Combinatorics Meets Ergodic Theory (15w5013)
Nikos Frantzikinakis (University of Crete)
Bryna Kra (Northwestern University)
Julia Wolf (University of Bristol)
Although the connection between the ergodic theoretic/dynamical systems side and the combiantorial/number theory side is now well established, there are only a few experts in these fields who are able to use tools from both of these disparate areas and pass from one to the other with ease. A systematic survey of the fields will therefore be of benefit to a wider community of researchers. As such, we aim to bring together mathematicians from ergodic theory, dynamics, combinatorics and number theory, with a view to exploring potential new connections, exploiting existing techniques and developing new tools.
To have a sense for the complexity of the interactions between the various fields, we consider a particular example. The Hardy-Littlewood circle method from number theory is a fundamental tool in proving subsequence ergodic theorems. Ergodic theoretic results motivated work on discrete singular integrals on lower dimensional manifolds, but required the extension of the circle method to situations which had not been studied by number theorists. These new developments in turn contributed to number theory and then came back again to ergodic theory in modified form. Szemeredi's Theorem is a more elaborate example of these interactions, with deeper connections still developing and drawing in new ideas.
We will focus on several key topics that emerge from these interactions: 1) Convergence Theorems. There are many variations on the mean and pointwise ergodic theorem, as well as versions with more terms (known as multiple ergodic averages). Although approaches to such problems were originally purely ergodic, such as in the work of Host and Kra, recent advances in the work of Tao and even more recently of Walsh have occurred by turning to combinatorial techniques.
2) Patterns in the primes. Ergodic theoretic techniques, especially the ergodic proof of Szemeredi's Theorem, feature prominently in the monumental work of Green and Tao and subsequent results on other patterns in the primes. As new ergodic theorems are proven, there are opportunities to study analogous results in the primes and further understand quantitative versions.
3) Connections between ergodic theory and additive combinatorics. While the formal similarities between the various approaches to the combinatorial results have been noted, the nature of the mathematical connection between the fields is yet to be fully understood. A particular example is the role of nilpotent objects in Host and Kra's proof of multiple convergence, the inverse theorem for the Gowers norms of Green, Tao and Ziegler, and the recently introduced nilspaces arising in the work of Szegedy. Another problem is the existence of a Furstenberg Correspondence Principle for graphs, the study of which has only begun in the last year.
4) Qualitative problems. Numerous open problems relate to further understanding what properties of a set (such as size) suffice to force it to contain particular patterns. Such problems arise both in the combinatorial setting (such as the famous unsolved conjecture of Erdos that any sequence of integers the sum of whose reciprocals diverges contains arbitrarily long arithmetic progressions) and in the ergodic setting (such as the existence of pointwise ergodic theorems along particular choices of sequences).
The proposed workshop covers an area in which major advances have been made over the past few years and in which a great deal of research is currently being carried out. It will allow us to bring together experts from different fields to approach the numerous outstanding problems, and introduce young researchers to this active area of mathematics. No text currently exists that draws together the different branches and it is difficult for people entering the field to master the needed techniques. We have focused on areas which are rich in open problems and for which interactions among the fields are key.