Victor Donnay’s research area is the ergodic properties of smooth dynamical systems, particularly geodesic flow on surfaces and billiard systems. He has found new examples of surfaces, including surfaces that are embedded in three dimensional Euclidean space, for which the geodesic flow is chaotic (ergodic). He has also studied conditions under which ergodicity breaks down due to the existence of elliptic islands and is now interested in exploring the boundary between ergodic and non-ergodic systems.

Donnay is also interested in conveying the beauty and excitement of mathematics to the general public. Together with his students, he created a computer generated video about the Costa minimal surface that became part of the mathematics exhibit "Beyond Numbers" at the Maryland Science Center. His team has also created computer generated pictures of the embedded surfaces with ergodic geodesic flow. These pictures were displayed at the Artist’s Market Gallery in Norwalk, Ct as part of the exhibit "M. C. Escher and Beyond". He continues to be interested in using computer graphics to present visual aspects of mathematics.

In his teaching, Donnay has worked to incorporate the new pedagogical tools of mathematics education: small group cooperative learning exercises, mathematical exploration via computer experiments, and projects. Among the courses he teaches are Multi-variable calculus, Probability, Real Analysis, Differential Equations, and Chaotic dynamical systems. Most recently, he has taught a course on Mathematical Modeling and the Environment and hopes to develop this into an interdisciplinary course to be team taught with faculty from other departments.

 

Selected Publications

1. Geodesic flow on the two-sphere, Part I: Positive measure entropy, Ergod. Th. &

Dynam. Sys. 8 (1988), 531-553.

2. Geodesic flow on the two-sphere, Part II: Ergodicity, Dynamical Systems, Springer

Lecture Notes in Math., Vol. 1342 (1988), 112-153.

3. Using integrability to produce chaos: billiards with positive entropy, Comm. Math. Phys. 141 (1991), 225-257.

4. Joint with C. Liverani, Potentials on the two-torus for which the Hamiltonian flow is

ergodic, Commun. Math. Phys. 135 (1991), 267-302.

  1. Elliptic islands in generalized Sinai billiards, Ergod. Th. & Dynam. Sys. (1996), 16,

975-1010.

6. Joint with K. Burns, Embedded surfaces with ergodic geodesic flow, Inter. J. of Bifurcation and Chaos, Vol. 7, No. 7 (1997) 1509-1527.

  1. Non-ergodicity of two particles interacting via a smooth potential, J. of Statistical Physics,

    2. Vol. 96, Nos. 5/6 (1999) 1021-1048.

  2. Joint with Charles Pugh, Finite horizon Riemannian structures and ergodicity, in preparation.