Number theory II.c - Selected topics

Summary

This year's topic is "Addictive combinatorics." We will introduce various methods from additive combinatorics, establish sum-product type theorems over finite fields and derive various applications (bounds for exponential sums, construction of expander graphs).

Content

Learning Prerequisites

Required courses

MATH-313 (Introduction to Analytic Number Theory).

MATH-337 (Combinatorial number theory).

Recommended courses

MATH-314 (Representation Theory I - Finite Groups)

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 + Exercise sessions

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

Resources

Virtual desktop infrastructure (VDI)

No

Bibliography

H. Iwaniec and E. Kowalski: Analytic Number Theory, Colloquium Publ. 53, A.M.S, 2004.

Kowalski, Emmanuel. An introduction to expander graphs.Cours Spéc., 26. Société Mathématique de France, Paris, 2019. x+276 pp.

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.

Moodle Link

• https://go.epfl.ch/MATH-417