## Columbia Mathematics Department Colloquium

# The symplectic topology of Stein manifolds*
*

by

# Mohammed Abouzaid

## MIT

Abstract:

Those complex manifolds which admit a proper embedding in affine

space are called Stein. In the early 90's, Eliashberg classified the

smooth manifolds of real dimension greater than 4 which admit a Stein

structure, leaving open the question of whether a manifold can admit two

Stein structures which are not deformation equivalent. By making full use

of the modern machinery of symplectic topology (i.e. Floer theory and the

Fukaya category), the last five years, starting with work of Seidel and

Smith, has seen much progress on this front. I will particularly focus on

the case of Stein structures on manifolds diffeomorphic to euclidean

space, and explain some ideas behind the proof that, in real dimensions

greater than 10, the set of equivalence classes of such Stein structures

(under deformation) maps surjectively to the set of sequences of prime

numbers. In particular, it is uncountable.

space are called Stein. In the early 90's, Eliashberg classified the

smooth manifolds of real dimension greater than 4 which admit a Stein

structure, leaving open the question of whether a manifold can admit two

Stein structures which are not deformation equivalent. By making full use

of the modern machinery of symplectic topology (i.e. Floer theory and the

Fukaya category), the last five years, starting with work of Seidel and

Smith, has seen much progress on this front. I will particularly focus on

the case of Stein structures on manifolds diffeomorphic to euclidean

space, and explain some ideas behind the proof that, in real dimensions

greater than 10, the set of equivalence classes of such Stein structures

(under deformation) maps surjectively to the set of sequences of prime

numbers. In particular, it is uncountable.