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.

     

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.

     

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.

     

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 .

     

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 .

     

How to do a -descent on an elliptic curve

In this paper, we describe an algorithm that reduces the computation of the (full) -Selmer group of an elliptic curve over a number field to standard number field computations such as determining the (-torsion of) the -class group and a basis of the -units modulo th powers for a suitable set of primes. In particular, we give a result reducing this set of `bad primes' to a very small set, which in many cases only contains the primes above . As of today, this provides a feasible algorithm for performing a full -descent on an elliptic curve over , but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of is favorable, simplifications are possible and -descents for larger are accessible even today. To demonstrate how the method works, several worked examples are included.

     

A new variational characterization of -dimensional space forms

A Riemannian manifold is associated with a Schouten -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with characterized the three-dimensional space forms.

     

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.

     

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.

     

Analysing finite locally -arc transitive graphs

We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms and are either locally -arc transitive for or -locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of . Given a normal subgroup which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of preserves both local primitivity and local -arc transitivity and leads us to study graphs where acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.