Combinatorial and Algebraic Structures in Dynamics

Project: Research project

Project Details

Description

Overview: The investigator proposes a program of research, education, and outreach in dynamics. The research entails problems on the topological properties of symbolic systems, including symmetries and algebraic invariants, and on the ergodic properties of symbolic systems, studying mixing properties, the simplex of invariant measures, and recurrence. The proposed work builds on previous results of the PI and lies within dynamics, but the problems are inspired by questions in arithmetic combinatorics, number theory, and computer science. The research goals are a deeper understanding of the connections among these fields and the outreach goals aim to broaden the cohort of researchers working in these areas.

Intellectual merit: A natural way to study dynamical systems is via a coding mechanism, creating symbolic models for complicated systems. The PI proposes building on past results to study topological and ergodic properties of symbolic systems. The first series of questions is on their automorphism groups, a way to capture the symmetries of the system, exploring the complicated group structure that arises for complex systems and the constraints on the group structure that arise in simple systems. The second area is on the measurable properties of symbolic systems, studying how the simplex of invariant measures depends on various notions of complexity. The third topic is higher dimensional systems, relating the complexity of configurations to the dynamical properties of the systems, and the fourth area is the study of nilpotent structures in recurrence and convergence problems. A running thread in all of these problems is understanding how various measurements of complexity of the system relate to its dynamical properties, combining dynamical methods with combinatorial and algebraic techniques to study these questions.

Broader impacts: The outreach portion of the project involves conference organization, mentoring, and advising. The PI organizes numerous conferences, including meetings aimed at junior researchers in dynamics, interdisciplinary meetings, mentoring meetings, education of the general public, and an innovative online series for the field. The PI continues to work with the graduate and undergraduate programs at Northwestern University, directing undergraduates in advanced topics and graduate students in doctoral research. The PI is involved in mentoring, running a group for women at all levels within the Mathematics, Statistics, and Applied Mathematics Departments at Northwestern University, supporting a large annual conference for undergraduate women from across the country designed to increase the number of women applying to graduate school in mathematics, and helping design and implement a postbaccalaureate program in mathematics for underrepresented groups in mathematics.
StatusNot started
Effective start/end date6/1/215/31/24

Funding

  • National Science Foundation (DMS-2054643)

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