2007 Lecturer: Karen Vogtmann
Automorphisms of Free Groups, Outer Space and Beyond
Abstract:
Outer space was introduced in the mid-1980s as a tool for studying the group Out(Fn) of outer automorphisms of a finitely-generated free group. The basic philosophy is that one should think of an automorphism of a free group as a topological object, either as a homotopy equivalence of a finite graph or as a diffeomorphism of a suitable three-manifold with free fundamental group. There are compelling analogies between the action of Out(Fn) on Outer space and the action of an arithmetic group on a homogeneous space or the action of the mapping class group of a surface on the associated Teichmuller space. In this talk I will first describe Outer Space and explain how it is used to obtain algebraic information about Out(Fn). I will then indicate how Outer Space is related to other areas, from infinite-dimensional Lie algebras to the mathematics of phylogenetic trees, and how ideas from Outer Space are currently expanding in new directions.
Brief Biography:
Inspired to pursue mathematics by an NSF summer program for high school students at the University of California, Berkeley, Karen Vogtmann received both her undergraduate and graduate degrees from Berkeley, investigating algebraic K-theory with Jack Wagoner. After wandering the academic world from Michigan to Brandeis, Columbia to the Institute for Advanced Studies, and back, she settled at Cornell University where she has been for the last twenty years. A profound mathematician, she has authored numerous articles, mentored eight PhD students, and averaged ten invited talks a year. Vogtmann has served as Vice President of the American Mathematical Society and on scientific advisory boards of the American Institute of Mathematics, the Mathematical Sciences Research Institute, the arXiv advisory board, the National Academy of Sciences Delegation to the International Mathematical Union General Assembly, and the Vietnam Education Foundation Panel for mathematics.
Vogtmann’s research views groups as symmetries of geometric objects. By understanding the geometry and topology of suitably chosen objects, she deduces algebraic information about the groups acting on them. Her work investigates orthogonal and symplectic groups, SL(2) of rings of imaginary quadratic integers, groups of automorphisms of free groups, and mapping class groups of surfaces. Vogtmann’s recent focus has been on the group of outer automorphisms of a free group where the appropriate geometric object is called Outer Space. This space turns out to have surprising connections with other areas of mathematics, for example with certain infinite-dimensional Lie algebras and even with the study of phylogenetic trees in biology.
