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Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? While the first results in this direction were obtained in the 1960-ies, various facets of this question remain largely open. I will review recent advances on this problem in the context of contact flows, which are closely related to Hamiltonian flows from classical mechanics. In particular, I'll discuss a proof of a conjecture of Irie stating that rotations of odd-dimensional ellipsoids admit a surprisingly large class of perturbations creating periodic orbits. The proof involves methods of modern symplectic topology including pseudo-holomorphic curves and contact homology. The talk is based on a joint work with Julian Chaidez, Ipsita Datta and Rohil Prasad, as well as a work in progress joint with Julian Chaidez.

Review of old and new results on the global structure of singular holomorphic codimension one foliations on projective manifolds and their moduli spaces.

ZOOM: https://weizmann.zoom.us/j/97197369277?pwd=c0kyTVVDaWhLQ1NaWDJLVE9MTzBVUT09 

 

The usual Poincare problem consists in determining whether or not there exist upper bounds for the degree of an invariant algebraic curve of a foliation of the complex projective plane in terms of the degree of the foliation. We are interested in the local analogue,  in which degrees are replaced with vanishing multiplicities at the origin. Since cusps of arbitrary degree can be invariant curves of foliations of degree one in the global setting and of multiplicity one in the local one, there are no universal upper bounds. In order to deal with this issue, extra hypotheses were considered, either on the invariant curve (Cerveau - Lins Neto,...) or on the desingularization of the foliation (Carnicer...). We consider an irreducible germ of invariant curve and no extra hypotheses. We provide a lower bound for the multiplicity of a germ of complex foliation in dimension 2 in terms of Puiseux characteristics of an irreducible invariant curve. Obviously, the lower bound is trivial for a cusp but it is valuable if the curve has at least two Puiseux characteristic exponents. This is a joint work with Jose Cano and Pedro Fortuny.  

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https://weizmann.zoom.us/j/9183356684?pwd=QlN4T1VtcXc4QXdXNEYxUlQ4dmR5dz09

 

I will present a joint work with A. Alvarez, J.L. Bravo and C. Christopher. 
We study the analogue of the classical infinitesimal center problem in the plane, but for zero cycles. We define the displacement function in this context and prove that it is identically zero if and only if the deformation has a composition factor. That is, we prove that here the composition conjecture is true, in contrast with the tangential center problem on zero cycles. Finally, we give examples of applications of our results.

Zoom link: https://us02web.zoom.us/j/88255165329

 

In this talk we introduce Newton non-degenerate codimension one foliations, that are given in terms of Newton polyhedra following the classical ideas of Kouchnirenko for hypersurfaces. We characterize Newton non-degenerate foliations in geometrical terms, as being foliations admitting a reduction of singularities of a combinatorial nature, the ones that we call (weak) toric type foliations. In addition, we provide a positive answer to Thom's question about the existence of invariant hypersurface for germs of codimension one foliations, in the three-dimensional case of toric type. In order to do it, we pass through a study of global character in dimension two. The Khovanskii-Kouchnirenko-Bernstein result that relates the number of solutions of a Laurent polynomial system in terms of the mixed volume of the associated polyhedra, leads us to conclude that isolated branches of Newton non-degenerate foliations defined over projective toric surfaces extend in a global way. This prolongation property is the key to spread the invariant surfaces in dimension three through the compact dicritical components.

Define a relation between labeled ideal polygons in the hyperbolic space by requiring that the complex distances (a combination of the distance and the angle) between their  respective sides equal c; the complex number c is a parameter of the relation. This defines a 1-parameter family of maps on the moduli space of ideal polygons in the hyperbolic space (or, in its real version, in the hyperbolic plane). I shall discuss complete integrability of this family of maps and mention related topics, including Coxeter's frieze patterns and elements of the theory of cluster algebras

The moduli space of quadratic differentials on punctured Riemann surfaces admits a integral piecewise linear structure. It gives rise to a measure whose total mass is finite and called the Masur-Veech volume.

These volumes are also related to the asymptotic growth of the number of multicurves on hyperbolic surfaces, once averaged against the Weil-Petersson metric. Adopting this point of view, we show that the Masur-Veech volumes can be obtained as the constant term in a family of polynomials computed by the topological recursion, and as integration of a class on the moduli space of curves. Based on work of Moeller, we can prove the same statement but for a different family of polynomial (and different initial data for the topological recursion). This is based on joint works with Jorgen Andersen, Severin Charbonnier, Vincent Delecroix, Alessandro Giacchetto, Danilo Lewanski and Campbell Wheeler.

