Number theory II.b - selected topics

Spring Semester

Course: Tuesdays 8:15-10:00 in room DIA 003

Exercices: Tuesdays 10:15-12:00 in room DIA 003

Instructor: philippe.michel@epfl.ch

Assistant: petru.constantinescu@epfl.ch

This year's topic is "Additive combinatorics" on how in  commutative (and non-commutative) groups, subsets growth under the application of the group law.

Besides the lectures, an active participation in discussions with the exercise sheets is also expected.

Zoom link when the course is on Zoom

https://epfl.zoom.us/j/62029476057

Next Zoom course: TBA

The recordings of the course will be available on a dedicated EPFL playlist:

MATH-417 Topics in number theory 2025

This week we discuss the Cauchy-Davenport Theorem.

Fourier analysis on finite commutative group cont'd.

Equidistribution modulo 1.

Weyl's criterion for equidistribution.

Equidistribution modulo 1 of small multiplicative subgroups of ${\mathbb F}_p^\times$

Growth in groups.

Approximate subgroups. Ruzsa distance.  Ruzsa controlled growth Lemma. Ruzsa covering Lemma.

Growth in groups. Pluennecke formula.

Energy vs Growth: the BGS theorems. The Haar measure notation.

Energy and growth: the BSG theorems (set theoretic and approximate group)

The subgroup recognition criterion and set theoretic BSG implies approximate group BSG

The BSG Theorem proof of the set-theoretic version

The Sum-Product Thm

Rough notations, Rusza calculus and warming-up

This week topic: Lattices and applications

This week's course will be on Zoom

https://epfl.zoom.us/j/62879204295

 Meeting ID: 628 7920 4295

Course Recording Part 1

Course Recording Part 2

Proof of the approximate version of the SL_2 product theorem.

Basics in Algebraic Geometry, algebraic varieties, linear algebraic groups

SL_2 versions of the Larsen-Pink inequalities

This week's the course has been recorded

Course Recording Part 1

Course Recording Part 2

Larsen-Pink inequalities for finite subgroups of SL_2 in special cases:

tori, unipotent subgroups and conjugacy classes.

Larsen-Pink inequalities for Conjugacy classes (end of Proof). Further LP inequalities.

Larsen-Pink inequalities for generating approximate subgroups: the "escape from Borels" wildcard.

LP inequalities for approximate subgroups (tori and conjugacy classes)

The "involved tori" dichotomy

Proof of Helfgott's product theorem.

Basics on graphs and their spectral properties.

Definition of expander graphs.

Mixing properties of expander graphs.

Expansion for Cayley graphs.

The Bourgain-Gamburd  expansion machine.