Number theory II.b - selected topics

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

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

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