Current Topics Seminar: Heather Lee

Abstract: Symplectic manifolds are real even-dimensional manifolds that generalize the phase space of classical Hamiltonian mechanical systems. In a 2N-dimensional symplectic manifold, we can simultaneously have at most N conserved quantities and they determine an N-dimensional submanifold which is an example of a Lagrangian submanifold. Studying Lagrangian submanifolds and how they intersect leads to powerful tools for understanding symplectic geometry and plays a crucial role in explaining mirror symmetry, which is a remarkable duality between symplectic geometries and complex geometries. We will focus on one formulation of mirror symmetry known as the homological mirror symmetry (HMS) conjecture.  HMS was originally formulated by Kontsevich for mirror pairs of compact Calabi-Yau manifolds. Since then, it has been extended to cover non-Calabi-Yau manifolds and open manifolds as well.  I will give a beginner's introduction to symplectic geometry, Floer thoery for Lagrangian intersections, and homological mirror symmetry.