The seminar is mainly intended for graduate students. (Faculty are also welcome.) This quarter we will have talks on a broad variety of topics given by several speakers. These topics include Morse theory and elements of symplectic topology, Hamiltonian dynamical systems (optical Hamiltonians and rigid body), Poisson geometry.

Abstract: Let F be a Lagrangian distribution on a symplectic manifold M, and H a smooth real function on M. The Hamiltonian flow generated by H is said to be optical if it "twists" all of the Lagrangian subspaces of F in the same direction.

We will sketch the proof of two formulas for the topological entropy of these flows in terms of the determinant of the differential of the flow restricted to the Lagrangian subspaces of F.

Abstract: We study the quotient of a symplectic manifold by a circle action of symplectomorphisms. The resulting space has the structure of a singular real algebraic variety with a Poisson bracket. We ask if this space can be "desingularized" by embedding it as a Poisson subspace of a Poisson manfold M. We find that the answer can depend both on the Poisson structure of the variety and the dimension of M.

Abstract: We will start with a brief comparison of classification theorems for circle homeomorphisms and the Denjoy theorem which states that an orientation preserving C^r-diffeomorphism of the circle with r>1 has either periodic points or all of its orbits are dense.

The main part of this talk will be the construction of the so-called Denjoy example - an orientation preserving C¹-diffeomorphism of the circle which has neither periodic points nor dense orbits. This example shows that the differentiability condition r>1 can not be relaxed to include r=1.

Abstract: The three dimensional rigid body is one of the best known classical mechanical systems and serves as a simple, but nontrivial, application of a wide variety of geometric constructions. The full rigid body equations can be regarded either as a Lagrangian system or canonical Hamiltonian system. Poisson reduction leads to the familiar Euler equations, which are a Lie-Poisson system; symplectic reduction leads to a Hamiltonian system on the two-sphere. The trajectories of the second order version of the equations are geodesics with respect to an appropriate Riemannian structure on the rotation group SO(3).

In the first part of the talk I will describe some of the basic features of the rigid body system, attempting to link some general geometric constructs to familiar vector and matrix identities. (Most actual calculations will be carried out using only techniques from undergraduate vector calculus.) In the second part of this talk I will use the rigid body system to motivate the introduction of moment maps and reduction in the context of symplectic geometry.

Abstract: This talk is an introduction to oscillatory integrals. We will explain how the Morse Lemma is applied in the study of these integrals. Any connection between this talk and symplectic geometry is accidental and is not intended by the speaker.

Abstract: We will give the definition of Morse functions and the Morse complex, outline a proof of the Morse inequalities, and illustrate the constructions by a number of examples.