Subgroups of acting transitively on -tuples

We consider subgroups of -diffeomorphisms of the circle which act transitively on -tuples of points. We show, in particular, that these subgroups are dense in the group of homeomorphisms of . A stronger result concerning -approximations is obtained as well. The techniques employed in this paper rely on Lie algebra ideas and they also provide partial generalizations to the differentiable case of some results previously established in the analytic category.

     

A class of -algebras generalizing both graph algebras and homeomorphism -algebras I, fundamental results

We introduce a new class of -algebras, which is a generalization of both graph algebras and homeomorphism -algebras. This class is very large and also very tractable. We prove the so-called gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem, and compute the -groups of our algebras.

     

Blakers-Massey elements and exotic diffeomorphisms of and via geodesics

We use the geometry of the geodesics of a certain left-invariant metric on the Lie group to find explicit related formulas for two topological objects: the Blakers-Massey element (a generator of ) and an exotic (i.e. not isotopic to the identity) diffeomorphism of (C. E. Durán, 2001). These formulas depend on two quaternions and their conjugates and we produce their extensions to the octonions through formulas for a generator of and exotic diffeomorphisms of , thus giving explicit gluing maps for half of the 15-dimensional exotic spheres expressed as the union of two 15-disks.

     

The -theory of is undecidable

The three quantifier theory of , the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of that lies between the two and three quantifier theories with but includes function symbols.

Theorem. The two quantifier theory of , the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on ) is undecidable.

The same result holds for various lattices of ideals of which are natural extensions of preserving join and infimum when it exits.

     

Coordinates in two variables over a -algebra

This paper studies coordinates in two variables over a -algebra. It gives several ways to characterize such coordinates. Also, various results about coordinates in two variables that were previously only known for fields, are extended to arbitrary -algebras.

     

Involutions fixing , II

This paper studies the equivariant cobordism classification of all involutions fixing a disjoint union of an odd-dimensional real projective space with its normal bundle nonbounding and a Dold manifold with a positive even and 0$">. The complete analysis of the equivariant cobordism classes of such involutions is given except that the upper and lower bounds on the codimension of may not be best possible. In particular, we find that there exist such involutions with nonstandard normal bundle to . Together with the results of part I of this title (Trans. Amer. Math. Soc. 354 (2002), 4539-4570), the argument for involutions fixing is finished.

     

On the -Minkowski problem

A volume-normalized formulation of the -Minkowski problem is presented. This formulation has the advantage that a solution is possible for all , including the degenerate case where the index is equal to the dimension of the ambient space. A new approach to the -Minkowski problem is presented, which solves the volume-normalized formulation for even data and all .

     

Surface superconductivity in dimensions

We study the Ginzburg-Landau system for a superconductor occupying a -dimensional bounded domain, and improve the estimate of the upper critical field obtained by K. Lu and X. Pan in J. Diff. Eqns., 168 (2000), 386-452. We also analyze the behavior of the order parameters. We show that, under an applied magnetic field lying below and not far from , order parameters concentrate in a vicinity of a sheath of the surface that is tangential to the applied field, and exponentially decay both in the normal and tangential directions away from the sheath in the sense. As the applied field decreases further but keeps in between and away from and , the superconducting sheath expands but does not cover the entire surface, and superconductivity at the surface portion orthogonal to the applied field is always very weak. This phenomenon is significantly different to the surface superconductivity on a cylinder of infinite height studied by X. Pan in Comm. Math. Phys., 228 (2002), 327-370, where under an axial applied field lying in-between and the entire surface is in the superconducting state.

     

Algebraic -actions of entropy rank one

We investigate algebraic -actions of entropy rank one, namely those for which each element has finite entropy. Such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery. This leads to numerous alternative characterizations of entropy rank one, both geometric and algebraic. We then compute the measure entropy of a class of skew products, where the fiber maps are elements from an algebraic -action of entropy rank one. This leads, via the relative variational principle, to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures. We use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms.

     

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 .

     

Analytic -adic cell decomposition and integrals

We prove a conjecture of Denef on parameterized -adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.

     

Homotopy groups of -contact toric manifolds

Contact toric manifolds of Reeb type are a subclass of contact toric manifolds which have the property that they are classified by the images of the associated moment maps. We compute their first and second homotopy group terms of the images of the moment map. We also explain why they are -contact.

     

Character sums and congruences with

We estimate character sums with , on average, and individually. These bounds are used to derive new results about various congruences modulo a prime and obtain new information about the spacings between quadratic nonresidues modulo . In particular, we show that there exists a positive integer such that is a primitive root modulo . We also show that every nonzero congruence class can be represented as a product of 7 factorials, , where , and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials with represent ``almost all' residue classes modulo p, and that products of 3 factorials with are uniformly distributed modulo .

     

Parametrized principles

We will present a collection of guessing principles which have a similar relationship to as cardinal invariants of the continuum have to . The purpose is to provide a means for systematically analyzing and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of and in models such as those of Laver, Miller, and Sacks.

     

The -module structure of -modules

Let be a regular ring, essentially of finite type over a perfect field . An -module is called a unit -module if it comes equipped with an isomorphism , where denotes the Frobenius map on , and is the associated pullback functor. It is well known that then carries a natural -module structure. In this paper we investigate the relation between the unit -structure and the induced -structure on . In particular, it is shown that if is algebraically closed and is a simple finitely generated unit -module, then it is also simple as a -module. An example showing the necessity of being algebraically closed is also given.

     

Cartan-decomposition subgroups of

For and , we give explicit, practical conditions that determine whether or not a closed, connected subgroup of has the property that there exists a compact subset of with . To do this, we fix a Cartan decomposition of , and then carry out an approximate calculation of for each closed, connected subgroup of .

     

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.

     

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.

     

Higher homotopy commutativity of -spaces and the permuto-associahedra

In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an -space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected -space has the finitely generated mod cohomology for a prime and the multiplication of it is homotopy commutative of the -th order, then it has the mod homotopy type of a finite product of Eilenberg-Mac Lane spaces s, s and s for .

     

Combinatorial properties of Thompson's group

We study some combinatorial consequences of Blake Fordham's theorems on the word metric of Thompson's group in the standard two generator presentation. We explore connections between the tree pair diagram representing an element of , its normal form in the infinite presentation, its word length, and minimal length representatives of it. We estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates. In addition we explore some properties of the Cayley graph of with respect to the two generator finite presentation. Namely, we exhibit the form of ``dead end' elements in this Cayley graph, and show that it has no ``deep pockets'. Finally, we discuss a simple method for constructing minimal length representatives for strictly positive or negative words.