#
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

## Original Announcement

This workshop 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

## Material from the workshop

A list of participants.