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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Bez Richard Neal Kinoshita Shinya
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- -
- Group
- -
- Course number
- MTH.E436
- Credits
- 2
- Academic year
- 2024
- Offered quarter
- 4Q
- Syllabus updated
- 2024/3/14
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
The main theme of the course is the Brascamp-Lieb inequality on euclidean spaces. Firstly, concrete examples such as Holder’s inequality, Young’s convolution inequality and the Loomis-Whitney inequality will be introduced, as well as some preliminaries on L^p spaces and interpolation theory for linear operators. After this, the general theory of the Brascamp-Lieb inequality will be discussed, including Lieb’s theorem and a characterisation of the finiteness of the Brascamp-Lieb constant due to J. Bennett, A. Carbery, M. Christ and T. Tao. As part of the general theory, a heat-flow proof of the geometric Brascamp-Lieb inequality will be studied.
The Brascamp-Lieb inequality has recently had a major influence on areas of mathematics such as convex geometry, harmonic analysis, geometric measure theory and number theory, but the main objective of this course is to obtain a deep understanding of the fundamental theory of the Brascamp-Lieb inequality.
Student learning outcomes
・Be able to handle special cases of the Brascamp-Lieb inequality
・Understand the general theory of the Brascamp-Lieb inequality
・Understand the role of the geometric Brascamp-Lieb inequality
・Understand how to prove Lieb’s theorem using heat-flow and factorisation of the Brascamp-Lieb constant
・Understand how to prove a characterisation of finiteness of the Brascamp-Lieb constant due to Bennett-Carbery-Christ-Tao
Keywords
Multilinear inequalities, Holder’s inequality, Young’s convolution inequality, Loomis-Whitney inequality, Brascamp-Lieb inequality, heat-flow
Competencies that will be developed
| ✔ 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/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | The following topics will be covered in this order : ・L^p spaces and an interpolation theorem ・Holder’s inequality ・Young’s convolution inequality ・Loomis-Whitney inequality ・Brascamp-Lieb inequality ・Lieb’s theorem ・Bennett-Carbery-Christ-Tao characterisation of finiteness ・Geometric Brascamp-Lieb inequality ・Heat-flow ・A heat-flow proof of the geometric Brascamp-Lieb inequality ・Proof of Lieb’s theorem by Bennett-Carbery-Christ-Tao ・Proof of Bennett-Carbery-Christ-Tao characterisation of finiteness | Details will be provided during each class session. |
Out-of-Class Study Time (Preparation and Review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required
Reference books, course materials, etc.
J. Bennett, A. Carbery, M. Christ, T. Tao, The Brascamp-Lieb inequalities: finiteness, structure and extremals, Geometric and Functional Analysis, Vol. 17 (2008), pp. 1343-1415
(arXiv version: https://arxiv.org/abs/math/0505065)
Assessment criteria and methods
Assignments (100%).
Related courses
- MTH.C351 ： Functional Analysis
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
- MTH.C341 ： Differential Equations I
- MTH.C342 ： Differential Equations II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None
