Udemy
    •  
    •  
    •  
    •  
    •  
    •  
    •  
    •  
Subscribe
Turn what you know into an opportunity and reach millions around the world.
Learn More
Your cart is empty.
Keep shopping
Riemann surfaces and their applications in integrable system
Rating: 4.9 out of 5(5 ratings)
1,738 students

Riemann surfaces and their applications in integrable system

Riemann surfaces and their applications in integrable system
Created byAndrey Mironov
Last updated 11/2020
English

What you'll learn

  • The theory of Riemann surfaces and its applications in integrable models of mathematical physics.

Course content

1 section • 20 lectures • 1h 59m total length

  • Introduction: Riemann surfaces1:56
  • Holomorphic function23:50
  • De finition of a Riemann surface20:34
  • Examples of Riemann surfaces and Basic theorem of algebra27:46
  • Complex forms on Rimann surfaces. Part 1
  • Complex forms on Riemann surfaces. Part 211:58
  • Residues of meromorphic 1-forms9:20
  • Sum of residues. Part 210:43
  • Sum of residues. Part 2 (Aplications)1:04
  • Divisors on Riemann surfaces12:15
  • Jacobi variety of a compact Riemann surface
  • Jacobi variety
  • Kadomtsev-Petviashvilli equation
  • Abel map
  • Abel theorem
  • Applications Riemann surfaces in integrable systems. Korteweg-de Vries equation
  • Lax representation of Korteweg-de Vries equation
  • Commuting ordinary difierential operators
  • Baker-Akhiezer function
  • Soliton solutions of the Koretweg-de Vries equation
  • The final assignment

Requirements

  • To understand this course, you need to know basic facts from the theory of function of complex variables and calculus

Description

In this course we discuss very interesting and beautiful object - Riemann surfaces. Riemann surfaces have many different applications in integrable systems. And one of our main aim is to explain how Riemann surfaces and their degenerations in singular algebraic curves help to solve problems from geometry and integrable models of mathematical physics. For example, one of such models is a famous Korteweg-de Vries equation:

ut = (6 u uxx +uxxx )/4, u = u(x, t).

This equation describes solitons, that is, solitary water waves in a channel. The theory of Riemann surfaces and its applications in integrable models of mathematical physics.

Sincerely, Andrey Mironov.


Описание курса на русском языке:

В этом курсе мы обсуждаем очень интересные и красивые объекты - римановы поверхности. Римановы поверхности имеют много различных применений в интегрируемых системах. И одна из наших главных целей-объяснить, как римановы поверхности и их вырождения в сингулярных алгебраических кривых помогают решать задачи из геометрии и интегрируемых моделей математической физики. Например, одной из таких моделей является знаменитое уравнение Кортевега-де Фриза:

ut = (6 u uxx +uxxx )/4, u = u(x, t).

Это уравнение описывает солитоны, то есть одиночные волны воды в канале.

Теория римановых поверхностей и ее приложения в интегрируемых моделях математической физики. Мы будем рады видеть Вас на нашем курсе!

Приятного изучения.

С уважением, Андрей Миронов.

Who this course is for:

  • Master's students in geometry and mathematical physics.

Report abuse