The PI recently proved the Strengthened Hanna Neumann Conjecture (SHNC), and developed a topological and analytic language that allows one to generalize its statement from free groups and graphs to more general groups and complexes. The more general property is called submultiplicativity. This restatement of SHNC shows that there is deep relation between SHNC and the Atiyah Conjecture, which comes from analysis. The PI introduced the deep-fall property, which is useful for each conjecture. This opens a new direction research, which the PI is planning to pursue: finding examples of group actions and matrices that satisfy the deep-fall property. Another part of the proposal is to investigate how PI's earlier constructions of geodesic flow and symmetric join help understanding the geometry of manifolds and more general metric spaces.
The PI's research interleaves themes and techniques from three core areas of mathematics: algebra, analysis, and geometry. A group is one of the central notions in algebra. The Hanna Neumann Conjecture, posed by Hanna Neumann in 1956-1957, is a question about a special class of groups called free groups. The PI recently proved this conjecture, and provided a more general point of view that allows asking similar questions in a more general setting, and using the tools from analysis. The PI's program is now to answer similar kinds of questions in this more general setting. Another goal is to investigate the structure of metric spaces, building upon PI's past results in this area. The goal is therefore, broadly, to relate these various branches of mathematics. The PI also plans to present his research results at national and international meetings and to invite research collaborators to the University of Illinois at Urbana-Champaign.
