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# Seminar in Geometry and Topology

We consider an algebraic variety $V$ and its foliation, both defined over a number field. Given a (compact piece of a) leaf $L$ of the foliation, and a subvariety $W$ of complementary codimension, we give an upper bound for the number of intersections between $L$ and $W$. The bound depends polynomially on the degree of $W$, the logarithmic height of $W$, and the logarithmic distance between $L$ and the locus of points where leafs of the foliation intersect $W$ improperly.

Using this theory we prove the Wilkie conjecture for sets defined using leafs of foliations under a certain assumption about the algebraicity locus. For example, we prove the if none of the leafs contain algebraic curves then the number of algebraic points of degree $d$ and log-height $h$ on a (compact piece of a) leaf grows polynomially with $d$ and $h$. This statement and its generalizations have many applications in diophantine geometry following the Pila-Zannier strategy.

I will focus mostly on the proof of the main statement, which uses a combination of differential-algebraic methods related to foliations with some ideas from complex geometry and value distribution theory. If time permits I will briefly discuss the applications to counting algebraic points and diophantine geometry at the end.

In 1981 P.Arnoux and J.-C.Yoccoz constructed their famous foliation on a surface of genus three with zero flux. Later it was shown that this example can be generalized, and in particular that there is an interesting fractal set of parameters that give rise to the foliations with similar properties. This fractal was named in honour of Gerard Rauzy.

In my talk I will briefly discuss how such a family of foliations appeared in different branches of mathematics (symbolic dynamics, Teichmuller dynamics, low-dimensional topology, geometric group theory) and even in theoretical physics (conductivity theory in monocrystals) and explain what do we know about this family from ergodic point of view.

The talk is based on joint work with Pascal Hubert and Artur Avila and on a work in progress with Ivan Dynnikov and Pascal Hubert.

We will consider two different types of hypergeometric decompositions of special data associated to five pencils of quartic surfaces. We see that over the complex numbers, the middle cohomology of these five pencils yield hypergeometric Picard-Fuchs equations. Using these parameters, we then consider the same pencils over finite fields, decomposing their rational point counts using the finite-field hypergeometric functions with the same parameters as above. This is joint work with Charles Doran, Adriana Salerno, Steven Sperber, John Voight, and Ursula Whitcher.

As it was observed a few years ago, there exists a certain signed count of real lines on real projective hypersurfaces of degree 2n+1 and dimension n that, contrary to the honest "cardinal" count, is independent of the choice of a hypersurface, and by this reason provides, as a consequence, a strong lower bound on the honest count. Originally, in this invariant signed count the input of a line was given by its local contribution to the Euler number of an appropriate auxiliary universal vector bundle.

The aim of the talk is to present other, in a sense more geometric, interpretations of the signs involved in the invariant count. In particular, this provides certain generalizations of Segre indices of real lines on cubic surfaces and Welschinger-Solomon weights of real lines on quintic threefolds.

This is a joint work with S.Finashin.

A classical theorem of Mittag-Leffler asserts that in a given Riemann surface X, for any pattern of multiplicities of poles and any configuration of residues (summing to zero), there is a meromorphic 1-form on X that realize them. The only obstruction is that residues at simple poles should be nonzero.

If we require that the multiplicity of the zeroes is also prescribed, the problem can be reformulated in terms of strata of meromorphic differentials. Using the dictionary between complex analysis and flat geometry, we are able to provide a complete characterization of configurations of residues that are realized for a given pattern of singularities. Two nontrivial obstructions appear concerning the combinatorics of the multiplicity of zeroes and the arithmetics of the residues. This is a joint work with Quentin Gendron.

We will discuss the field of definition of a rational function and in what ways it can change under iteration, in particular when the degree over the base field drops. We present two families of rational functions with the property above, and prove that in the special case of polynomials, only one of these families is possible. We also explain how this relates to Ritt's decomposition theorem on polynomials. Joint work with Francesco Veneziano (SNS Pisa).

The talk will be about two classical problems in the theory of Carleman classes of smooth functions. The first one is to describe the image of a Carleman class under the Borel map (which maps a smooth function to its jet of Taylor coefficients at a given point). The second one concerns possible ways to construct a function in a given Carleman class with prescribed Taylor coefficients. I will present solutions to both problems. If time permits, I will also discuss related problems which originate in the singularity theory of Carleman classes.

