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2016　Special Lecture on Science in English (Mathematics 3)

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Tonegawa Yoshihiro 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Intensive (H201)  
Group
-
Course number
MTH.E441
Credits
1
Academic year
2016
Offered quarter
3Q
Syllabus updated
2016/9/20
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

This course is an introduction to the Monge-Ampere equation. After introducing basic concepts such as normal mapping and Monge-Ampere measure, we will study important properties of the Monge-Ampere measure including: weak continuity and invariance property. We then study the celebrated Aleksandrov's maximum principle, the comparison principle, John's lemma and applications. Finally, we will discuss the Dirichlet problem and sections of convex solutions to the Monge-Ampere equation.
The Monge-Ampere equation appears in many areas and applications including affine geometry, convex geometry, optimal transportation and meteorology. The Monge-Ampere equation and its applications is a very active area of research. This course hopes to provide solid background and motivate interested students entering this research area.

Student learning outcomes

・Be familiar with basic concepts in the Monge-Ampere equation such as normal mapping, Monge-Ampere measure, Aleksandrov's solution
・Be familiar with modern tools and concepts in the Monge-Ampere theory such as John's lemma and sections of solutions
・Understand and be able to use maximum principles in the Monge-Ampere equation

Keywords

normal mapping, Monge-Ampere measure, weak continuity, invariance property, Aleksandrov's solution, Aleksandrov's maximum principle, comparison principle, John's lemma, Dirichlet problem, sections of convex functions.

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

This is a standard lecture course. Homework will be assigned every week.

Course schedule/Required learning

  Course schedule Required learning
Class 1 The Monge-Ampere equation in different contexts; normal mapping, Monge-Ampere measure, Legendre transform Details will be provided during each class
Class 2 Aleksandrov's solution, Examples, Weak continuity of Monge-Ampere measure, Details will be provided during each class
Class 3 Invariances of the Monge-Ampere equation, maximum principles Details will be provided during each class
Class 4 Aleksandrov's maximum principle Details will be provided during each class
Class 5 John's lemma, comparison principle Details will be provided during each class
Class 6 The Dirichlet's problem: uniqueness and solvability by the Perron method Details will be provided during each class
Class 7 Sections of convex functions Details will be provided during each class
Class 8 Geometric properties of sections of solutions to the Monge-Ampere equation Details will be provided during each class

Textbook(s)

None required

Reference books, course materials, etc.

The course will be based on Part 3 of the instructor's lecture notes “N. Q. Le, The second boundary value problem of the prescribed affine mean curvature equation and related linearized Monge-Ampere equation”, available at: http://pages.iu.edu/~nqle/SBVP_lectures.pdf

Assessment criteria and methods

Final exam 50%, assignments 50%.

Related courses

  • MTH.C305 ： Real Analysis I
  • MTH.C351 ： Functional Analysis
  • MTH.C305 ： Real Analysis I
  • MTH.C351 ： Functional Analysis

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Basic knowledge on advanced analysis is essential

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