2019 Lecturer: Bryna Kra
Dynamics of Systems with Low Complexity
One way to classify dynamical systems is by their entropy, which roughly speaking gives a measure of the disorder in the system. Deterministic systems have zero entropy, but in spite of this structure, many basic questions about systems with zero entropy remain open. Even when placing strong constraints on the complexity of the system, easily formulated questions remain intractable. I will give an overview of the relations among complexity, algebraic properties, and dynamical characteristics of the system (such as periodicity, minimality, and transitivity), and their relations to combinatorial problems.
Bryna Kra is the Sarah Rebecca Roland Professor of Mathematics at Northwestern University. She has been selected as the 2019 Noether Lecturer for her profound impact on mathematics, both through her work in the fields of dynamical systems and ergodic theory and through her service to the profession.
Dr. Kra received her AB from Harvard University and both her MA and PhD degrees in mathematics from Stanford University, under the guidance of Yitzhak Katznelson. Before joining the Northwestern University faculty in 2004, Kra was an assistant professor at Pennsylvania State University, an NSF-NATO Fellow (University of Marne-la-Vallée), a Raymond and Beverly Sackler Fellow (Institut des Hautes Études Scientifiques), a Golda Meir Postdoctoral Fellow (Hebrew University of Jerusalem) as well as a postdoc at the University of Michigan and The Ohio State University.
Kra is best known for her fundamental contributions to ergodic theory. Her 2005 paper joint with Bernard Host titled “Nonconventional ergodic averages and nilmanifolds” (Annals of Mathematics) settled a long-standing open problem on the existence of the limit of certain multiple ergodic averages, uncovering the role of nilpotent groups and their homogeneous spaces in analyzing configurations in sets of integers. The work inspired many further developments, including structure theorems in ergodic theory, in topological dynamics, and in combinatorics, convergence results for numerous multiple ergodic averages, and the uncovering of recurrence phenomena that imply the existence of patterns in sufficiently large sets of integers. In further work joint with Vitaly Bergelson and Host, they introduce the notion of a nilsequence and use it to provide further structural results in dynamics. It has been adapted to the combinatorial setting, playing an important role in studying patterns in smaller subsets of the integers, for example the set of primes. Continuing her work at the intersection of dynamics and combinatorics, Kra’s more recent research lies in topological and symbolic dynamics, studying systems of low complexity. In joint work with Van Cyr, she has the strongest work to date on Nivat’s Conjecture, relating a global property of periodicity of a two-dimensional configuration to a locally checkable property on the complexity.
In addition to becoming a recent Fellow of the American Academy of Arts and Sciences, Kra is also an AMS Fellow and was awarded an AMS Centennial Fellowship as well as the AMS Levi L. Conant Prize. She was an invited speaker at the International Congress of Mathematicians (2006) and has given numerous invited lectures, including an AMS-MAA Invited Address (2007), Arnold Ross Lecture of the AMS (2013), National Museum of Mathematics (2014), Bartlett Lecture (2015), Dresden Lectures (2015) and the Coven-Wood Lectures (2017). Kra is currently on the Board of Trustees of the AMS and has previously served on the AWM Executive Committee, the Council and Executive Committee of the AMS, the Board of Trustees of the Institute for Pure and Applied Mathematics, the Advisory Board for the Young Mathematicians Conference, and the Steering Committee for the Park City Mathematics Institute. She holds several editorial positions, including those with Ergodic Theory and Dynamical Systems and Discrete Analysis.