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# Seminar in Geometry and Topology

Schwartz functions are classically defined as smooth functions such that they, and all their (partial) derivatives, decay at infinity faster than the inverse of any polynomial. This was formulated on $\mathbb{R}^n$ by Laurent Schwartz, and later on Nash manifolds (smooth semi-algebraic varieties) by Fokko du Cloux and by Rami Aizenbud and Dima Gourevitch. In a joint work with Boaz Elazar we have extended the theory of Schwartz functions to the category of (possibly singular) real algebraic varieties. The basic idea is to define Schwartz functions on a (closed) algebraic subset of $\mathbb{R}^n$ as restrictions of Schwartz functions on $\mathbb{R}^n$.

Both in the Nash and the algebraic categories there exists a very useful characterization of Schwartz functions on open subsets, in terms of Schwartz functions on the embedding space: loosely speaking, Schwartz functions on an open subset are exactly restrictions of Schwartz functions on the embedding space, which are zero "to infinite order" on the complement to this open subset. This characterization suggests a very intuitive way to attach a space of Schwartz functions to an arbitrary (not necessarily semi-algebraic) open subset of $\mathbb{R}^n$.

In this talk, I will explain this construction, and more generally the construction of the category of tempered smooth manifolds. This category is in a sense the "largest" category whose objects "look" locally like open subsets of $\mathbb{R}^n$ (for some $n$), and on which Schwartz functions may be defined. In the development of this theory some classical results of Whitney are used, mainly Whitney type partition of unity (this will also be explained in the talk). As time permits, I will show some properties of Schwartz functions, and describe some possible applications. This is a work in progress.

Let X be a set definable in some o-minimal structure. The Pila-Wilkie theorem (in its basic form) states that the number of rational points in the transcendental part of X grows sub-polynomially with the height of the points. The Wilkie conjecture stipulates that for sets definable in $R_\exp$, one can sharpen this asymptotic to polylogarithmic.

I will describe a complex-analytic approach to the proof of the Pila-Wilkie theorem for subanalytic sets. I will then discuss how this approach leads to a proof of the "restricted Wilkie conjecture", where we replace $R_\exp$ by the structure generated by the restrictions of $\exp$ and $\sin$ to the unit interval (both parts are joint work with Dmitry Novikov). If time permits I will discuss possible generalizations and applications.

Cauchy-Riemann maps (shortly: CR-maps) occur in complex analysis as boundary values of maps holomorphic in a domain in complex space. As a rule, CR-mappings of real-analytic hypersurfaces appear to be analytic as well. However, we recently showed in a joint work with Rasul Shafikov the existence of Stokes Phenomenon in CR-geometry: there exist real-analytic hypersurfaces, which are equivalent formally, but not holomorphically.

Despite of this, it appears that in complex dimension 2, CR-maps necessarily posses appropriate weaker regularity properties. Namely, components of such maps necessarily belong to the well known Gevrey classes. The latter statement has the following remarkable application: if two real-analytic hypersurfaces in complex two-space are equivalent formally, then they are also equivalent smoothly.

The proof of all these facts employs the recent multi-summability theory in Dynamical Systems. It as well employs the recent CR-DS technique that we developed, which connects CR-manifolds and certain Dynamical Systems. In this talk, I will outline the technique, as well as some details of the proof.

For some rectangular Hardy classes of analytic functions,an optimal method of interpolation has been previously found, within the framework of Optimal Recovery. It will be shown that this method of interpolation, based on the Abel-Jacobi elliptic functions, is also optimal, according to corresponding criteria of Nonparametric Regression and Optimal Design.

In a non-asymptotic setting, the maximal mean squared error of the optimal interpolant is evaluated explicitly, for all noise levels away from 0. In these results, a pivotal role is played by an interference effect, in which both the stochastic and deterministic parts of the interpolant exhibit an oscillating behavior, with the two oscillating processes mutually subduing each other.

We consider the following problem: given a set of algebraic conditions on an $n$-tuple of functions and their first $l$ derivatives, admitting finitely many solutions (in a differentiably closed field), can one give an upper bound for the number of solutions?

I will present estimates in terms of the degrees of the algebraic conditions, or more generally the volumes of their Newton polytopes (analogous to the Bezout and BKK theorems). The estimates are singly-exponential with respect to $n,l$ and have the natural asymptotic with respect to the degrees or Newton polytopes, sharpening previous doubly-exponential estimates due to Hrushovski and Pillay. I will also discuss some diophantine applications to counting transcendental lattice points on algebraic varieties.

The spectral flow is a well-known invariant of a 1-parameter family of self-adjoint Fredholm operators. It is defined as the net number of operator’s eigenvalues passing through 0 with the change of parameter.

Let S be a compact surface with non-empty boundary. Consider the space Ell(S) of first order self-adjoint elliptic differential operators on S with local boundary conditions. The first part of the talk is devoted to the computing of the spectral flow along loops in Ell(S), and also along paths with conjugated ends.

After that we consider more general situation: a family of elements of Ell(S) parameterized by points of a compact space X. We define the topological index of such a family and show that it coincides with the analytical index of the family. Both indices take value in K^1(X). When X is a circle, this result turns into the formula for the spectral flow from the first part of the talk.