Number theory II.b - selected topics
MATH-417 / 5 credits
Teacher: Michel Philippe
Language: English
Remark: Cours donné en alternance tous les deux ans
Summary
This year's topic is "Additive combinatorics and applications." We will introduce various methods from additive combinatorics, establish the sum-product theorem over finite fields and derive various applications (exponential sums, Cayley graphs attached to algebraic groups, etc...).
Content
This year we will discuss various techniques from additive combinatorics mostly in the context of finite fields.
After introducing several general techniques and results (Rusza calculus, the Balog-Gowers-Szmeredy theorem, ...), we will establish the sum-product phenomenon discovered by Bourgain-Katz-Tao in the context of finite fields and will derive several applications.
These will include:
The study of the mixing properties of certain Cayley graphs for some algebraic group over finite fields due to Helfgott and Bourgain-Gamburd.
Bounds for exponential sums along very small subgroups of the multiplicative group of finite fields due to Bourgain-Gilibichuk-Konyagin.
Keywords
Product sets
Characters of finite abelian groups
Arithmetic progressions
Approximate subgroup
Rusza calculus
Sum-product phenomenon
Learning Prerequisites
Required courses
MATH-313: Introduction to Analytic Number Theory.
MATH-337: Combinatorial number theory
Recommended courses
- Some knowledge of modular forms (such as MATH-511 "Modular forms and applications" ) may be useful.
Learning Outcomes
By the end of the course, the student must be able to:
- Demonstrate a good mastery of the basics of additive combinatorics
- Solve basic problems of additive combinatorics
Transversal skills
- Access and evaluate appropriate sources of information.
- Make an oral presentation.
- Demonstrate the capacity for critical thinking
Teaching methods
Ex-Cathedra Course
Expected student activities
We expect a proactive attitude during the courses and the exercises sessions (possibly with individual presentation of the solution of various problems).
Assessment methods
Oral Exam
Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.
Supervision
| Office hours | No |
| Assistants | Yes |
| Forum | No |
| Others | a moodle with ressources for the course will be maintained |
Resources
Virtual desktop infrastructure (VDI)
No
Bibliography
H. Iwaniec and E. Kowalski: Analytic Number Theory, Colloquium Publ. 53, A.M.S, 2004.
K. Soundararajan: Finite fields, with applications to combinatorics, Student Math. Library 99, American Math. Soc., 2022.
T. Tao and V. Vu: Additive combinatorics, Cambridge Studies in Advanced Math. 105, Cambridge Univ. Press, 2006.
Notes/Handbook
Typed lecture notes will be made available as the course progress.
Prerequisite for
Current research in number theory
In the programs
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Number theory II.b - selected topics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Number theory II.b - selected topics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Number theory II.b - selected topics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional