Abstract: Sarnak's conjecture is a hot topic in number theory and ergodic theory in recent years, which says that the Mobius function is orthogonal to any sequence coming from a dynamical system of low complexity. Nowadays, many special cases of the Sarnak's conjecture have been proven and many applications have been found.
In this talk, we will talk about generalizations of Sarnak's conjecture to the ring of integers of an arbitrary number field, and new results in this direction. We will also talk about its application in combinatorics (called the partition regularity problem). For example, we show that for any finite coloring of Gaussian integers there exist x and y distinct and non-zero of the same color, such that x^2-y^2=n^2 for some Gaussian integer n .
