From December 12 through December 20, 2022, the following special lecture by Professor Katrin Wendland (Trinity College Dublin, Ireland) will be delivered as part of the international educational activities of the Multiscale Analysis, Modelling and Simulation Unit.
Dates
- December 12, 2022: 14:45-16:15 + 16:30-18:00
- December 13, 2022: 14:45-16:15 + 16:30-18:00
- December 14, 2022: 14:45-16:15 + 16:30-18:00
- December 15, 2022: 14:45-16:15 + 16:30-18:00
- December 19, 2022: 14:45-16:15 + 16:30-18:00
- December 20, 2022: 14:45-16:15 + 16:30-18:00
Venue
Room 1-07A, Bldg. 62W, Nishi-Waseda Campus, Waseda University
Lecturer
Professor Katrin Wendland (Trinity College Dublin, Ireland)
Prof. Dr. Wendland is well known for her work on the relations between geometry and quantum field theory, in particular singularity theory and conformal field theory. In recent work she has investigated the role of K3 surfaces for a special class of conformal field theories with extended supersymmetry. Symmetries and dualities of these theories have been central to her work, as well as the role of modular forms and their cousins, in particular the complex elliptic genus. She is an excellent expositor, and in these lectures she will give an introduction to modular forms and their applications in representation theory and conformal field theory.
Summary of topics
Modular forms are special holomorphic functions on the complex upper half-plane. They are characterized by a functional equation which depends on an integer k, called the weight. More precisely, we have a natural action of the so-called modular group by integral Möbius transforms on the upper half-plane; the characterizing functional equation for a modular form F then implies that F naturally defines a tensor field on the complex upper-half plane which is invariant under the action of the modular group. Modular functions have beautiful properties – for example, the space of modular forms of a given weight is finite dimensional. Modular forms are naturally connected with many areas in mathematics and physics, including the theory of lattices and quadratic forms, various problems in number theory, and conformal field theory.
The course will give an elementary introduction to modular forms, beginning with the modular group and its natural action on the complex upper half-plane. We will interpret the fundamental domain of this action in terms of biholomorphism classes of elliptic curves. This will naturally lead to the definition of modular forms. We will see that the space of modular forms of a given weight is finite dimensional, by calculating this space explicitly. We will also study concrete examples of modular forms and their properties in detail, including the so-called Eisenstein series and the discriminant function. We will investigate some of their beautiful number theoretic properties, and we will take a look at how these examples feature as building blocks of characters in conformal field theory.
Language
English
Audience
Students/Graduate Students, Researchers
Participation
Free
Registration
Not required
Contact
Martin Guest
Professor, Faculty of Science and Engineering, Waseda University
martin [at] waseda.jp
