课程信息
4.7
11 个评分
3 个审阅
This is a master course given in Moscow at the Laboratory of Algebraic Geometry of the National Research University Higher School of Economics by Valery Gritsenko, a professor of University Lille 1, France. Jacobi forms are holomorphic functions in two complex variables. They are modular in one variable and abelian (or double periodic) in another variable. The theory of Jacobi modular forms became an independent research subject after the famous book of Martin Eichler and Don Zagier “Jacobi modular forms” (Progress in Mathematics, vol. 55, 1985) which was cited more than a thousand times in research papers. This is due to many applications of Jacobi forms in arithmetic, topology, algebraic and differential geometry, mathematical and theoretical physics, in the theory of Lie algebras, etc. The list of mentioned subjects shows that my course might be useful for master and Ph.D. students working in different directions. Motivated undergraduate students can also study this subject. To follow the course one has to know only elementary basic facts from the theory of modular forms (for example, the paragraphs 1-4 of the chapter VII of Serre’s “A Course in Arithmetic” are enough). The main hero of the course is the Jacobi theta-series. Using it we will construct a lot of concrete examples of Jacobi forms in one or many abelian variables, in particular, Jacobi forms for root systems. For some of you, who will be successful with the theoretical exercises of the course, I am ready to formulate research problems for Master or Ph.D. thesis. (Ph.D. support might be available at CEMPI in Lille or at the Faculty of Mathematics of National Research University Higher School of Economics in Moscow)...
Globe

100% 在线课程

立即开始,按照自己的计划学习。
Calendar

可灵活调整截止日期

根据您的日程表重置截止日期。
Advanced Level

高级

Clock

Approx. 53 hours to complete

建议:12 weeks of study, 3-5 hours per week...
Comment Dots

English

字幕:English...
Globe

100% 在线课程

立即开始,按照自己的计划学习。
Calendar

可灵活调整截止日期

根据您的日程表重置截止日期。
Advanced Level

高级

Clock

Approx. 53 hours to complete

建议:12 weeks of study, 3-5 hours per week...
Comment Dots

English

字幕:English...

教学大纲 - 您将从这门课程中学到什么

Week
1
Clock
完成时间为 1 小时

Introduction to the Course

Welcome to the course! I hope you have an opportunity to reserve some time to explore the course content, course logic and our grading policy. The course consists of 12 lectures. This course will help you to start your progress in the field of the theory of Jacobi modular forms. Best regards, Valery Gritsenko...
Reading
1 个视频(共 14 分钟), 5 个阅读材料
Video1 个视频
Reading5 个阅读材料
Pre-Course Survey10分钟
Course Overview10分钟
Grading and Logistics10分钟
Suggested Readings10分钟
About the Instructor10分钟
Clock
完成时间为 3 小时

Jacobi modular forms: motivations

This module is devoted to motivations to study Jacobi forms. We provide some first examples including theta-functions. Also there is a peer review in the end of this module....
Reading
5 个视频(共 62 分钟), 1 个测验
Video5 个视频
Motivations12分钟
Theta-function12分钟
Modular and abelian transformations12分钟
Pullbacks of theta-function12分钟
Modular forms12分钟
Week
2
Clock
完成时间为 3 小时

Jacobi modular forms: the first definition

This module is devoted to the first definition of Jacobi forms. In this module we also define Jacobi modular group. Also there is a peer review in the end of this module....
Reading
6 个视频(共 75 分钟), 1 个测验
Video6 个视频
Definition of Jacobi forms (part 2)13分钟
Basic properties of Jacobi forms12分钟
Jacobi modular group11分钟
Symplectic group11分钟
Jacobi modular group (part 2)13分钟
Week
3
Clock
完成时间为 3 小时

Jacobi modular group and the second definition of Jacobi forms. Special values of Jacobi modular forms

This module is devoted to the second definition of Jacobi forms. In this module we also consider special values of Jacobi forms. Also there is a peer review in the end of this module....
Reading
6 个视频(共 86 分钟), 1 个测验
Video6 个视频
The action of Jacobi modular group14分钟
The action of Jacobi modular group (part 2)10分钟
The second definition of Jacobi forms15分钟
Special values of Jacobi forms11分钟
The first theorem20分钟
Week
4
Clock
完成时间为 3 小时

Zeros of Jacobi forms. The Jacobi theta-series, the Dedekind eta-function and the first examples of Jacobi modular forms

This module is devoted to zeros of Jacobi modular forms, their Taylor extensions and the first examples of Jacobi forms. Using classical Jacobi theta-series and Dedekind eta-function we construct a series of Jacobi forms. Also there is a peer review in the end of this module....
Reading
6 个视频(共 76 分钟), 1 个测验
Video6 个视频
The zeros of Jacobi forms (part 2)11分钟
Taylor expansion of Jacobi forms12分钟
Taylor expansion of Jacobi forms (part 2)13分钟
Dimensions of some spaces of Jacobi forms10分钟
Examples of Jacobi modular forms14分钟

讲师

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

关于 National Research University Higher School of Economics

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communications, IT, mathematics, engineering, and more. Learn more on www.hse.ru...

常见问题

  • Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.

  • When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.

还有其他问题吗?请访问 学生帮助中心