Symmetry and topology - the mathematics of group theory and homotopy theory - play special roles in atomic, nuclear, high-energy and condensed matter physics. The first half of this course will focus on Lie groups that play central roles in quantum theory of atomic and nuclear structure and processes, the standard model of particle physics and quantum field theories of condensed matter systems.
The second part of the course will explore the role of spontaneous symmetry breaking (SSB) in theories ranging from phase transitions to mass generation in quantum field theory, particularly as SSB relates to the emergence of new elementary particles and quanta, as well as novel excitations, "topological defects". We will develop homotopy theory related to the underlying topology of broken symmetry ground states and use it to classify and study the stable topological defects in a wide range of broken symmetry states of quantum field theories.
