Problems in combinatorics and number theory via ergodic theoretic methods

Overview The main objectives of this project are concentrated in the following three topics, which lie at the interface of dynamics, combinatorics, and number theory: - Addressing pending conjectures about the qualitative behavior of orbits in discrete-time dynamical systems. Related to this are questions about the convergence and the positivity of multiple ergodic averages which naturally arise in connection to problems in additive combinatorics and Ramsey theory. -The development of new ideas and techniques involving ultrafilters and the algebra of the Stone-Cech compactification of the natural numbers. This has the potential to unlock solutions to unsolved problems in combinatorics where more conventional approaches have so far failed, in particular in connection with the Erdős sumset conjecture. - To improve our understanding of the dynamical aspects of Sarnak's and Chowla's conjecture in number theory. The main goal is the investigation of the interplay between dynamical systems of multiplicative group actions and additive group actions. This aims to push Sarnak's conjecture to a new generality that juxtaposes ideas and methods from dynamics and number theory. Intellectual Merit The study of the long-term behavior of dynamical systems has far-reaching applications to other areas of mathematics. The employment of analytic tools coming from measurable, topological, and symbolic dynamics offers new possibilities for analyzing seemingly static number-theoretic and combinatorial situations and has proven to be a powerful method in solving numerous open problems in other areas of mathematics. The main challenges in this area range from recasting open problems in discrete mathematics in a dynamical language to developing new analytic and dynamical tools to tackle those reformulations. This project intents to address, and potentially settle, some of these problems, and, more generally, promote an interdisciplinary discourse between dynamical systems, combinatorics and number theory. Broader Impacts Of The Proposed Work The theory of dynamical systems is a rich and fruitful branch of mathematics that finds many real-world applications, such as modeling stock market movements, oxygen and carbon dioxide transport in our bodies and the spreading of diseases, to name a few. Moreover, dynamical methods have yielded powerful methods for solving problems in other areas of mathematics, as well. This project seeks to improve on the tools and techniques available in dynamical systems, which will ultimately impact a wide range of areas.