Mariusz Lemanczyk: Furstenberg Disjointness, Ratner Properties and Sarnak’s Conjecture

Explore the intricate connections between Furstenberg disjointness, Ratner properties, and Sarnak's conjecture in this 46-minute lecture by Mariusz Lemanczyk. Delve into the world of multiplicative functions, examining key concepts such as Mobius orthogonality, the Prime Number Theorem, and the Riemann Hypothesis. Investigate the Chowla conjecture on autocorrelations and analyze various strategies, including multiplicative weights and ergodic weights approaches. Examine topological dynamics, visible measures, and quasi-generic points, while exploring the relationship between the Chowla conjecture and Furstenberg systems. Gain insights into reductions in the logarithmic Sarnak conjecture case and understand the implications of Veech's conjecture in this comprehensive mathematical exploration.

Intro
Mobius orthogonality and Sarnak's conjecture
Multiplicative functions
General motivations: mean of p, Prime Number Theorem (PNT). Riemann Hypothesis (RH) Thecrem Landau The following conditions are equivalent
What makes it a difficult problem?
Chawla conjecture on autocorrelations Chowla conjecture
Multiplicative weights (MW-) strategy
Ergodic weights (EW-) strategy
MW-strategy: discussion on "controversies"
MW-strategy, basics 2
Sarnak's conjecture - selected examples
Topological dynamics. Visible measures and (quasi-)generic points
Chowla conjecture and Furstenberg systems
log S holds for deterministic uniquely ergodic systems
Reductions in the logarithmic Sarnak conjecture case
Further reductions for the logarithmic Sarnak conjecture
Veech's conjecture.III