For X a symplectic manifold and L a Lagrangian submanifold, genus zero open Gromov-Witten (OGW) invariants count configurations of pseudoholomorphic disks in X with boundary conditions in L and various constraints at boundary and interior marked points. In a joint work with Jake Solomon from 2016, we define OGW invariants using bounding chains, a concept that comes from Floer theory. In a recent work, also joint with Solomon, we find that the generating function of OGW satisfies a system of PDE called open WDVV equation. This PDE translates to an associativity relation for a quantum product we define on the relative cohomology H^*(X,L). For the projective space, open WDVV gives rise to recursions that, together with other properties, allow the computation of all OGW invariants.

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.

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.

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.

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.

 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.

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). 

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.

We revisit how the Euler and Burgers equations arise as geodesics on the groups of diffeomorphisms. It turns out that the Euler hydrodynamics is in a sense dual to problems of optimal mass transport. We also describe L^2 and H^1 versions of the the Wasserstein space of volume forms. It turns out that for the homogeneous H^1 metric the Wasserstein space is isometric to (a piece of) an infinite-dimensional sphere and it leads to an integrable generalization of the Hunter-Saxton equation.

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.

A conjecture of Victor Ivrii (1980) says that in every billiard with smooth boundary the set of periodic orbits has measure zero. This conjecture is closely related to spectral theory. Its particular case for triangular orbits was proved by M. Rychlik (1989, in two dimensions), Ya. Vorobets (1994, in any dimension) and other mathematicians. The case of quadrilateral orbits in dimension two was treated in our joint work with Yu. Kudryashov (2012). We study the complexified version of planar Ivrii's conjecture with reflections from a collection of planar holomorphic curves. We present the classification of complex counterexamples with four reflections and partial positive results. The recent one says that a billiard on one irreducible complex algebraic curve without too complicated singularities cannot have a two-dimensional family of periodic orbits of any period. The above complex results have applications to other problems on real billiards: Tabachnikov's commuting billiard problem and Plakhov's invisibility conjecture.

The talk provides a new perspective of the global bifurcation theory on the plane. Theory of planar bifurcations consists of three parts: local, nonlocal and global ones. It is now clear that the latter one is yet to be created. Local bifurcation theory (in what follows we will talk about the plane only) is related to transfigurations of phase portraits of differential equations near their singular points. This theory is almost completed, though recently new open problems occurred. Nonlocal theory is related to bifurcations of separatrix polygons (polycycles). Though in the last 30 years there were obtained many new results, this theory is far from being completed. Recently it was discovered that nonlocal theory contains another substantial part: a global theory. New phenomena are related with appearance of the so called sparkling saddle connections. The aim of the talk is to give an outline of the new theory and discuss numerous open problems. The main new results are: existence of an open set of structurally unstable families of planar vector fields, and of families having functional invariants (joint results with Kudryashov and Schurov). Thirty years ago Arnold stated six conjectures that outlined the future development of the global bifurcation theory in the plane. All these conjectures are now disproved. Though the theory develops in quite a different direction, this development is motivated by the Arnold's conjectures.

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.

We describe how persistence modules, a notion originating in data sciences, can be applied to obtain new results on Hofer's geometry, related in particular to autonomous Hamiltonian flows. Joint work with Leonid Polterovich.

A spherical quadrilateral (membrane) is a bordered surface homeomorphic to a closed disc, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that at most three angles at the corners are not multiples of Pi. This is a very old problem, related to the properties of solutions of the Heun's equation (an ordinary differential equation with four regular singular points). The corresponding problem for the spherical triangles, related to the properties of solutions of the hypergeometric equation, has been solved by Klein, with some gaps in Klein's classification filled in by Eremenko in 2004. The general quadrilateral case remains open. This is joint work with V. Tarasov (IUPUI). The first part is by A. Eremenko (Purdue), 13:00-14:00, room 1. The second is by A. Gabrielov (Purdue), 16:00-17:15, room 261.

Tropical geometry is a powerful technique to study enumerative problems in algebraic geometry. The theory of Berkovich spaces gives us a natural framework to apply tropical techniques in a much wider context. I will begin by explaining motivations from mirror symmetry. Then I will introduce a notion of Khaler structures in non-archimedean geometry. I will explain the construction of the moduli stack of non-archimedean stable maps and an analog of Gromov-Fs compactness theorem in the non-archimedean setting. They are the first steps of enumerative non-archimedean geometry. I will also discuss the tropicalization of the space of stable maps. They are based on arXiv 1401.6452 and 1407.8444. If time permits, I will discuss a related joint work with M. Porta concerning higher non-archimedean stacks and GAGA theorems.