RIGIDITY · Rigidity: Groups, Geometry and Cohomology

Our proposal has three components:1. Unitarizable representations.2. Spaces and groups of non-positive curvature.3. Bounds for characteristic classes.The three parts are independent and each one is justified by major well-known conjectures and/or ambitious goals. Nevertheless, there is a unifying theme: Group Theory and its relations to Geometry, Dynamics and Analysis.In the first part, we study the Dixmier Unitarizability Problem. Even though it has remained open for 60 years, it has witnessed deep results in the last 10 years. More recently, the PI and co-authors have obtained new progress. Related questions include the Kadison Conjecture. Our methods are as varied as ergodic theory, random graphs, L2-invariants.In the second part, we study CAT(0) spaces and groups. The first motivation is that this framework encompasses classical objects such as S-arithmetic groups and algebraic groups