Algebras, representations, and branes.

I will give a leisurely (and hopefully accessible) introduction to some connections between ideas from geometry, algebra, and theoretical physics. I will focus on a particular setting which connects constructions in string theory to algebras and categories of modules over those algebras. This is related to some new research developments in the areas of twisted supergravity and twisted holography. If time permits, I will try to end with a specific example, relating representations of the double affine Hecke algebra to the geometry of Lagrangian submanifolds of the moduli space of Higgs bundles on the punctured torus.