L6

I am interested in integer solutions to equations of the form $f(x)=0$ where $f$ is a transcendental, globally analytic function defined in a neighbourhood of $\infty$ in $\mathbb{R}^n \cup \{\infty\}$. These notions will be defined precisely, and clarified in the wider context of globally semi-analytic and globally subanalytic sets.

The case $n=1$ is trivial (the global assumption forces there to be only finitely many (real) zeros of $f$) and the case $n=2$, which I shall briefly discuss, is completely understood: the number of such integer zeros of modulus at most $H$ is of order $\log\log H$. I shall then go on to consider the situation in higher dimensions.

A major desideratum in transcendental number theory is a simple sufficient condition for a given real number to be irrational, or better yet transcendental. In this talk we consider various forms such a criterion might take, and prove the existence or non-existence of them in various settings.

In {\it Was sind und was sollen die Zahlen?} (1888), Dedekind proves the Recursion Theorem (Theorem 126), and applies it to establish the categoricity of his axioms for arithmetic (Theorem 132). It is essential to these results that mathematical induction is formulated using second-order quantification, and if the second-order quantifier ranges over all subsets of the first-order domain (full second-order quantification), the categoricity result shows that, to within isomorphism, only one structure satisfies these axioms. However, the proof of categoricity is correct for a wide class of non-full Henkin models of second-order quantification. In light of this fact, can the proof of second-order categoricity be taken to establish that the second-order axioms of arithmetic characterize a unique structure?

Ax's theorem on the dimension of the intersection of an algebraic subvariety and a formal subgroup (Theorem 1F in "Some topics in differential algebraic geometry I...") implies Schanuel type transcendence results for a vast class of formal maps (including exp on a semi-abelian variety). Ax stated and proved this theorem in the characteristic 0 case, but the statement is meaningful for arbitrary characteristic and still implies positive characteristic transcendence results. I will discuss my work on positive characteristic version of Ax's theorem.

The transfer principle of Ax-Kochen-Ershov says that every first order sentence φ in the language of valued fields is, for p sufficiently big, true in ℚ_p iff it is true in \F_p((t)). Motivic integration allowed to generalize this to certain kinds of non-first order sentences speaking about functions from the valued field to ℂ. I will present some new transfer principles of this kind and explain how they are useful in representation theory. In particular, local integrability of Harish-Chandra characters, which previously was known only in ℚ_p, can be transferred to \F_p((t)) for p >> 1. (I will explain what this means.)

This is joint work with Raf Cluckers and Julia Gordon.

We consider a class of CR manifold which are defined as asymptotically

Heisenberg,

and for these we give a notion of mass. From the solvability of the

$\Box_b$ equation

in a certain functional class ([Hsiao-Yung]), we prove positivity of the

mass under the

condition that the Webster curvature is positive and that the manifold

is embeddable.

We apply this result to the Yamabe problem for compact CR manifolds,

assuming positivity

of the Webster class and non-negativity of the Paneitz operator. This is

joint work with

J.H.Cheng and P.Yang.

A new nematic phase has recently been discovered and characterized experimentally. It embodies a theoretical prediction made by Robert B. Meyer in 1973 on the basis of mere symmetry considerations to the effect that a nematic phase might also exist which in its ground state would acquire a 'heliconical' configuration, similar to the chiral molecular arrangement of cholesterics, but with the nematic director precessing around a cone about the optic axis. Experiments with newly synthetized materials have shown chiral heliconical equilibrium structures with characteristic pitch in the range of 1o nanometres and cone semi-amplitude of about 20 degrees. In 2001, Ivan Dozov proposed an elastic theory for such (then still speculative) phase which features a negative bend elastic constant along with a quartic correction to the nematic energy density that makes it positive definite. This lecture will present some thoughts about the possibility of describing the elastic response of twist-bend nematics within a purely quadratic gradient theory.

We show that the spatial norm in any critical homogeneous Besov

space in which local existence of strong solutions to the 3-d

Navier-Stokes equations is known must become unbounded near a singularity.

In particular, the regularity of these spaces can be arbitrarily close to

-1, which is the lowest regularity of any Navier-Stokes critical space.

This extends a well-known result of Escauriaza-Seregin-Sverak (2003)

concerning the Lebesgue space $L^3$, a critical space with regularity 0

which is continuously embedded into the spaces we consider. We follow the

``critical element'' reductio ad absurdum method of Kenig-Merle based on

profile decompositions, but due to the low regularity of the spaces

considered we rely on an iterative algorithm to improve low-regularity

bounds on solutions to bounds on a part of the solution in spaces with

positive regularity. This is joint work with I. Gallagher (Paris 7) and

F. Planchon (Nice).