2023 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on current topics in Mathematics K
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Masato Hoshino / Syoiti Ninomiya
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive (本館2階201数学系セミナー室)
- Class
- -
- Course Code
- MTH.E641
- Number of credits
- 200
- Course offered
- 2023
- Offered quarter
- 2Q
- Syllabus updated
- Jul 8, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The main subject of this course is the theory of regularity structures. First, we recall the basics of the rough path theory and introduce its application to stochastic differential equations. Next we introduce the basics of regularity structures as an extension of the rough path theory. Finally we also introduce its application to renormalizations of singular stochastic partial differential equations (SPDEs).
While the theory of regular structures is powerful enough to apply to many singular SPDEs, it is written in a lot of abstract notions and is not easy to comprehend. In this course, we explain basic concepts such as rough paths, Hopf algebras, and renormalizations which are necessary to understand the regularity structures.
Course description and aims
・Be familiar with renormalizations of singular SPDEs
・Understand the essentials of the rough path theory and regularity structures
・Be familiar with algebraic notions required for regularity structures
・Understand applications of regularity structures to singular SPDEs
Keywords
Singular SPDE, regularity structure, rough path, branched rough path, model, modelled distribution, reconstruction, rooted decorated tree, BPHZ model
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The following topics will be covered in this order : -- singular SPDEs -- rough path theory -- branched rough path theory -- a relation between Hopf algebra and branched rough paths -- abstract of the theory of regularity structures -- regularity structures, model, modelled distribution -- reconstruction theorem -- renormalization of models -- renormalization of SPDEs -- BPHZ model -- convergence of BPHZ models | Details will be provided during each class session |
Study advice (preparation and review)
Textbook(s)
None required
Reference books, course materials, etc.
M. Hairer, ""A theory of regularity structures"", Invent, Math. 198 (2014), 269–504.
Y. Bruned, M. Hairer and L. Zambotti, ""Algebraic renormalisation of regularity structures"", Invent. Math, 215 (2019), 1039–1156.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.C341 ： Differential Equations I
- MTH.C342 ： Differential Equations II
Prerequisites
None