Random curves in space and how they are knotted give an insight into the behavior of "typical" knots and links. They have been studied by biologists and physicists in the context of the structure of random polymers. Several randomized models have been suggested and investigated, and many results have been obtained via computational experiment. The talk will begin with a review of this area.

In work with Hass, Linial, and Nowik, we study knots based on petal projections, potholder diagrams, and more. We have found explicit formulas for the limit distribution of finite type invariants of random knots and links in the Petaluma model. I will discuss these results and sketch proof ideas as time permits.

In the last 35 years, geometric flows have proven to be a powerful tool in geometry and topology. The Mean Curvature Flow is, in many ways, the most natural flow for surfaces in Euclidean space.

In this talk, which will assume no prior knowledge, I will illustrate how mean curvature flow could be used to address geometric questions. I will then explain why the formation of singularities of the mean curvature flow poses difficulties for such applications, and how recent new discoveries about the structure of singularities (including a work joint with Kyeongsu Choi and Robert Haslhofer) may help overcome those difficulties.

In this talk, I will discuss a question which originates in complex analysis but is really a problem in non-linear elliptic PDE. A finite Blaschke product is a proper holomorphic self-map of the unit disk, just like a polynomial is a proper holomorphic self-map of the complex plane. A celebrated theorem of Heins says that up to post-composition with a M\"obius transformation, a finite Blaschke product is uniquely determined by the set of its critical points. Konstantin Dyakonov suggested that it may interesting to extend this result to infinite degree. However, one must be a little careful since infinite Blaschke products may have identical critical sets. I will show that an infinite Blaschke product is uniquely determined by its "critical structure" and describe all possible critical structures which can occur. By Liouville's correspondence, this question is equivalent to studying nearly-maximal solutions of the Gauss curvature equation $\Delta u = e^{2u}$. This problem can then be solved using PDE techniques, using the method of sub- and super-solutions.

The Baum-Bott indexes are important local invariants for singular holomorphic foliations by curves with isolated singularities. On a compact complex manifold, we consider foliations with fixed cotangent bundle. Then, the Baum-Bott map associates to a foliation its Baum-Bott indexes on its singularities. We focus on foliations on the projective space and we are interested in the generic rank of that map. The generic rank for foliations on the projective plane is known. For high-dimensional projectives spaces, we give an upper bound and in some cases we determine the generic rank.

An extremal metric, as defined by Calabi, is a canonical Kahler metric: it minimizes the curvature within a given Kahler class. According to the Yau-Tian-Donaldson conjecture, polarized Kahler manifolds admitting an extremal metric should correspond to stable manifolds in a Geometric Invariant Theory sense.

In this talk, we will explain that a projective extremal Kahler manifold is asymptotically relatively Chow stable. This fact was conjectured by Apostolov and Huang, and its proof relies on quantization techniques. We will explain various implications, such that unicity or splitting results for extremal metrics.

Joint work with Yuji Sano ( Fukuoka University).

Being the gradient flow of the area functional, the mean curvature flow can be thought of as a greedy algorithm for simplifying embedded shapes. But how successful is this algorithm?

In this talk, I will describe three examples for how mean curvature flow, as well as its variants and weak solutions, can be used to achieve this desired simplification.

The first is a short time smoothing effect of the flow, allowing to smooth out some rough, potentially fractal initial data.

The second is an application of mean curvature flow with surgery to smooth differential topology, allowing to conclude Schoenflies-type theorems about the moduli space of smooth embedded spheres and tori, satisfying some curvature conditions.

The third is an application of (weak,modified) mean curvature flow to differential geometry, allowing to relate bounds on the gaussian entropy functional to the topology of a closed hypersurface.

In this talk, which will assume no prior knowledge in PDE or mean curvature flow, I will try to highlight the relation between the analysis of the flow and in particular, its singularity formation, to both ''time dependent'' and "classical" geometry.

Some of the results described in this talk are joint works with Reto Buzano, Robert Haslhofer and Brian White.

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.