The Moduli spaces that parametrise the finite energy solution to elliptic partial differential equations from physics provide deep information for geometry and physics at both small and large scales. In this project, basic tools (regularities, Fredholmness, compactness, gluing and virtual fundamental cycles) will be introduced to a large class of PDEs including Yang-Mills equations, Seiberg-Witten equations, Vortex and monopole equations, Gromov-Witten equations, Hamiltonian Gromov-Witten equations, Kapustin-Witten equations.
Some exciting relations to geometry of low dimensional manifolds, dualities in quantum field theories and string theory will be explored.
You will need basic familiarities in differential geometry and differential topology such as de Carmo's book on differential forms, Hirsch's book on differential topology are required. Backgrounds in quantum field theory or string theory are helpful, but not essentially required.