Interactions of harmonic analysis, combinatorics and number theory

Lecturers

Joaquim Ortega-Cerdà (BGSMath/UB)
Email: jortega@ub.edu

Juanjo Rué (BGSMath/UPC)
Email: juan.jose.rue@upc.edu

Jorge Jímenez (BGSMath/UPC)
Email: jjimenez@ma4.upc.edu

Oriol Serra (BGSMath/UPC)
Email: juan.jose.rue@upc.edu

Pablo Candela (UAM)
Email: pablo.candela@uam.es

Schedule

25th April-19th May (24 hours)

Week 1: Joaquim Ortega-Cerdà: Harmonic analysis and applications

Week 2: Juanjo Rué: Interactions of harmonic analysis and combinatorics

Week 3: Jorge Jímenez: Interactions of harmonic analysis and analytic number theory

Week 4: Oriol Serra i Pablo Candela: Interactions of harmonic analysis and combinatorial number theory

Location

Universitat de Barcelona (UB)


Summary

The main purpose of this course is to explore several applications of Fourier analytic techniques in number theory and combinatorics, as well as their interactions. The course is designed to be of interest for a wide variety of graduate students with different backgrounds and motivations, including students in the area of analysis, discrete mathematics and number theory.

Contents

 

Week 1 (Joaquim Ortega-Cerdà), 4 hours:

– Session 1 (2 hours, 15:30h-17:30h Tuesday 25/04, Main room IMUB): Introduction

– Session 2 (2 hours, 15:30h-17:30h, Wednesday 26/04, Room T2, UB): tilings on groups and Fuglede Conjecture

Week 2 (Juanjo Rué), 5 hours:

– Session 1 (2,5 hours, 15:30h-18:00h, Wednesday 3/05, Room T2, UB): Boolean functions

– Session 2 (2,5 hours, 16h:18:30h, Friday 5/05, Main Room IMUB): Further combinatorial results

Week 3 (Jorge Jiménez), 6 hours:

– Session 1 (2 hours,15:30h-17:30h, Monday 8/05, Main Room IMUB): Exponential sums

– Session 2 (2 hours, 15:30h-17:30h, Wednesday 10/05, Room T2, UB): The circle Method

– Session 3 (2 hours, 15:30h-17:30h, Friday 12/05, Main Room IMUB): Applications

Week 4 (Oriol Serra and Pablo Candela), 7 hours:

– Session 1 (2 hours,15:30h-17:30h, Monday 15/05, Main Room IMUB ) Roth Theorem (I)

– Session 2 (2 hours, 15:30h-17:30h, Wednesday 17/05 Room T2, UB) Roth Theorem (II) and Chang’s Lemma

– Session 3 (1,5 hours, 15:30-17:00 h), Thursday 18/05, Main Room IMUB) Higher order Fourier analysis: Gowers norms (I)

– Session 4 (1,5 hours, 15:30h-17:00h, Friday 19/05, Main Room IMUB) Higher order Fourier analysis: Gowers norms (II)