Dynamical Systems and Additive Combinatorics

Time: the fourth semester

Course Classification: elective course

Credit: 3 hours

Period: 54 hours

Syllabus:

(1) Topological dynamics: minimality, recurrence and van der Waerden's theorem

(2) Measure-preserving systems: ergodicity, mixing, Poincaré recurrence

(3) Ergodic proof of Roth's theorem and Szemerédi's theorem

(4) Topological dynamics: topological entropy

(5) Sarnak's Mobius disjointness conjecture

(6) Gowers uniformity norms and inverse theorems

(7) The Green-Tao theorem and nilsequences

(8) Further progressions on Sarnak's Mobius disjointness conjecture

References:

(1) M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, GTM 259, Springer.

(2) T. Tao and Van Vu, Additive Combinatorics, Cambridge University Press, 2009.

(3) P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, IAS Lecture Notes.

(4) B. Green and T. Tao, Linear equations in primes, Ann. of Math., 2010.