Abstract: In this talk we shall discuss some of the results found in Martin Krupa's paper, "Bifurcations of Relative Equilibria". Let G be a compact Lie group acting on Rⁿ and let f be a smooth G-equivariant vector field. Group orbits in Rⁿ which are invariant under the flow of f are called relative equilibria. A very useful thereom in this paper states that the flow of f near a relative equilibrium can be decomposed into two G-equivariant flows: one along the group orbit direction and another that is normal to the orbit. We shall present a precise description and proof of how this decomposition is obtained. Next, suppose that x(t) is a relative equilibrium of a parameter dependent vector field F. A bifurcation of F is the emergence of new relative equilibria branching from x(t) as the parameter is varied. We shall show that the above decomposition of the vector field can be used to analyze certain types of bifurcations of relative equilibria.

Abstract: Floer homology was originally invented to solve Arnold's conjecture. One way to phrase this conjecture is that the number of fixed points of a non-degenerate Hamiltonian diffeomorphism on a compact symplectic manifold M is bounded from below by the number of critical points of a Morse function on M. In 1988 Floer proved Arnold's conjecture for a broad class of symplectic manifolds. Floer's proof is based on a variational principle on the space of contractible loops of M which, in turn, led him to the concept of Floer homology. In a series of talks, we will discuss the underlying ideas of this theory.

Abstract: The Seifert conjecture is the question due to Seifert (1950) whether or not every smooth non-vanishing vector field on the 3-dimensional sphere has a periodic orbit. This problem led to many remarkable results concerning a similar question for other manifolds or other classes of vector fields (e.g., finitely smooth). However, the original Seifert conjecture was finally settled only in 1994 by Krystyna Kuperberg when she found a "counterexample" to the Seifert conjecture.

In a similar vein, the Hamiltonian Seifert conjecture is the question whether or not there exists a proper function on R²ⁿ whose Hamiltonian flow has no periodic orbits on at least one regular level set. An essential difference of the Hamiltonian case from the general one is manifested by the almost existence theorem of Hofer-Zehnder and Struwe, which asserts that almost all regular levels of a proper Hamiltonian must carry periodic orbits. In other words, regular levels without periodic orbits are exceptional in the sense of measure theory.

For 2n>4, smooth "counterexamples" to the Hamiltonian Seifert conjecture have been known since 1995. Recently, in a joint work with Viktor Ginzburg, we constructed a C²-smooth "counterexample" in dimension four, i.e., a C²-smooth function on R⁴ with at least one regular level set having no periodic orbits. In this talk, I will give an outline of the construction and show how it fits into the general context of the Seifert conjecture.

Abstract: For a dynamical system with symmetries, a relative equilibrium is an integral curve that coincides with the action of a one-parameter subgroup of symmetries. Example: the motion of a planet on a circular orbit about a star is a relative equilibrium; symmetries are rotations about the star. If the initial condition of the relative equilibrium possesses a nontrivial isotropy, the one-parameter group that generates the relative equilibrium is not unique. This allows us to use isotropy as bifurcation parameters in order to find other relative equilibria nearby.

This approach is especially useful for Lagrangian systems, where relative equilibria can be characterized as critical points of a function, called locked Lagrangian. I will talk about a test on the locked Lagrangian that can be used to detect bifurcating branches of relative equilibria. Then I will talk about a recent result on the stability of these branches. This result explains why, for some well-known examples, the relative equilibria bifurcating from a stable one retain stability in a striking contrast with the classical pitchfork bifurcation.

Abstract: This talk will be a quick introduction to some of the basic results about bifurcations of equilibria and some standard techniques for analyzing bifurcations in smooth dynamical systems. The talk is intended to be a warm-up to the next talk and also to give a general, although brief, overview of the theory, focusing on main ideas and examples.

Some of the topics that will be covered are: saddle-node and pitchfork bifurcations, transfer of stability, symmetry, and methods to reduce the dimension of a bifurcation problem.

Abstract: The topological entropy of a dynamical system is a conjugacy invariant that measures the complexity of the global orbit structure of the system.

Geodesic flows on Riemannian manifolds provide natural examples of dynamical systems with interesting properties that arise due to the interaction between geometry and dynamics. For them, Ricardo Mane proved a formula for the topological entropy, in terms of the exponential growth rate of the average number of geodesics joining two points of the manifold. His proof makes use of a natural symplectic structure on TM, which turns the geodesic flow into a Hamiltonian one, the Hamiltonian being the kinetic energy.

In our talk we will prove, for magnetic flows, a result analogous to Mane's. These flows (also known as twisted geodesic flows) are obtained as Hamiltonian ones for the kinetic energy and a perturbation of the natural symplectic structure in TM.

Abstract: These are the first two talks in a series of talks in symplectic topology we plan to have this quarter. Arnold's conjecture is the question of existence of fixed points for Hamiltonian maps of symplectic manifolds. This problem has had tremendous influence on many areas of symplectic geometry and mathematics in general. In the first talk, we will discuss Arnold's conjecture and its connections with other questions. In the second talk we will prove the Conley-Zehnder theorem which asserts that Arnold's conjecture holds for the 2n-dimensional torus. This proof will help us to understand better the nature of Floer homology which we will discuss in the forthcoming talks.

Winter 2001: Symplectic Geometry and Mechanics Seminar

Fall 2000: Symplectic Geometry Seminar