Explore symplectic manifolds and bordism theory in this advanced mathematics lecture from the University of Chicago's Department of Mathematics. Delve into the evolution of symplectic topology over the four decades following Andreas Floer's groundbreaking contributions, tracing its origins from Poincaré's early investigations into Hamiltonian dynamical systems. Examine how the revolutionary work of Gromov and Floer in the 1980s provided the essential tools to extend Poincaré's insights from low-dimensional cases to higher-dimensional contexts. Discover how these foundational methods have been enhanced with increasingly sophisticated algebraic structures, enabling mathematicians to solve numerous long-standing problems, including determining the homotopy types of Lagrangian submanifolds across various classes of examples. Learn about the significant developments of the past five years, during which the field has undergone a paradigm shift by integrating homotopy theory methods with generalizations of Morse theory, thereby generating novel research questions in both mathematical domains.