This semester’s RTG lecture will be split between a course by Roman Sauer on “Torsion invariants” and a course by Andy Sanders on “Harmonic maps”. On every RTG Day there will be one lecture of each course.

**Title: Torsion Invariants (Roman Sauer)**

*Abstract:* Torsion invariants in algebraic topology are secondary invariants in the sense that they are only defined if some primary homotopy invariants, like Betti numbers, vanish. Torsion invariants often reveal more about a space than its homotopy-theoretic properties. We will discuss examples of lens spaces that are homotopy equivalent but not homeomorphic - the homeomorphism type is distinguished via Reidemeister torsion. We also discuss situations where torsion invariants reveal geometric properties of Riemannian manifolds which is quite surprising given their algebraic-topological definition. Later in the course more recent developments around L2-torsion will be addressed.

**Title: Harmonic maps (Andy Sanders)**

*Description:* A harmonic map between two Riemannian manifolds is defined as a critical point of an energy functional, generalizing the Dirichlet energy functional arising in the study of harmonic functions. In particular, harmonic functions, geodesics, minimal submanifolds, and isometries all give examples of harmonic maps. Generally, a harmonic map satisfies a second order elliptic semilinear partial differential equation. As a result, harmonic maps have a rich existence and uniqueness theory, leading to a wide variety of applications in differential geometry, topology and representation theory.

In these lectures, we will give a broad overview of the general theory, with a focus on the role of harmonic maps as a tool to probe the geometry of manifolds. We will be particularly interested in harmonic maps from two dimensional Riemannian manifolds, where the harmonic map equation becomes conformally invariant, leading to a rich interplay between harmonic and holomorphic objects on Riemann surfaces.