Hilbert's sixteenth problem remains among the most persistent in his famous list, yielding first place in this regard only to the Riemann Hypothesis. This project focuses on Hilbert's sixteenth problem for quadratic vector fields, the infinitesimal version of Hilbert's sixteenth problem, and several related topics. The latter include the following: algebraic solutions of polynomial differential equations in higher dimensions; rigidity of complex polynomial foliations; relations between moduli of elliptic curves and rotation numbers; new problems about analytic families of germs of conformal maps, related to so-called mixed families. The first part of the project is based on two major results obtained under the principal investigator's previous award. The first of those results is an almost complete solution to the restricted Hilbert problem for quadratic vector fields. This solution is to be completed in the current project. In its final form, this solution would provide a "covering function" on the space of quadratic vector fields, a covering function with the following properties: (1) it is finite and lower semicontinuous on the set of vector fields that have no polycycles (separatrix polygons); (2) its restriction to this set has an infinite limit on the exclusive set of vector fields with polycycles; (3) it majorizes the number of limit cycles outside the exclusive set. The principal investigator proposes to find an upper bound on the number of limit cycles by establishing a local persistence property for limit cycles and then replacing the covering function with a cut-off function that still majorizes the number of limit cycles. The maximum of the cut-off function will be the desired upper bound. The second major objective of the current project is the complete solution of the restricted infinitesimal Hilbert sixteenth problem. This solution involves another covering function, this time on the set of all ultra-Morse polynomials of given degree, with the following property: it majorizes the number of real zeros of an Abelian integral of any polynomial one-form of degree smaller than the given one over the real ovals of a real ultra-Morse polynomial. The covering function has poles on the set of non-ultra-Morse polynomials. Once more, the goal is to prove a local persistence theorem for zeros of Abelian integrals and to derive an upper bound for the number of these zeros by replacing the covering function with a cut-off function. This will solve the infinitesimal Hilbert problem. The problems described in the latter part of the proposal (algebraic solutions, rigidity, moduli of elliptic curves, mixed families) lie at the interface of differential equations, complex analysis, and algebraic geometry. While they are of independent interest, at least half of them are related to Hilbert's problem.
The theory of dynamical systems is divided into two parts: multidimensional systems (the realm of chaos) and two-dimensional systems (the realm of order). Hilbert's sixteenth problem is a central one in the theory of two-dimensional systems. It is well known that two-dimensional dynamical systems provide models for various problems in physics, engineering, and biology (e.g., predator-prey models in biology). The understanding of real two-dimensional dynamics is therefore a subject of general scientific interest. On the other hand, the study of complex extensions of real dynamical systems provides important new information about real systems and is interesting in its own right. Indeed, some experts even say that the "Book of Nature" is written in the language of complex analysis.
Intellectual merit The project is dedicated to the study of the Hilbert’s 16th problem and related topics of analysis and geometry. The Hilbert’s 16th problem asks: "What may be said about the number and location of limit cycles of a polynomial differential equation in the plane?" Differential equations describe the evolutionary processes that surround us in the real world. When the process may be described by two parameters, it corresponds to a differential equation in the plane. Limit cycles describe the periodic solutions. According to Poincare, these solutions, like torches, enlighten the global behavior of the system. Polynomial systems are the basic ones. From this the importance of the Hilbert’s problem. The main question related to the problem is: what is an upper estimate of the number of limit cycles for a polynomial differential equation of given degree? This question, in its full generality, remains widely open even for the degree two. Together with his coauthors and students, the PI obtained upper estimates for "almost all" polynomial differential equation of degree 2. Excluded is only a shy set in the space of all such equations. Closely related are geometric questions about polynomial differential equations in the complex domain. Together with his students, the PI established so called rigidity property of such equations of degree two. Roughly speaking, rigidity means the implication: topological equivalence implies the analytic one. For the realm of real differential equations, the topological equivalence is a mild equivalence relation, and the analytic one is very tough. The coincidence of these two kinds of equivalence in the complex domain is a surprising fact. Together with his students, the PI organized a study of a special differential equation on a torus that is important for the theory of superconductivity, and is described on the heuristic level in several physical monographs. Limit cycles of this equation were studied in a rigorous manner, and some effects, not anticipated by physicists, were discovered. Several other results obtained are beyond the scope of this short survey. During the report period, the PI published 18 papers, plus one accepted; his students published under his advisory 23 papers plus one accepted. Broader impact In the reported p[eriod, 17 of the PI’s graduate students defended their PhD thesis, two at Cornell and 15 at the Moscow State University. The PI was an invited speaker at 20 scientific conferences at this period. In addition, he was a main speaker at seven scientific schools: 1 – 6) From 2008 to 2013 the PI organized 6 Summer Schools ``Dynamical Systems'', the last five international. They took place at Stola, Slovakia, June 24 --- July 10, 2009 and June 26 --- July 9, 2010, and in Dubna, Russia, June 23 -- July 7, 2011, June 23 -- July 5, 2012, June 20 -- July 2, 2013 7) "Limit cycles in the complex perspective", 10 lectures in the School of holomorphic foliations and dynamical systems, Mexico-City, Mexico, August 2--6, 2010 and one of the three lecturers at the Spanish Winter School, "Recent Trends in Nonlinear Science", 25--29 January 2010, with a course of ten lectures "Attractors and limit cycles". The PI is a Director of the Math in Moscow program that is supported by the NSF through the AMS. The program makes a reasonable contribution to the exchange or the American and Russian mathematical cultures. The PI is a President of the Independent University of Moscow (IUM), one of the best institution for training future research mathematicians. The IUM is one of the attracting centers of the Moscow mathematical life. Many visitors from the West, in particular, from the US, come there and give talks and crash courses.
