Applications are closed
for this workshop

Sarnak's conjecture

December 10 to December 14, 2018

at the

American Institute of Mathematics, San Jose, California

organized by

Mariusz Lemanczyk and Maksym Radziwill

This workshop, sponsored by AIM and the NSF, will be devoted to the recent progress on Chowla and Sarnak's conjecture. Chowla's conjecture postulates the lack of correlation of the Liouville function with its shifts and is widely seen as an analogue of the twin prime conjecture. Sarnak's conjecture asserts that the Liouville function is asymptotically orthogonal to any sequence of topological entropy zero. In recent years it has emerged that the two conjectures are deeply related and indeed a large amount of progress has been made on both in tandem. We are at a stage in which it appears that both conjectures are within grasp, perhaps with a few new ideas. The objective of the workshop is to highlight where we stand on those questions and to chart a path for their proofs.

The main topics for the workshop are

  • Tao's entropy decrement argument and its place within analytic number theory
  • The local Fourier uniformity conjecture for the Liouville function, its relation with additive combinatorics, and Sarnak and Chowla's conjecture
  • Recent work of Frantzikinakis-Host on the logarithmic Sarnak conjecture for ergodic weights
  • Recent work of Tao-Teravainen on the odd cases of Chowla's conjecture
  • Work of Matomaki-Radziwill on multiplicative functions in short intervals and its applications to Sarnak's and Chowla's conjectures
  • Recent progress on specific cases of Sarnak's conjecture: for automatic sequences (Mullner), analytic skew products (Wang), etc.
  • The classical techniques (Daboussi's criterion, joinings in ergodic theory): the relationship between them and their limitations

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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