Abstract: Given a vector bundle, S, over a compact manifold M which allows a Dirac operator, D, we can associate a heat equation. The fundamental solution operator of this equation has a kernel which is often called the heat kernel. By taking an asymptotic expansion of this kernel and by using the McKean-Singer Formula we can derive an Index Theorem which is a major step on the way to a proof of the Atiyah-Singer Index Theorem. This "pre-Atiyah-Singer Theorem" is not as useful as Atiyah-Singer, but yields nice "heat equation proofs" for famous index theorems in low dimensions (e.g. Gauss-Bonnet and Riemann Roch for dimension 2).

In this talk I plan to sketch the proof of this theorem assuming as little as possible.

Light refreshments will be served before the talk

Abstract: Calabi-Yau n-folds are Kahler manifolds of complex dimension n equipped with an (n,0)-form. A submanifold L of real dimension n is Lagrangian if the Kahler form vanishes on it. If the (n,0)-form also vanishes, it is called a special Lagrangian. In this talk we will explore the moduli space of special Lagrangians. It turns out the the the moduli space can be realized as a Lagrangian of VxV*. Here V=H¹(L) and V*=H^(n-1)(L), where L is the original special Lagrangian.

Lastly we try to see the connection to mirror symmetry and the Strominger-Yau-Zaslow conjecture, the reason special Lagrangians are studied. Since VxV* can be viewed as either T*V or T*V*, its Lagrangian submanifold(i.e.the moduli space of the special Lagrangian) is locally defined by a function on either V or V*. This symmetry which is Legendre transform can be viewed as a manifestation of mirror symmetry.

Light refreshments will be served before the talk

Abstract: We study some asymptotic properties of the sequences of symplectic Lefschetz pencils constructed by Donaldson. In particular we prove that the vanishing spheres of these pencils are, for large degree, conjugated under the action of the symplectomorphism group of the fiber. This implies the non-existence of homologically trivial vanishing spheres in these pencils. Moreover we show some basic topological properties of the space of asymptotically holomorphic transverse sections.

Light refreshments will be served before the talk

Abstract: For finite-dimensional Hamiltonian systems, a well known criterion for determining nonlinear stability of an equilibrium is to show that the Hamiltonian H is convex in some neighborhood of this point. A sufficient condition for this to occur is for the hessian D²H to be positive-definite at the equilibrium. In this talk we shall discuss some of the functional analysis issues that arise when this criterion is applied for infinite dimensional systems. We shall demonstrate, by example, that sufficient convexity estimates of H can depend on a norm that is weaker than the norm required to guarantee existence and uniqueness of solutions to the Hamiltonian vector field. In this context, we present the notion of ``conditional stability'' which incorporates stability criteria (with respect to a weak norm) with the existence and uniqueness requirements imposed by a stronger norm.

Abstract: Contact homology was introduced by Eliashberg, Givental and Hofer. In this theory, we count holomorphic curves in the symplectization of a contact manifold, which are asymptotic to periodic Reeb orbits. These closed orbits are assumed to be nondegenerate and, in particular, isolated. This assumption makes practical computations of contact homology very difficult.

In this talk, I will discuss an extension of contact homology to Morse-Bott situations, in which closed Reeb orbits form submanifolds of the contact manifold. Then, I will explain how to use this method to compute contact homology and obtain further applications of these contact invariants.

Light refreshments will be served before the talk

Spring 2001: Symplectic Geometry and Mechanics Seminar

Winter 2001: Symplectic Geometry and Mechanics Seminar

Fall 2000: Symplectic Geometry Seminar