Accompanying workshops will be held on Monday, 30th June 2025, in room 1016.
Workshop schedule:
11:00 - 11:45 Borys Kuca, Ergodic methods in additive combinatorics
12:05 - 12:50 Tomasz Downarowicz, Positive entropy system uncorrelated to the Möbius function
13:10 - 13:55 Mariusz Lemańczyk, Furstenberg systems, characteristic classes and Sarnak's conjecture
Abstracts:
Borys Kuca (UJ), Ergodic methods in additive combinatorics
Additive combinatorics studies arithmetic patterns in large sets of numbers while ergodic theory examines long-term statistical behavior of dynamical systems. Although very different on the surface, these two areas are very deeply intertwined. Their rich interaction started in 1977 when Furstenberg provided an ergodic proof of the Szemerédi theorem from additive combinatorics. Since then, ergodic methods have proved to be extraordinarily successful in resolving problems in combinatorics and number theory. In this talk, I will describe how ergodic techniques can be used in combinatorics together with some recent breakthroughs in the area.
Tomasz Downarowicz (PWr), Positive entropy system uncorrelated to the Möbius function
The Sarnak conjecture asserts that all zero entropy systems are uncorrelated to the Möbius function. One can reverse the question: are zero entropy systems the only systems that are uncorrelated to the Möbius function? It may seem surprising, but the answer to this question is negative and I will sketch the construction of a suitable example.
Mariusz Lemańczyk (UMK), Furstenberg systems, characteristic classes and Sarnak's conjecture
The talk will be concentrated on the ergodic theory approach to Sarnak's conjecture. We will sketch the proof of Veech's conjecture which predicted the equivalence of Sarnak's conjecture with a special property of so-called Furstenberg systems of the Moebius function. We will show to what extent this allows us to verify Sarnak's conjecture when the zero entropy class is replaced with some (smaller) characteristic class of automorphisms.
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