Research and Teaching

About Me: I'm a professor in the Mathematics Department at the United States Naval Academy.

Email: medynets@usna.edu Phone: 410-293-6727

Mailing Address: U.S. Naval Academy, Chauvenet Hall, 572C Holloway Road, Math Dept Stop 9E, Annapolis, MD 21402

Positions

Arrived at USNA: 2012

Professor, 2023-present

Associate Professor, 2017-2023

Assistant Professor, 2012-2017

Zassenhaus Assistant Professor, The Ohio State University, 2009-2012

Junior Research Fellow, The Erwin Schrödinger International Institute for Mathematical Physics, Vienna, Austria, Spring 2009

Education

Ph.D., Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 2008.

M.S., Mathematics, Summa Cum Laude, Kharkiv National University, Kharkiv, Ukraine, 2003

B.S., Mathematics, Summa Cum Laude, Kharkiv National University, Kharkiv, Ukraine, 2002

Research InTERESTS

Ergodic Theory, Topological Dynamics, Tiling and Symbolic Dynamical Systems
Stability Theory of Dynamical Systems
Orbit Equivalence and Operator Algebras Associated to Dynamical Systems
Transformation Groups
Representation Theory of Infinite Groups
Geometric Group Theory and Random Walks on Groups

To learn more about my research, go to my ArXiv page, see the AMS MathSciNe profile, Google Scholar, or see my publications below.

Research Students

Sarabeth Joyner, Class of 2025. Research Topic: Picard's Method for ODEs and PDEs
Cami Herman, Class of 2022. Research Topic: Swarm Dynamics
Josh Dye, Class of 2021. Research Topic: The Grigorchuk Group.
Aidan Sabetymass, Class of 2019. Research Topic: Cellular Automata, Computational Group Theory. Project Website
Carl Kolon, Class of 2018. Research Topic: Swarms of Agents with Spring-like Interactions and Self-propulsion, Stability Theory, Limit Cycles, Project Website
James Talisse, Class of 2017. Research Topic: Multidimensional Symbolic Dynamical Systems, Centralizers, Topological Entropy.

Publications

** = publications with undergraduate co-authors

On Spatial Cohesiveness of Second-Order Self-Propelled Swarming Systems, (jointly with I. Popovici), SIAM Journal on Applied Mathematics, Vol. 83, Iss. 6, 2169-2188, download pdf
**On the Stability of Rotating States in Second-Order Self-Propelled Multi-Particle Systems (jointly with C. Kolon and I. Popovici), 2023, download pdf
On Invariant Random Subgroups of Block-Diagonal Limits of Symmetric Groups (jointly with A. Dudko), Proc. Amer. Math. Soc. 147 (2019), 2481-2494 , download pdf.
**Toeplitz Subshifts with Trivial Centralizers and Positive Entropy (jointly with J.P. Talisse), Involve, a journal of mathematics, Vol. 12, No. 3, 2019, download pdf.
Presentations of Topological Full Groups by Generators and Relations (jointly with R.I. Grigorchuk), Journal of Algebra, Volume 500 (2018), 46-68. download pdf.
Cantor Systems and Quasi-Isometry of Groups, joint with R. Sauer and A. Thom, Bulletin of the London Mathematical Society (4) 49 (2017), 709-724. download pdf
Orbit Equivalence Rigidity of Equicontinuous Systems, Journal of the London Mathematical Society (2) 94 (2016), 545-556, joint work with M.I. Cortez, download pdf.
Second Order Ergodic Theorem for Self-Similar Tiling Systems (jointly with B. Solomyak), Ergodic Theory & Dynamical Systems, 34 (2014), no. 6, 2018-2053, download pdf.
On the algebraic properties of topological full groups (jointly with R. Grigorchuk). Mat. Sb. 205 (2014), no. 6, 87 - 100. download pdf.
Finite Factor Representations of Higman-Thompson's groups (jointly with A. Dudko), Groups, Geometry, and Dynamics 8 (2014), no. 2, 375 - 389. download pdf.
On Characters of Inductive Limits of Symmetric Groups (jointly with A. Dudko), Journal of Functional Analysis, 264 (2013), no. 7, 1565 - 1598. download pdf.
Finite Rank Bratteli diagrams: Structure of Invariant Measures (jointly with S. Bezuglyi, J. Kwiatkowski, and B. Solomyak) Transactions of the AMS, 365 (2013), no. 5, 2637 - 2679. download pdf.
Reconstruction of Orbits of Cantor Systems from Full Groups, Bulletin of the London Mathematical Society, Volume 43 Part 6 (2011), 1104–1110. This file slightly differs from the published version. download pdf
Invariant measures on stationary Bratteli diagrams (jointly with S. Bezuglyi, J. Kwiatkowski, and B. Solomyak). Ergodic Theory and Dynamical Systems, 30 (2010), no. 4, 973-1007. download pdf
Aperiodic substitution systems and their Bratteli diagrams (jointly with S. Bezuglyi and J. Kwiatkowski). Ergodic Theory and Dynamical Systems 29 (2009), no. 1, 37–72. download pdf
Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems (jointly with S. Bezuglyi). Colloquium Mathematicum 110 (2008), no. 2, 409–429. download pdf.
On approximation of homeomorphisms of a Cantor set. Fundamenta Mathematicae 194 (2007), no. 1, 1–13. download pdf.
Cantor aperiodic systems and Bratteli diagrams. Comptes Rendus Mathematique 342 (2006), no. 1, 43–46. download pdf.
Approximation in ergodic theory, Borel, and Cantor dynamics (jointly with S. Bezuglyi and J. Kwiatkowski). Algebraic and topological dynamics, 39–64, Contemporary Mathematics, 385. download pdf.
The Rokhlin lemma for homeomorphisms of a Cantor set (jointly with S. Bezuglyi and A. H. Dooley). Proceedings of the American Mathematical Society 133 (2005), no. 10, 2957–2964. download pdf
Smooth automorphisms and path-connectedness in Borel dynamics (jointly with S. Bezuglyi). Indagationes Mathematicae 15 (2004), no. 4, 453–468. download pdf

slides

Dynamics on Bratteli Diagrams. download pdf.
The Second Order Ergodic Theorem for Substitution Tiling Systems. download pdf.
Characters of Thompson's Group. download pdf.
Orbit Equivalence Rigidity for Equicontinuous Systems. download pdf.
USNA Applied Math Seminar, download pdf
ONR Nonlinear Physics, 2022, keynote, powerpoint, pdf

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