授業の目的 【日本語】 Goals of the Course(JPN)   The title of this course is "Modular forms and their combinatorial variants". Modular forms are functions appearing in several areas of mathematics as well as mathematical physics. There are two cardinal points about them which explain why they are interesting. First of all, the space of modular forms of a given weight is finite dimensional and algorithmically computable. Secondly, modular forms occur naturally in connection with problems arising in many areas of mathematics. Together, these two facts imply that modular forms have a huge number of applications. In this course, we will start with a basic introduction to modular forms of arbitrary level. In the second part of the course we will explain current research themes around combinatorial and formal variants of modular forms. 


授業の目的 【英語】 Goals of the Course   The title of this course is "Modular forms and their combinatorial variants". Modular forms are functions appearing in several areas of mathematics as well as mathematical physics. There are two cardinal points about them which explain why they are interesting. First of all, the space of modular forms of a given weight is finite dimensional and algorithmically computable. Secondly, modular forms occur naturally in connection with problems arising in many areas of mathematics. Together, these two facts imply that modular forms have a huge number of applications. In this course, we will start with a basic introduction to modular forms of arbitrary level. In the second part of the course we will explain current research themes around combinatorial and formal variants of modular forms. 


到達目標 【日本語】 Objectives of the Course(JPN))   The objectives of this course is to give the students the fundamental knowledge and basic tools to use modular forms from the perspective of number theory. The first part is following a standard textbook style introduction to modular forms and the second part gives a unique combinatorial interpretation of them closely related to new and current research. 


到達目標 【英語】 Objectives of the Course   The objectives of this course is to give the students the fundamental knowledge and basic tools to use modular forms from the perspective of number theory. The first part is following a standard textbook style introduction to modular forms and the second part gives a unique combinatorial interpretation of them closely related to new and current research. This will prepare students to have basic knowledge in this area but also to have insight on active research, which can be used to inspire their own research projects. 


授業の内容や構成 Course Content / Plan   We will cover at least the following topics: The action of the modular group on the complex upper halfplane and modular forms, Eisenstein series and their Fourier expansion, Cusp forms and Ramanujan's Delta function, The space of modular and its dimension, Modular forms of level N, Period polynomials, Connection to double zeta values, Combinatorial proofs for idendities among Eisenstein series.
An updated and more detailed course plan will appear on https://www.henrikbachmann.com/mf_2023.html shortly before the semester starts. 


履修条件 Course Prerequisites   Basic knowledge of (linear) algebra, calculus and complex analysis are expected. 


関連する科目 Related Courses   Some talks in the small class seminar of the instructor might be related to the course. If this is the case, then it will be announced during the lecture. 


成績評価の方法と基準 Course Evaluation Method and Criteria   The evaluation will be based on homework assignments. 


教科書・テキスト Textbook   We will not follow a specific textbook but will provide lecture notes. A list of references will be provided, but the students do not need to purchase any books. 


参考書 Reference Book   A list of reference and lecture notes will be provided at the beginning of the course. 


課外学習等（授業時間外学習の指示） Study Load(Selfdirected Learning Outside Course Hours)   There will be homework assignments every 23 weeks. The students are expected to recall the previous lecture before attending the lecture. 


注意事項 Notice for Students   Please check https://www.henrikbachmann.com/mf_2023.html for updated informations. 


他学科聴講の可否 Propriety of Other department student's attendance   

他学科聴講の条件 Conditions of Other department student's attendance   

レベル Level   

キーワード Keyword   

履修の際のアドバイス Advice   

授業開講形態等 Lecture format, etc.   The lecture will be facetoface. The language used will be English. 


遠隔授業（オンデマンド型）で行う場合の追加措置 Additional measures for remote class (ondemand class)   All information on this course will be available on the course homepage: https://www.henrikbachmann.com/mf_2023.html 

