If we have a sequence starting with 4, 7, 10, 13, what will come next? Questions of this sort are meant to check pattern detection skills; for that reason, they are often used to test mathematical abilities. In this example, each subsequent term is greater by 3 than the previous one; hence the next term will be 16. Sequences of this form, in which the difference between consecutive terms is constant, are called arithmetic progressions.

The search for patterns like arithmetic progressions in sets of numbers is a central task of the subfield of pure mathematics called combinatorial number theory. It is known that large subsets of natural numbers contain plenty of “reasonable” patterns, which includes arithmetic progressions. One of the main objectives of the area is to quantify these statements, either by showing that all sets of a certain size contain a pattern or by estimating the number of patterns in sets of a given size.

By contrast, ergodic theory examines the statistical behaviour of processes over long time, and its origins lie in the study of large systems of particles in statistical mechanics. The systems to study range from relatively simple ones, such as rotations on a circle, to highly complicated flows on surfaces and their higher-dimensional generalisations. If possible, one wants to obtain a definitive form (e.g. space average) for the long-term behaviour of the system (i.e. its time average).

On the surface, there seems to be little connection between these two apparently distant areas, both in terms of the scope of research and available techniques. Yet there is a strong link between them, forged by Furstenberg's ergodic proof of the Szemerédi theorem from combinational number theory. Szemerédi proved that each sufficiently large subset of natural numbers contains arithmetic progressions of arbitrary length; Furstenberg rederived this result by relating it to the problems of multiple recurrence in ergodic theory. The interaction between combinatorial number theory and ergodic theory initiated by Furstenberg has proved enormously fruitful, creating a virtuous cycle that lead to countless new developments in both branches of mathematics. The goal of this project is to tackle a number of open problems at the interface between ergodic theory and combinatorial number theory. 

 News

22 April 2024

My paper Multidimensional polynomial patterns over finite fields: bounds, counting estimates and Gowers norm control has been accepted in Advances in Mathematics.

17 April 2024

I gave a colloquium talk at IMPAN titled "The Szemerédi theorem and beyond" in which I outlined recent developments in ergodic theory and additive combinatorics that take their roots in the Szemerédi theorem.

13 March 2024

To comemmorate the International Day of Pi taking place on 14 March, I gave an expository talk on the mysteries of pi and the history of numbers in IV Liceum Ogólnokształcące in Kraków. 

29 February 2024

I have been interviewed by Radio Kraków on the everyday nature of a mathematician's job, the applications of pure mathematics to real-life problems, and the upsides and downsides of mathematical education.

I was the main guest at a two-hour long discussion hosted by Tarnowski Klub Dyskusyjny. We talked about the nature of pure mathematics, its connection with other sciences, real-life applications, benefits of studying and teaching mathematics, and many other topics. Big thanks to Fundacja Alegoria for hosting the discussion.

Photo: Franciszek Kiełbasa

January-February 2024

I have visited my collaborator Sebastián Donoso at University of Chile, where we worked on our joint project on the joint ergodicity of Hardy sequences. I also gave a talk in the dynamical systems seminar.

07-14.12.2023

I had the pleasure of hosting my two collaborators from Greece, Andreas Koutsogiannis and Konstantinos Tsinas, as a part of collaborative work on the joint ergodicity conjecture for Hardy sequences. 

04.12.2023

I gave a talk at the dynamical systems seminar at IMPAN (Warsaw) on my recent works with Nikos Frantzikinakis concerning the limiting behaviour of multiple ergodic averages along polynomials for systems of commuting transformations. The results discussed in this talk are an inspiration a starting point for several research tasks in this Polonez Bis project. 

09.11.2023

I gave a talk at the number theory seminar at the Jagiellonian University in which I summarised recent developments in the polynomial Szemerédi theorem. This is one of the main topics of this Polonez BIS project.

This research is part of the project No. 2022/47/P/ST1/00854 within the POLONEZ BIS programme co-funded by the National Science Centre and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 945339.