• A
  • A
  • A
  • АБB
  • АБB
  • АБB
  • А
  • А
  • А
  • А
  • А
Обычная версия сайта
2019/2020

Научно-исследовательский семинар "Введение в комплексную динамику и аналитическую теорию дифференциальных уравнений"

Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус: Дисциплина общефакультетского пула
Когда читается: 3, 4 модуль
Язык: английский
Кредиты: 3
Контактные часы: 36

Course Syllabus

Abstract

Complex dynamics and analytic theory of ordinary differential equations are situated on crossing of many domains of contemporary mathematics. The analytic theory of ordinary differential equations and its global extension, the theory of holomorphic foliations were born in the first half of XX-th century, in studying the Painlevé equations and the second part of Hilbert 16-th Problem about limit cycles of real planar polynomial vector fields. Studying the 16-th Hilbert Problem led to a lot of important results in local dynamics, normal forms and global properties of holomorphic foliations that became classical. Now holomorphic foliations and complex dynamics are quickly developing areas on the crossing of several domains in mathematics, including dynamical systems, analysis, complex and Riemannian geometry, ergodic theory. For example, both foliations and holomorphic dynamics arise in classification problems in complex geometry and in some problems of mathematical physics. The course will present selected classical results on local dynamics, with an accent on moduli of analytic classification, Stokes phenomena and also applications of Stokes phenomena and holomorphic foliations in other domains of mathematics.
Learning Objectives

Learning Objectives

  • Learning basic theory of analytic normal forms, moduli of analytic classification related to Stokes phenomena: Stokes matrices of germs of linear differential equations at irregular singular points, Ecalle—Voronin moduli of germs one-dimensional conformal mappings tangent to the identity, Martinet—Ramis moduli of germs of two-dimensional holomorphic vector fields at saddle-node singularity.
  • Studying applications in complex dynamics, holomorphic foliations, geometry and real dynamics.
Expected Learning Outcomes

Expected Learning Outcomes

  • To know main definitions and theorems in the theory of analytic normal forms, including linearization and Poincaré—Dulac theorems
  • To know analytic classification theorems in the above-mentioned context of Stokes phenomena
  • To know how to use almost complex structures to prove realization theorems for moduli of analytic classification
  • To know some basic applications of the above-mentioned theory in complex and real dynamics, holomorphic foliations and geometry
Course Contents

Course Contents

  • Germs of holomorphic mappings at fixed points
  • Resonances. Non-resonant formal normal forms for maps
  • Applications of the Stokes phenomena to real dynamics: model of Josephson effect
  • Irregular singularities. Stokes phenomena
  • Resolution of singularities of two-dimensional holomorphic vector fields: Bendixon–Seidenberg theorem without proof
  • Saddle node singularities and their monodromy; parabolic germs of conformal mappings
  • Holomorphic vector fields, singular points
  • Resonances. Non-resonant formal normal forms for vector fields
  • Analytic normal forms of singular (fixed) points with linear parts in the Poincaré domain
  • Linear equations, Fuchsian singularities, normal forms
  • The Riemann–Hilbert problem
  • Analytic classification of parabolic germs. Ecalle–Voronin moduli
  • Analytic classification of saddle-node singularities of holomorphic vector fields: Martinet–Ramis moduli
  • One-dimensional holomorphic foliations. Main conjectures on minimal sets and topology of leaves
  • Density of leaves of a generic foliation with an invariant line
  • A very new striking unexpectable result by Alvarez and Deroin: an example of structurally stable foliation on complex projective plane without dense leaves
  • Simultaneous uniformization of leaves
Assessment Elements

Assessment Elements

  • non-blocking problem solutions
  • non-blocking final exam
    oral
Interim Assessment

Interim Assessment

  • Interim assessment (4 module)
    0.7 * final exam + 0.3 * problem solutions
Bibliography

Bibliography

Recommended Core Bibliography

  • llyashenko, Y., & Yakovenko, S. (2008). Lectures on Analytic Differential Equations. Providence, Rhode Island: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=971274

Recommended Additional Bibliography

  • Milnor, J. W. (1990). Dynamics in one complex variable: introductory lectures. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.math%2f9201272