Three rather ambitious projects are proposed in the area of geometric group theory. The first is inspired by Zlil Sela's geometric solution of Tarski's logic problem. The PI and Mladen Bestvina plan to complete their program to solve by geometric means two forty-year-old problems of Mal'cev. They also propose to generalize their methods and provide an alternate proof to the Tarski problem. The second project, joint with Michael Handel, is to solve the conjugacy problem for the outer automorphism group Out(F) of a free group F. Much of the research in this area is driven by the similarities between linear groups, surface mapping class groups, and Out(F). The conjugacy problem has been solved for these other two classes. The final project, again joint with Mladen Bestvina, is to better understand the geometry of outer automorphism groups of free groups. In particular, it is proposed to show that a complex analogous to the curve complex in the mapping class group setting is word hyperbolic.
Three projects are proposed in the area of geometric group theory. Geometric group theory is a relatively young branch of mathematics in which problems from other areas of mathematics are reformulated and then solved in geometric terms. This approach has been successful in areas that are sometimes viewed as distant from geometry. For example, Zlil Sela used geometric methods to solve an old logic problem of Tarski. Inspired by Sela's methods, the PI and Mladen Bestvina propose to solve two old logic problems of Mal'cev. Group theory is another area where these methods have been particularly successful. Groups are ubiquitous in math and the sciences. The set of symmetries of a molecule is an example. There is an important class of groups called "free groups" from which all other groups can be constructed. The set of symmetries of a free group F is another important group, denoted Out(F), which has been the subject of much current interest. The PI proposes two other projects focusing on Out(F). In one, he and Michael Handel propose to solve an old and fundamental problem on Out(F), namely that its conjugacy problem is solvable. In the other project, the PI and Bestvina propose to better understand the intrinsic geometry of Out(F).