Courses in Fall 2024:
Math 39 - Discrete Mathematics with an Introduction to Proof
Research:
My area of research is in dynamical systems. These originally arose in the study of systems of differential equations and were used to model physical phenomena. One simplification of this is to discretize time, which leads to the study of iterations of a single transformation acting on a space. One can then study the global behavior of the system and the probabilistic happenings of this space. Different assumptions can be placed on the space and the map, and these lead to different areas of study in dynamical systems. For instance, ergodic theory studies the statistical and qualitative behavior of group actions on a measure space. It's origins are in the work of Boltzmann in statistical mechanics, where time- and space-distribution averages are equal. Symbolic dynamics studies spaces of infinite sequences of symbols with the dynamics given by the shift operator. It originated with Hadamard's study of geodesics on surfaces of negative curvature and lead to the study of information systems.
Papers:
Finite Odometer Factors of Rank One Group Actions
Joint work with David McClendon
Indagationes Mathematicae (2024)
Topological Speedups of Zd Actions
Joint work with David McClendon
Dynamical Systems: An International Journal, 37 2 (2022)
The Complexity Threshold for the Emergence of Kakutani Inequivalence [pdf]
Joint work with Van Cyr, Bryna Kra, and Ayse Sahin
Israel Journal of Mathematics (2021)
Speedups and Orbit Equivalence of Finite Extensions of Ergodic Zd Actions [pdf]
Joint work with David McClendon
New York Journal of Mathematics 21 (2015), 1371-1387
Directional Recurrence for Infinite Measure Preserving Zd Actions [pdf]
Joint work with Ayse Sahin
Ergodic Theory and Dynamical Systems, 35 7 (2015), 2138-2150
Speedups of Ergodic Group Extensions of Zd Actions [pdf]
Joint work with David McClendon
Dynamical Systems: An International Journal, 29 2 (2014)
Projectional Entropy in Higher Dimensional Shifts of Finite Type [pdf]
Joint work with Steve Kass and Kathleen Madden
Complex Systems, 17 (2007), 243-257
The Relative Growth of Information in Two-Dimensional Partitions [pdf]
Joint work with Karma Dajani and Martijn de Vries
Real Analysis Exchange, 31 No 2 (2005/2006)
Factoring Higher Dimensional Shifts of Finite Type onto the Full Shift [pdf]
Joint work with Kathleen Madden
Ergodic Theory and Dynamical Systems, 25 (2005)
The Symbolic Dynamics of Multidimensional Tilings Systems [pdf]
Joint work with Ethan Coven, Natasa Jonaska, and Kathleen Madden
Ergodic Theory and Dynamical Systems, 23(2003), 447 - 460
Renewal Systems, Sharp Eyed Snakes, and Shifts of Finite Type [pdf]
Joint work with Kathleen Madden
American Mathematical Monthly 109 No 3 (2002), 258 - 272
Even Kakutani Equivalence and the Loose Block Independence Property for Positive Entropy Zd Actions [pdf]
Joint work with Ayse Sahin
Illinois Journal of Mathematics, 45 No 2 (2001), 495 - 516
Isometric Extensions of Zero Entropy Zd Loosely Bernoulli Transformations [pdf]
Joint work with Ayse A. Sahin
Transactions of the American Mathematical Society, 352 No 3 (2000), 1329-1343
Recurrent Sequences and IP Sets [pdf]
Joint work with Kamel Haddad
Topology and its Applications, 98 Vol. 1-3 (1999), 203 - 210
The Decomposition Theorem for Two-Dimensional Subshifts of Finite Type [pdf]
Joint work with Kathleen Madden
Proceedings of the AMS, 127 N0. 5 (1999), 1533 - 1543
Finite Rank Z^d Actions and the Loosely Bernoulli Property [pdf]
Joint work with Ayse A. Sahin
New York Journal of Mathematics, 3A (1998), 125 - 134
Rank One and Loosely Bernoulli in Zd [pdf]
Joint work with Ayse A. Sahin
Ergodic Theory and Dynamical Systems, 18 (1998), 1159 - 1172
IP Cluster Points, Idempotents, and Recurrent Sequences [pdf]
Joint work with Kamel N. Haddad
Topology Proceedings, 22 (1997), 213 - 226
Putting the Pieces Together: Understanding Robinson's Nonperiodic Tilings [pdf]
Joint work with Kathleen Madden
The College Mathematics Journal, 28 No. 3 (1997), 172 - 181
Auslander Systems [pdf]
Joint work with Kamel N. Haddad
Proceedings of the AMS, 125 No. 7 (1997), 2161 - 2179
A Dynamical System with a Z2 Centralizer [pdf]
Joint work with Kyewon Koh Park
Journal of Mathematical Analysis and Applications, 210 (1997), 337 - 359
Convergence under xp of xq Invariant Measures on the Circle [pdf]
Joint work with Daniel J. Rudolph
Advances in Mathematics, 115 (1995), 117 - 140
Commuting Endomorphisms of the Circle [pdf]
Joint work with Daniel J. Rudolph
Ergodic Theory and Dynamical Systems 12 (1992), 743 - 748
Measures on the Circle Invariant under Multiplication by a Nonlacunary Subsemigroup of the Integers [pdf]
Israel Journal of Mathematics 77 (1992), 211 - 240
Book:
Discovering Discrete Dynamical Systems
Joint work with Ayse A. Sahin and Kathleen Madden
MAA Press, Classroom Resource Materials (2017)
Articles:
The Art of Teaching Mathematics [pdf]
Joint work with G. Campbell, J. Jacobsen, M.Orrison
MAA Focus 28 (1) (2008)
Pictures:
At the Wandering Seminar in Wroclaw, Poland March 2017
At Conference in honor of Mike Boyle: Maryland, Spring 2011
At Oxtoby Centennial Conference, Bryn Mawr College Oct 2010
At an Expanding Your Horizons Workshop
At Pingree Park (Colorado State University) conference in August of 2008
At Dynamics Conference, University of Maryland, Spring 2008
At conference in memory of Daniel J. Rudolph, University of Maryland, Spring 2010
