context of classifying the maximal graded subalgebras of affine Kac-Moody algebras, and the maximal graded subalgebras of loop toroidal Lie algebras. Indeed, our main results complete this classification in most cases. Finally, we briefly discuss the analogous question for bounded filtrations with respect to other Archimedean ordered groups.

     

On almost one-to-one maps

A continuous map of topological spaces is said to be almost -to- if the set of the points such that is dense in ; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and -compact spaces (e.g., -manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if is a minimal self-mapping of a 2-manifold , then point preimages under are tree-like continua and either is a union of 2-tori, or is a union of Klein bottles permuted by .

     

Stable geometric dimension of vector bundles over even-dimensional real projective spaces

In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles of order over if is even and sufficiently large and . In this paper, we use the Bendersky-Davis computation of to show that the 1981 result extends to all (still provided that is sufficiently large). If , the result is often different due to anomalies in the formula for when , but we also determine the stable geometric dimension in these cases.

     

A generalization of Marshall's equivalence relation

For prime and for a field containing a root of unity of order , we generalize Marshall's equivalence relation on orderings to arbitrary subgroups of of index . The equivalence classes then correspond to free pro- factors of the maximal pro- Galois group of . We generalize to this setting results of Jacob on the maximal pro- Galois group of a Pythagorean field.

     

Stationary isothermic surfaces and uniformly dense domains

We establish a relationship between stationary isothermic surfaces and uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain in the -dimensional Euclidean space is said to be uniformly dense in a surface of codimension if, for every small the volume of the intersection of with a ball of radius and center does not depend on for

We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary , and we show that the principal curvatures of satisfy certain identities.

The case in which the surface coincides with is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if , it must be either a circle or a straight line and (ii) if it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.

     

A density theorem on automorphic -functions and some applications

We establish a density theorem on automorphic -functions and give some applications on the extreme values of these -functions at and the distribution of the Hecke eigenvalue of holomorphic cusp forms.

     

Root invariants in the Adams spectral sequence

Let be a ring spectrum for which the -Adams spectral sequence converges. We define a variant of Mahowald's root invariant called the `filtered root invariant' which takes values in the term of the -Adams spectral sequence. The main theorems of this paper are concerned with when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the -Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime . We use the filtered root invariants to compute some low-dimensional root invariants of -periodic elements at the prime . We also compute the root invariants of some infinite -periodic families of elements at the prime .

     

The automorphism tower of groups acting on rooted trees

The group of isometries of a rooted -ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in . This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group studied by R. Grigorchuk, and the group studied by N. Gupta and the second author.

In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as and . We describe this tower for all subgroups of which decompose as infinitely iterated wreath products. Furthermore, we fully describe the towers of and .

More precisely, the tower of is infinite countable, and the terms of the tower are -groups. Quotients of successive terms are infinite elementary abelian -groups.

In contrast, the tower of has length , and its terms are -groups. We show that is an elementary abelian -group of countably infinite rank, while .

     

A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal

Let be a locally compact group, and let denote the space of weakly almost periodic functions on . We show that, if is a -group, but not compact, then the dual Banach algebra does not have a normal, virtual diagonal. Consequently, whenever is an amenable, non-compact -group, is an example of a Connes-amenable, dual Banach algebra without a normal, virtual diagonal. On the other hand, there are amenable, non-compact, locally compact groups such that does have a normal, virtual diagonal.

     

Algebraic Goodwillie calculus and a cotriple model for the remainder

Goodwillie has defined a tower of approximations for a functor from spaces to spaces that is analogous to the Taylor series of a function. His order approximation at a space depends on the values of on coproducts of large suspensions of the space: .

We define an ``algebraic' version of the Goodwillie tower, , that depends only on the behavior of on coproducts of . When is a functor to connected spaces or grouplike -spaces, the functor is the base of a fibration

whose fiber is the simplicial space associated to a cotriple built from the cross effect of the functor . In a range in which commutes with realizations (for instance, when is the identity functor of spaces), the algebraic Goodwillie tower agrees with the ordinary (topological) Goodwillie tower, so this theory gives a way of studying the Goodwillie approximation to a functor in many interesting cases.

     

Quivers with relations arising from clusters case)

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let be a cluster algebra of type . We associate to each cluster of an abelian category such that the indecomposable objects of are in natural correspondence with the cluster variables of which are not in . We give an algebraic realization and a geometric realization of . Then, we generalize the ``denominator theorem' of Fomin and Zelevinsky to any cluster.

     

Isovariant Borsuk-Ulam results for pseudofree circle actions and their converse

In this paper we shall study the existence of an -isovariant map from a rational homology sphere with pseudofree action to a representation sphere . We first show some isovariant Borsuk-Ulam type results. Next we shall consider the converse of those results and show that there exists an -isovariant map from to under suitable conditions.

     

Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems

Given an orbit whose linearization has invariant subspaces satisfying some non-resonance conditions in the exponential rates of growth, we prove existence of invariant manifolds tangent to these subspaces. The exponential rates of growth can be understood either in the sense of Lyapunov exponents or in the sense of exponential dichotomies. These manifolds can correspond to ``slow manifolds', which characterize the asymptotic convergence.

Let be a regular orbit of a dynamical system . Let be a subset of its Lyapunov exponents. Assume that all the Lyapunov exponents in are negative and that the sums of Lyapunov exponents in do not agree with any Lyapunov exponent in the complement of Denote by the linear spaces spanned by the spaces associated to the Lyapunov exponents in We show that there are smooth manifolds such that and . We establish the same results for orbits satisfying dichotomies and whose rates of growth satisfy similar non-resonance conditions. These systems of invariant manifolds are not, in general, a foliation.

     

Poisson structures on complex flag manifolds associated with real forms

For a complex semisimple Lie group and a real form we define a Poisson structure on the variety of Borel subgroups of with the property that all -orbits in as well as all Bruhat cells (for a suitable choice of a Borel subgroup of ) are Poisson submanifolds. In particular, we show that every non-empty intersection of a -orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.