Danny Calegari will study the relationship between geometry, group theory and dynamics in a number of contexts where the common theme will be the use of homological tools. Low dimensional topological spaces can be very effectively probed by maps of surfaces. The linearization of this study leads to homology and its variants (rational homotopy, algebraic K-theory, etc.); the interaction with geometry leads to further refinements (bounded cohomology, quasimorphisms) which are at the heart of phenomena such as universal circles, Thurston norm, the 11/8 conjecture, etc. The interaction of bounded cohomology with number theory leads to new and largely unstudied phenomena, with connections to fixed point problems in dynamics, and topological problems like the simple loop conjecture. By exploring these phenomena in terms of a single underlying abstract tool (namely stable commutator length) Calegari intends to develop machinery with which to explain their unity, and to settle several open conjectures. The main objects of study are spectral gaps and rationality of stable commutator length in word-hyperbolic (and CAT(0)) groups, and central limit theorems for quasimorphisms on arbitrary groups.
Topology is the qualitative study of geometry. After centuries of work, and some spectacular progress in recent years (especially Perelman's proof of the Poincare Conjecture, and Brock-Canary-Minsky's proof of the ending lamination conjecture) some very fundamental questions are still very mysterious. If you pull a loop of string tight on a surface, how will it cross itself? What is the most efficient (abstract) way to encode a data structure in a computer program? How do vortices in fluids or strands of DNA in a cell become knotted, and how does this knottiness affect their structural stability? Sometimes the best "metrics" are qualitative; this project studies such qualitative metrics of size and efficiency in the simplest possible topological contexts, since the results we obtain will be universal in their applicability to other, more complicated contexts.
