Ergodic theory and its applications to number theory and combinatorics 

Research description 

Ergodic theory, originating from statistical and celestial mechanics, describes the behavior of complex evolving systems and finds applications in various mathematical fields. The word ergodic, which was introduced by Boltzmann, is commonly thought to derive from two Greek words ergon (work) and hodos (path). Boltzmann made the ergodic hypothesis: the time average along an orbit is equal to the space average. However, the hypothesis as it was stated was false, and the systems possessing this property are now termed ergodic systems.

In ergodic theory, the fundamental object of study is a measure-preserving system, comprising a phase space as a probability space and a measure-preserving action of a group on this space. The first key result in ergodic theory is Poincare’s recurrence theorem, indicating that any event with positive probability will occur again. In the mid-70s, Furstenberg established a deep and far reaching generalization of Poincare's recurrence theorem - the multiple recurrence theorem. He then used his recurrence theorem to derive a new proof of the Szemeredi's celebrated theorem on arithmetic progressions. His ergodic approach has led to numerous new results in additive combinatorics, some of which still do not have a conventional proof. Furstenberg's method is also used in the spectacular work of Green and Tao on arithmetic progressions in primes.

My area of research is ergodic theory and its applications to number theory and combinatorics. Here are some of the research topics I have worked on: 

1. Joint normality of representations of numbers

2. Joint ergodicity of piecewise monotone interval maps

3. Equidistribution of generalized polynomials and applications to sets of recurrence

4. Uniform distribution of subpolynomial functions and applications to sets of recurrence