Principal Investigator: Kiyoshi Igusa
The principal investigator proposes to develop and use his three new formulas for the Miller-Morita-Mumford classes on the cohomology of the mapping class group to investigate some of the main problems in geometric topology, in particular the question of whether these invariants are nontrivial on the Torelli group. Igusa also plans to integrate the basic ideas into the first year graduate topology course at Brandeis. The fundamental concept being used is that of an A-infinity functor. Since ordinary homology and cohomology have intrinsic A-infinity structures and since A-infinity functors give an alternative approach to spectral sequences, it would be both feasible and appropriate to introduce this concept at an early stage and to engage beginning graduate students and gifted undergraduates into research related topics at an early stage. The lecture notes would be made available online and expository papers would also be posted on preprint servers to explain and disseminate the current uses of this old concept.
This research is about graphs and surfaces. The study of the space of all graphs and the space of surfaces is an active area of research in Topology and Algebraic Geometry which has captured the attention of mathematicians and theoretical physicists in recent years because of the work of Ed Witten and Maxim Kontsevich both of whom received Fields Medals for their influential work. Igusa brings an innovative approach to this field by apply new methods in high dimensional Algebraic Topology to the study of these one and two dimensional objects. Igusa's methods have already solved several previously unanswered questions, both theoretical and computational. In this project Igusa plans to tackle some of the fundamental theoretical problems in this field which his preliminary research has found to be amenable to his methods. He also proposed to educate the Mathematical public to demonstrate the accessibility of these exotic methods. The impact of this research will be to accelerate the expansion of this important area of research and to make basic Topology more exciting to the students who study it by incorporating topics of current research.
