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Summer School: Additive Combinatorics

06/10/2024 - 06/14/2024
Big Lecture Hall, Rényi Institute

Description

The summer school is dedicated to graduate students and young researchers, and aims to give an introduction to recent techniques and topics of additive combinatorics. The lectures of the summer school will concentrate on recent developments of the polynomial method, some combinatorial methods of additive combinatorics, and the introduction of Fourier analytic techniques connected to them. The main topics will be presented by top researchers of the area.

Lecture sessions

Christian ElsholtzLarge structures without certain patterns

Abstract: We study point sets in (Z/mZ)^n, when m is fixed and n tends to infinity, especially with the constraint that they do not contain any arithmetic progression of length k, or that there are no three collinear points (so called caps). In these lectures the emphasis is on different methods to achieve lower bounds.

In the case of caps (over prime fields) we will similarly improve previous lower bound constructions and achieve in F_5 a "density" which supersedes the one of the most frequently studied case F_3. We also discuss results on progression-free sets in the integers.

In another part we intend to discuss improvements on the longest gaps between integers which are sums of two squares.

Péter Pál PachPolynomial method

Abstract: During the last few decades the polynomial method has developed into a powerful tool in combinatorics. It involves encoding combinatorial problems to the nonvanishing of some polynomials, and then investigating the resulting polynomial question. Usually a common step is to show that a low-degree (non-zero) polynomial can not vanish on a certain set (which is sufficiently large). We will start with exploring several applications of the Combinatorial Nullstellensatz, including classical problems like the Cauchy-Davenport theorem and other questions. The solution of the finite field Kakeya conjecture is also going to be discussed. Finally, we will study the CLP polynomial method which was first used to prove that sets avoiding 3-term arithmetic progressions in groups like Z_4^n and Z_3^n are exponentially small (compared to the size of the group). We will also look at the slice rank formulation of the technique and explore some further applications, for instance, the solution of the Erdős-Szeméredi sunflower conjecture.

Sean Prendiville: Nonlinear Fourier analysis and nonlinear progressions

Abstract: We introduce some of the ideas behind higher order Fourier analysis, in particular the use of this theory to locate patterns in sets of arithmetic interest (e.g. dense sets or primes). As an example application, we detail how to count certain polynomial progressions in dense subsets of a finite field - an application which exhibits the essential features of the theory in an elementary manner.

Julia Wolf: Linear methods in additive combinatorics

Abstract: In Lecture 1, we introduce Fourier analysis on finite abelian groups. As an application, we shall prove Bogolyubov’s lemma, and then move on to illustrating the limitations of the technique. In Lecture 2, we shall cover a probabilistic technique due to Croot and Sisask, which allows us to detect ‘almost periodicity’ in certain subsets of finite abelian groups. As an application, we give a simple proof of a Roth-type theorem. In Lecture 3, we intend to give a sketch of the recent breakthrough result of Kelley and Meka on the size of 3-term progression-free sets in the integers, carried out in the finite-field setting.

Registration 

The registration period has ended. 

Reminder: 

The registration is not automatic, after filling the form we have to approve it. We will send confirmation that the registration is successful.

If you are an MSc or PhD student, we ask in the registration form the name and email address of advisor / mentor for asking a brief reference email. 

If you are a postdoc, please send your CV by email to faap.semester@renyi.hu, with the subject "[Summer school: Additive Combinatorics] CV - APPLICANT'S FULL NAME"

Fund

We can cover the accommodation for a limited number of participants. Please, indicate on the registration form whether you need such financial support. In case you do, please, do not arrange your accommodation yet, as we may do it for you. We are going to make the decision of support before the end of October.  

Important information

The following link gives practical information (e.g. local traveling): 

https://erdoscenter.renyi.hu/practicalities

Organizers

Gergely Kiss (Rényi Institute)
Máté Matolcsi (Rényi Institute & BME)
Imre Ruzsa (Rényi Institute)
Oriol Serra (UPC, Spain)
Gábor Somlai (ELTE)

Invited Speakers

Elsholtz, Christian

Pach, Péter Pál

Prendiville, Sean

Wolf, Julia

Participants

Altman, Daniel

Angelidou, Maria

Aydogdu, Hazal Sena

Banerjee, Agnijo

Basak, Debmalya

Basu, Ayush

Bedert, Benjamin

Beker, Adrian

Bhowmik, Pablo

Charamaras, Dimitrios

D'orville, Jourdan

Deakin, Joseph

Ermoshin, Ivan

Führer, Jakob

Guo, David

Guo, Jing

Haith Rowlatt, Merlin

Havlas, Rok

Hollom, Lawrence

Johnston, Griffin

Kolossváry, István

Lee, Yongjin

Leng, James

Luo, Sammy

Maclean, Sophie

Martin, Sarah

Meilin, Tal

Miklósi, Roland Botond

Mond, Adva

Mountakis, Andreas

Mulrenin, Eion

Müllner, Károly

Negrini, Pietro

Nguyen, The

O'Keeffe, Max

Oganesyan, Kristina

Pekmezci, Can Devrim

Perea, Sinuhe

Pilatte, Cédric

Portier, Julien

Poursalidis, Nikos

Raghavan, Rushil

Rakhmonov, Firdavs

Raz, Hillel

Ritoprovo, Roy

Shala, Besfort

Souza, Victor

Stadlmann, Julia

Stavropoulos, Sokratis

Szőnyi, Laura

Thomas, Kate

Tyrrell, Fred

Vazquez Saez, Santiago

Wessel, Mieke

Wurzinger, Lena

Yu, Ke

Yu-Chen, Sun

Zheyu, Zeng