Model theory of difference fields

A difference field is a field with a distinguished automorphism . This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is .

     

Trivial, strongly minimal theories are model complete after naming constants

We prove that if is any model of a trivial, strongly minimal theory, then the elementary diagram is a model complete -theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are -decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is .

     

Phase transitions in phylogeny

We apply the theory of Markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies.

We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least , and the net transition on each edge is bounded by . Motivated by a conjecture by M. Steel, we show that if 1$">, then for balanced trees, the topology of the underlying tree, having leaves, can be reconstructed from samples (characters) at the leaves. On the other hand, we show that if , then there exist topologies which require at least samples for reconstruction.

Our results are the first rigorous results to establish the role of phase transitions for Markov random fields on trees, as studied in probability, statistical physics and information theory, for the study of phylogenies in mathematical biology.

     

On model complete differential fields

We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODE . We show that this sub-family is usually definable (in particular if lives on a curve of positive genus). As a corollary, we show the existence of many model-complete, superstable theories of differential fields.

     

Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms

In this paper, we propose a generalization of the algorithm we developed previously. Along the way, we also develop a theory of quaternionic -symbols whose definition bears some resemblance to the classical -symbols, except for their combinatorial nature. The theory gives a more efficient way to compute Hilbert modular forms over totally real number fields, especially quadratic fields, and we have illustrated it with several examples. Namely, we have computed all the newforms of prime levels of norm less than 100 over the quadratic fields and , and whose Fourier coefficients are rational or are defined over a quadratic field.

     

Rational -equivariant homotopy theory

We give an algebraicization of rational -equivariant homotopy theory. There is an algebraic category of `` -systems' which is equivalent to the homotopy category of rational -simply connected -spaces. There is also a theory of ``minimal models' for -systems, analogous to Sullivan's minimal algebras. Each -space has an associated minimal -system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.

     

A closed model category for ()-connected spaces

For each integer , we give a distinct closed model category structure to the category of pointed spaces such that the corresponding localized category is equivalent to the standard homotopy category of -connected CW-complexes. The structure of closed model category given by Quillen to is based on maps which induce isomorphisms on all homotopy group functors and for any choice of base point. For each , the closed model category structure given here takes as weak equivalences those maps that for the given base point induce isomorphisms on for .

     

A note on groups definable in difference fields

We prove that a group definable in a model of is virtually definably embeddable in an algebraic group. We give an improved proof of the same result for groups definable in differentially closed fields. We also extend to the difference field context results on the unipotence of definable groups on affine spaces.

     

On the computability-theoretic complexity of trivial, strongly minimal models

We show the existence of a trivial, strongly minimal (and thus uncountably categorical) theory for which the prime model is computable and each of the other countable models computes . This result shows that the result of Goncharov/Harizanov/Laskowski/Lempp/McCoy (2003) is best possible for trivial strongly minimal theories in terms of computable model theory. We conclude with some remarks about axiomatizability.

     

A model category structure for equivariant algebraic models

In the equivariant category of spaces with an action of a finite group, algebraic `minimal models' exist which describe the rational homotopy for -spaces which are 1-connected and of finite type. These models are diagrams of commutative differential graded algebras. In this paper we prove that a model category structure exists on this diagram category in such a way that the equivariant minimal models are cofibrant objects. We show that with this model structure, there is a Quillen equivalence between the equivariant category of rational -spaces satisfying the above conditions and the algebraic category of the models.

     

Weak axioms of choice for metric spaces

In the framework of ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice AC, we show that if the family of all non-empty, closed subsets of a metric space has a choice function, then so does the family of all non-empty, open subsets of . In addition, we establish that the converse is not provable in ZF.

We also show that the statement ``every subspace of the real line with the standard topology has a choice function for its family of all closed, non-empty subsets" is equivalent to the weak choice form ``every continuum sized family of non-empty subsets of reals has a choice function".

     

Regularity of Solution Maps of Differential Inclusions Under State Constraints

Consider a differential inclusion under state constraints

Finite state automata: A geometric approach

Recently, finite state automata, via the advent of hyperbolic and automatic groups, have become a powerful tool in geometric group theory. This paper develops a geometric approach to automata theory, analogous to various techniques used in combinatorial group theory, to solve various problems on the overlap between group theory and monoid theory. For instance, we give a geometric algorithm for computing the closure of a rational language in the profinite topology of a free group. We introduce some geometric notions for automata and show that certain important classes of monoids can be described in terms of the geometry of their Cayley graphs. A long standing open question, to which the answer was only known in the simplest of cases (and even then was non-trivial), is whether it is true, for a pseudovariety of groups , that a -trivial co-extension of a group in must divide a semidirect product of a -trivial monoid and a group in . We show the answer is affirmative if is closed under extension, and may be negative otherwise.

     

Hyperbolic minimizing geodesics

We apply the intersection theory for Lagrangian submanifolds to obtain a Sturm type comparison theorem for linearized Hamiltonian flows. Applications to the theory of geodesics are considered, including a sufficient condition that arclength minimizing closed geodesics, for an -dimensional Riemannian manifold, are hyperbolic under the geodesic flow. This partially answers a conjecture of G. D. Birkhoff.

     

Elliptic planar vector fields with degeneracies

This paper deals with the normalization of elliptic vector fields in the plane that degenerate along a simple and closed curve. The associated homogeneous equation is studied and an application to a degenerate Beltrami equation is given.

     

Contact topology and hydrodynamics III: knotted orbits

We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct a steady nonsingular solution to the Euler equations on a Riemannian whose flowlines trace out closed curves of all possible knot and link types. Using careful contact-topological controls, we can make such vector fields real-analytic and transverse to the tight contact structure on . Sufficient review of concepts is included to make this paper independent of the previous works in this series.

     

Couples contacto-symplectiques

We introduce a new geometric structure on differentiable manifolds. A contact-symplectic pair on a manifold is a pair where is a Pfaffian form of constant class and a -form of constant class such that is a volume form. Each form has a characteristic foliation whose leaves are symplectic and contact manifolds respectively. These foliations are transverse and complementary. Some other differential objects are associated to it. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds and principal torus bundles. As a deep application of this theory, we give a negative answer to the famous Reeb's problem which asks if every vector field without closed 1-codimensional transversal on a manifold having contact forms is the Reeb vector field of a contact form.

     

Closed timelike geodesics in compact spacetimes

Let be a compact spacetime which admits a regular globally hyperbolic covering, and a nontrivial free timelike homotopy class of closed timelike curves in We prove that contains a longest curve (which must be a closed timelike geodesic) if and only if the timelike injectivity radius of is finite; i.e., has a bounded length. As a consequence among others, we deduce that for a compact static spacetime there exists a closed timelike geodesic within every nontrivial free timelike homotopy class having a finite timelike injectivity radius.

     

Pure Picard-Vessiot extensions with generic properties

Given a connected linear algebraic group over an algebraically closed field of characteristic 0, we construct a pure Picard-Vessiot extension for , namely, a Picard-Vessiot extension , with differential Galois group , such that and are purely differentially transcendental over . The differential field is the quotient field of a -stable proper differential subring with the property that if is any differential field with field of constants and is a Picard-Vessiot extension with differential Galois group a connected subgroup of , then there is a differential homomorphism such that is generated over as a differential field by .