Title: Making static theorems dynamic: Connections between dynamics and combinatorial number theory
Speaker: John Johnson
Speaker Info: The Ohio State University
Van der Waerden's theorem on arithmetic progressions states for any finite coloring of the positive integers, there is one color that has arbitrarily long arithmetic progressions. It was the remarkable insight of Furstenberg that we can translate combinatorial problems into the language of dynamical systems.Date: Friday, January 26, 2018
One challenging facet of combinatorics is that we typically have to produce some ad hoc method and apply it in a clever way to solve a combinatorial problem. Dynamics, on the other hand, has a large collection of techniques, well-developed theory, and numerous motivating examples to apply when solving a dynamics problem.
In the context of van der Waerden's theorem, we can outline Furstenberg's argument in three steps: construct a metric space X and a continuous map T : X → X; prove this dynamical system (X, T) has a certain type of "recurrence"; and show that recurrence implies van der Waerden's theorem.
Ultrafilters on the positive integers are roughly "combinatorial measures" that shares some of the formal properties of van der Waerden's theorem. Like many other combinatorial problems, studying and understanding ultrafilters is challenging.
However, motivated by the principle that combinatorial problems become easier to understand when translated into dynamics, in this talk we'll state a few combinatorial theorems; define ultrafilters and develop some intuition on the connection between ultrafilters and combinatorics; define dynamical systems and give a few examples of such systems; and develop the connection between dynamics and combinatorics related to ultrafilters.