Mathematical and Numerical Methods, PHYS 777, September 9 - December 11, 2026

This course introduces mathematical and numerical tools that are useful for various fields of theoretical physics. In the numerical part of the course, students will develop introductory programming skills in both Python and Mathematica. The mathematical part of the course covers three main topics: First, we will study basics of smooth manifolds, define forms on them, and then de Rham cohomology. Second, we will introduce homology theory, and explain in what sense it can be understood as dual to cohomology. Lastly, we will study homotopy theory. Throughout the course, we will emphasize many applications to physics.