Informações sobre o curso
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Aprox. 54 horas para completar

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

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Legendas: Inglês

100% online

Comece imediatamente e aprenda em seu próprio cronograma.

Prazos flexíveis

Redefinir os prazos de acordo com sua programação.

Nível avançado

Aprox. 54 horas para completar

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

Inglês

Legendas: Inglês

Programa - O que você aprenderá com este curso

Semana
1
1 hora para concluir

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 vídeo (total de (Total 14 mín.) min), 4 leituras
1 vídeos
4 leituras
Course Overview10min
Grading and Logistics10min
Suggested Readings10min
About the Instructor10min
3 horas para concluir

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 vídeos (total de (Total 62 mín.) min), 1 teste
5 videos
Theta-function12min
Modular and abelian transformations12min
Pullbacks of theta-function12min
Modular forms12min
Semana
2
3 horas para concluir

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 vídeos (total de (Total 75 mín.) min), 1 teste
6 videos
Definition of Jacobi forms (part 2)13min
Basic properties of Jacobi forms12min
Jacobi modular group11min
Symplectic group11min
Jacobi modular group (part 2)13min
Semana
3
3 horas para concluir

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 vídeos (total de (Total 86 mín.) min), 1 teste
6 videos
The action of Jacobi modular group14min
The action of Jacobi modular group (part 2)10min
The second definition of Jacobi forms15min
Special values of Jacobi forms11min
The first theorem20min
Semana
4
3 horas para concluir

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 vídeos (total de (Total 76 mín.) min), 1 teste
6 videos
The zeros of Jacobi forms (part 2)11min
Taylor expansion of Jacobi forms12min
Taylor expansion of Jacobi forms (part 2)13min
Dimensions of some spaces of Jacobi forms10min
Examples of Jacobi modular forms14min
Semana
5
3 horas para concluir

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 vídeos (total de (Total 84 mín.) min), 1 teste
6 videos
Properties of Jacobi theta-function9min
Properties of Jacobi theta-function (part 2)13min
Properties of Jacobi theta-function (part 3)15min
Construction of Jacobi forms using theta-series15min
Construction of Jacobi forms using theta-series (part 2)13min
Semana
6
4 horas para concluir

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 vídeos (total de (Total 96 mín.) min), 1 teste
7 videos
Theta-blocks (part 2)13min
Theta-quarks14min
Dimension of the space of Jacobi forms of odd weight14min
Theta-quarks (part 2)11min
The proof of theorem about theta-quarks16min
The first Jacobi cusp form of weight 210min
Semana
7
4 horas para concluir

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 vídeos (total de (Total 93 mín.) min), 1 teste
6 videos
Jacobi theta-series and Jacobi forms in many variables (part 2)15min
Hyperbolic reformulation14min
Examples of Jacobi forms in many variables14min
Examples of Jacobi forms in many variables (part 2)16min
Examples of Jacobi forms in many variables (part 3)16min
Semana
8
3 horas para concluir

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 vídeos (total de (Total 90 mín.) min), 1 teste
6 videos
Theta-quarks as pull-backs of Jacobi forms (part 2)16min
Theta-quarks as pull-backs of Jacobi forms (part 3)13min
The splitting principle15min
The splitting principle (part 2)16min
Weak Jacobi forms in many variables13min
Semana
9
3 horas para concluir

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 vídeos (total de (Total 85 mín.) min), 1 teste
6 videos
The Weil representation (part 2)15min
Jacobi forms of singular weight14min
Jacobi forms of singular weight (part 2)15min
Jacobi forms of critical weight15min
Jacobi forms of critical weight (part 2)12min
Semana
10
3 horas para concluir

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 vídeos (total de (Total 84 mín.) min), 1 teste
6 videos
Differential operator D15min
Quasi-modular Eisenstein series15min
Automorphic correction and Taylor expansions13min
Automorphic correction of Jacobi forms in many variables12min
Automorphic correction of Jacobi forms in many variables (part 2)12min
Semana
11
3 horas para concluir

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 vídeos (total de (Total 90 mín.) min), 1 teste
6 videos
Heat operator14min
Heat operator (part 2)15min
Applications of differential operators for Jacobi forms11min
The graded ring of the weak Jacobi modular forms17min
Jacobi type forms in many variables14min
Semana
12
3 horas para concluir

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 vídeos (total de (Total 52 mín.) min), 1 teste
4 videos
A new proof of Cohen-Kuznetsov-Zagier theorem13min
A new proof of Cohen-Kuznetsov-Zagier theorem (part 2)9min
Cohen-Kuznetsov-Zagier theorem in case of many variables14min

Instrutores

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Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

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

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