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National Research University Higher School of Economics

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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)...

立即开始，按照自己的计划学习。

根据您的日程表重置截止日期。

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

字幕：English...

立即开始，按照自己的计划学习。

根据您的日程表重置截止日期。

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

字幕：English...

Week

1Welcome 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...

1 个视频（共 14 分钟）, 5 个阅读材料

About the Course14分钟

Pre-Course Survey10分钟

Course Overview10分钟

Grading and Logistics10分钟

Suggested Readings10分钟

About the Instructor10分钟

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....

5 个视频（共 62 分钟）, 1 个测验

Motivations12分钟

Theta-function12分钟

Modular and abelian transformations12分钟

Pullbacks of theta-function12分钟

Modular forms12分钟

Week

2This 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....

6 个视频（共 75 分钟）, 1 个测验

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

3This 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....

6 个视频（共 86 分钟）, 1 个测验

Heisenberg group13分钟

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

4This 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....

6 个视频（共 76 分钟）, 1 个测验

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

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...

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