Abstract

In topology one uses Morse theory to prove lower bounds for the number of 0-dimensional objects, namely critical points of smooth functions on smooth manifolds. In symplectic topology one uses Floer theory to prove lower bounds for the number of 1-dimensional objects, namely solutions to Hamiltonian ODEs. In this talk I will outline how the framework of symplectic topology can be generalized to study 2-dimensional or even higher-dimensional objects, leading to a class of first-order (Hamiltonian) PDEs sharing similar rigidity properties. In the same way as the Hamiltonian ODEs provide a generalized framework for classical mechanics, our class of Hamiltonian PDEs shall provide a generalized framework for studying equilibrium states of reaction-diffusion systems. This is joint work with my PhD student Ronen Brilleslijper.