**Title:** Inequalities in Topology Motivated by the Schwarz Lemma

**Abstract:** We will discuss a number of inequalities that can be stated in purely topological terms, but their proofs may involve other ideas. The protoptype is Knesers inequality for the degree of a map of Riemann surfaces, published in 1930: if *X*, *Y* are Riemann surfaces of genus (*g*_{X},g_{Y}) > 1 and *f* : *X* → *Y* is a continuous map, then |*degree*(*f*)| ≤ (*g*_{X} − 1)/(*g*_{Y} − 1). If *X*, *Y* have complex structures and *f* is holomorphic, then the inequality would be an immediate consequence of the Schwarz Lemma: holomorphic maps of the unit disk do not increase length in the Poincaré metric. We will discuss proofs of this inequality and related ones by various methods: bounded cohomology, harmonic maps, Higgs bundles. We will also indicate, as time permits, how these inequalities have motivated much work on the structure of the space of representations of the fundamental group of a surface in various Lie groups. This will be an exposition of work of Milnor, Wood, Dupont, Goldman, Gromov, Hitchin, and many others. It should be mentioned that some of the current work on this subject uses Ahlfors generalization of the Schwarz Lemma in a very essential way.

**Date and time:** Tuesday, January 16 at 4:00 in room 135 TMCB.