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高級

完成時間大約為54 小時

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

英語(English)

字幕:英語(English)

100% 在線

立即開始,按照自己的計劃學習。

可靈活調整截止日期

根據您的日程表重置截止日期。

高級

完成時間大約為54 小時

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

英語(English)

字幕:英語(English)

教學大綱 - 您將從這門課程中學到什麼

1
完成時間為 1 小時

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...
1 個視頻 (總計 14 分鐘), 4 個閱讀材料
1 個視頻
4 個閱讀材料
Course Overview10分鐘
Grading and Logistics10分鐘
Suggested Readings10分鐘
About the Instructor10分鐘
完成時間為 3 小時

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....
5 個視頻 (總計 62 分鐘), 1 個測驗
5 個視頻
Motivations12分鐘
Theta-function12分鐘
Modular and abelian transformations12分鐘
Pullbacks of theta-function12分鐘
Modular forms12分鐘
2
完成時間為 3 小時

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....
6 個視頻 (總計 75 分鐘), 1 個測驗
6 個視頻
Definition of Jacobi forms (part 2)13分鐘
Basic properties of Jacobi forms12分鐘
Jacobi modular group11分鐘
Symplectic group11分鐘
Jacobi modular group (part 2)13分鐘
3
完成時間為 3 小時

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....
6 個視頻 (總計 86 分鐘), 1 個測驗
6 個視頻
The action of Jacobi modular group14分鐘
The action of Jacobi modular group (part 2)10分鐘
The second definition of Jacobi forms15分鐘
Special values of Jacobi forms11分鐘
The first theorem20分鐘
4
完成時間為 3 小時

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....
6 個視頻 (總計 76 分鐘), 1 個測驗
6 個視頻
The zeros of Jacobi forms (part 2)11分鐘
Taylor expansion of Jacobi forms12分鐘
Taylor expansion of Jacobi forms (part 2)13分鐘
Dimensions of some spaces of Jacobi forms10分鐘
Examples of Jacobi modular forms14分鐘
5
完成時間為 3 小時

The Jacobi theta-series as Jacobi modular form. The basic Jacobi modular forms

This module is devoted to detailed study of Jacobi theta-series. We will discuss abelian, modular and some other properties of this function. Also there is a peer review in the end of this module....
6 個視頻 (總計 84 分鐘), 1 個測驗
6 個視頻
Properties of Jacobi theta-function9分鐘
Properties of Jacobi theta-function (part 2)13分鐘
Properties of Jacobi theta-function (part 3)15分鐘
Construction of Jacobi forms using theta-series15分鐘
Construction of Jacobi forms using theta-series (part 2)13分鐘
6
完成時間為 4 小時

Theta-blocks, theta-quarks and the first Jacobi cusp form of weight 2

This module is devoted to very important notion of theta-blocks and theta-quarks. In this module we also construct the first Jacobi form of weight 2. Also there is a peer review in the end of this module....
7 個視頻 (總計 96 分鐘), 1 個測驗
7 個視頻
Theta-blocks14分鐘
Theta-blocks (part 2)13分鐘
Theta-quarks14分鐘
Dimension of the space of Jacobi forms of odd weight14分鐘
Theta-quarks (part 2)11分鐘
The proof of theorem about theta-quarks16分鐘
The first Jacobi cusp form of weight 210分鐘
7
完成時間為 4 小時

Jacobi forms in many variables and the Eichler-Zagier Jacobi forms

This module is devoted to Jacobi forms in many variables. In this module we also define classical Eichler-Zagier Jacobi forms in terms of Jacobi forms in many variables. Also there is a peer review in the end of this module....
6 個視頻 (總計 93 分鐘), 1 個測驗
6 個視頻
Jacobi theta-series and Jacobi forms in many variables (part 2)15分鐘
Hyperbolic reformulation14分鐘
Examples of Jacobi forms in many variables14分鐘
Examples of Jacobi forms in many variables (part 2)16分鐘
Examples of Jacobi forms in many variables (part 3)16分鐘
8
完成時間為 3 小時

Jacobi forms in many variables and the splitting principle. Theta-quarks as a pull-back. Weak Jacobi forms in many variables

In this module we continue studying Jacobi forms in many variables. Among other things we discuss splitting principle and realize theta-quarks as a pull-back. Also there is a peer review in the end of this module....
6 個視頻 (總計 90 分鐘), 1 個測驗
6 個視頻
Theta-quarks as pull-backs of Jacobi forms (part 2)16分鐘
Theta-quarks as pull-backs of Jacobi forms (part 3)13分鐘
The splitting principle15分鐘
The splitting principle (part 2)16分鐘
Weak Jacobi forms in many variables13分鐘
9
完成時間為 3 小時

The Weil representation and vector valued modular forms. Jacobi forms of singular weight

This module is devoted to very useful notion of the Weil representation and vector-valued modular forms. In this module we also define Jacobi forms of singular weight. Also there is a peer review in the end of this module....
6 個視頻 (總計 85 分鐘), 1 個測驗
6 個視頻
The Weil representation (part 2)15分鐘
Jacobi forms of singular weight14分鐘
Jacobi forms of singular weight (part 2)15分鐘
Jacobi forms of critical weight15分鐘
Jacobi forms of critical weight (part 2)12分鐘
10
完成時間為 3 小時

Quasi-modular Eisenstein series. The automorphic correction of Jacobi forms and Taylor expansions

This module is devoted to Quasi-modular Eisenstein series. In this module we also define the automorpic correction of Jacobi forms and its Taylor expansion that gives us the way to construct the series of Jacobi forms. Also there is a peer review in the end of this module....
6 個視頻 (總計 84 分鐘), 1 個測驗
6 個視頻
Differential operator D15分鐘
Quasi-modular Eisenstein series15分鐘
Automorphic correction and Taylor expansions13分鐘
Automorphic correction of Jacobi forms in many variables12分鐘
Automorphic correction of Jacobi forms in many variables (part 2)12分鐘
11
完成時間為 3 小時

Modular differential operators. The graded ring of the weak Jacobi modular forms

This module is devoted to Modular differential operators. In this module we also consider the Jacobi forms as the space with the structure of the bigraded ring. Also there is a peer review in the end of this module....
6 個視頻 (總計 90 分鐘), 1 個測驗
6 個視頻
Heat operator14分鐘
Heat operator (part 2)15分鐘
Applications of differential operators for Jacobi forms11分鐘
The graded ring of the weak Jacobi modular forms17分鐘
Jacobi type forms in many variables14分鐘
12
完成時間為 3 小時

Jacobi type forms and the generalisation of the Cohen-Kuznetsov-Zagier operator

The last module is devoted to Jacobi type forms. In this module we also consider the generalisation of the Cohen-Kuznetsov-Zagier operator. Also there is a peer review in the end of this module....
4 個視頻 (總計 52 分鐘), 1 個測驗
4 個視頻
A new proof of Cohen-Kuznetsov-Zagier theorem13分鐘
A new proof of Cohen-Kuznetsov-Zagier theorem (part 2)9分鐘
Cohen-Kuznetsov-Zagier theorem in case of many variables14分鐘

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Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

關於 国立高等经济大学

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 communicamathematics, engineering, and more. Learn more on www.hse.ru...

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