Program Description
The program involves two hours of lectures each morning followed by informal or problem sessions in the afternoons. Every Monday evening there is a social event. The intense month long program allows the students to learn from and interact with the visiting speakers and one another. A certain camaraderie and sense of inclusion is fostered among the students by the chance to work closely with distinguished mathematicians over a lengthy period in a warm social and intellectual atmosphere. Student evaluations of the program are very positive. This is a good example of an excellent program available through the PIs that no one institution could provide.
Hamiltonian Dynamics, Variational Principles and Symplectic Invariants: Helmut Hofer (New York University), June 7-11
There is a very intricate relationship between (dynamical systems) questions in Hamiltonian dynamics and (geometric) questions in symplectic geometry. For example, the problem of finding periodic orbits on a prescribed energy surface is closely related to the problem of designing an energy-efficient transport for open sets in phase space. Exploring this mysterious connection leads to important concepts One is that of symplectic capacities. These are non-volume related invariants of open sets which are preserved under Hamiltonian flows. The first such invariant was introduced by Gromov in 1985. The second concept is concerned with the notion of energy of a symplectic map. This idea leads to a bi-invariant metric on the (infinite-dimensional) group of compactly supported Hamiltonian diffeomorphisms. Autonomous Hamiltonian flows are geodesics for this metric (but not the unique ones) and periodic orbits are related to conjugate points. Already the mere existence of a bi-invariant metric has surprising consequences. Exploring the relationship between capacity and energy leads to interesting applications to Hamiltonian dynamics. It also stresses the practical use of the variational principles underlying Hamiltonian dynamics.
This course will be based on the Lecture Notes (c) below. As a prerequisite knowledge of the material in b) below, pages 35-66, or the equivalent material in (a) would be very helpful.
Suggested Literature:
a) H. Hofer and E. Zehnder. Symplectic invariants and Hamiltonian dynamics. Birkhuser Advanced Texts. Birkhuser Verlag, Basel, 1994. xiv+341 pp. ISBN 3-7643-5066-0
b) D. McDuff and D. Salamon. Introduction to Symplectic Topology. Oxford Mathematical Monographs. Oxford University Press, 1995, ISBN: 0 19 851177 9
c) H. Hofer. Hamiltonian Dynamics, Variational Principle and Symplectic Invariants Lecture Notes
Geometric Introduction to Control Theory: Dmitri Burago (Pennsylvania State University), June 14-18
Consider an n-dimensional Riemannian manifold (you may think of a region in Rn) together with a collection of vector fields Vi, i=1,2, ... m, m< n. We will be concerned with the following questions:
a. Given two points in M, does there exist a (smooth) path connecting the points and such that its tangent vector at every point is a linear combination of Vi's?
b. If such paths exist, what is the shortest one?
c. What are the properties of the metric spaces whose distance function is defined as the length of shortest path tangent to the span of Vi's at every point?
d. How can we analyze particular examples?
One of the motivations for this set-up comes from applied problems with the number of controls smaller than the dimension of the configuration space of objects to control. A classical example is parallel parking: the driver has only steering wheel and acceleration pedal in his/her disposal, while the space of positions of the car is three-dimensional. It is even more striking for a truck with several trailers: the configuration space of a trailer train with k trailers is (3+k)-dimensional. We will discuss many other examples of such systems (planographers, bicycles, rolling a ball, falling cats, particles in magnetic field etc.) My main reason to choose Control Theory for my mini-course is that it belongs to the intersection of many mathematical topics, giving an excellent opportunity to give geometric introduction into these disciplines and then show how they can work if one puts them together. These mathematical topics include: geometry of Length Spaces; theory of Connections; Variational Methods; Nilpotent Groups. At the same time, there are many nice real-life examples and applications.
This is an approximate plan of the course:
- Our main model example: the three-dimensional Euclidean coordinate space viewed as the Heisinger group, and a distribution of two-planes invariant under the group action. In this example we will see how it is possible that, having only two-dimensional space of available directions at every point, one still can find a path connecting any two given points (and such that its velocity vector at every point belongs to the two-plane of our distribution at that point.)
- Lie brackets of vector fields and integrability/nonintegrability conditions of Frobenius and Chow.
- Important examples (and related notions and theories): connections in vector bundles (holonomies and curvature); contact structures.
- Length spaces. Carnot-Caratheodory spaces (length spaces arising from control theory). Local structure of Carnot-Caratheodory spaces (box-ball theorem, metric tangent cones).
- Examples.
I do not want to suggest reading any books on this topic before the conference. It will be helpful, however, if students refresh their knowledge of differential equations (existence and uniqueness of solutions, smooth dependence of initial data and parameters), and multi-dimensional calculus (inverse and implicit function theorems, basics in differential forms and smooth manifolds). However, our exposition is planned to be almost self-contained, we will begin from the scratch and review almost everything that we need to use.
Introduction to the Modern Theory of Dynamical Systems: Anatole Katok (Pennsylvania State University), June 21-25
The book is written by A.Katok and B.Hasselblatt and is published by Cambridge University Press. It is currently available in the paperback edition. The following parts constitute an essential background: Chapter 1.
- Sections 2.1-2.6
- Sections 3.1-3.2
- Sections 5.2-5.5
- Section 6.1, 6.2 (omit details of proofs), 6.4 Section 15.1
- Sections 17.4.17.5
At the very least the participants should be familiar with definitions and statements of results from these parts and understand the principal examples. I am planning to use some other parts of the book as a text. I am also planning to prepare extra hanouts from articles and preprints.
Celestial Mechanics: Jeff Xia (Northwestern University), June 28-July 2
