My main research interest is a long-term project to make fundamental computations in motivic homotopy theory. This research is supported by a grant from the National Science Foundation.  

High-dimensional spheres are the basic building blocks of geometry.  More complicated geometric objects can be constructed by fitting these spheres together.  It turns out that spheres of different dimensions can fit together in only a few different ways.  Enumerating these combinations of spheres is one of the fundamental questions of stable homotopy theory.  

This chart represents one possible approach to the problem.  Each dot represents a particular combination of spheres.  The lines indicate certain relationships between these different combinations.  You can view a presentation that I gave on this topic in January 2014 at the Mathematical Sciences Research Institute in Berkeley, California, USA, and another presentation in August 2017 at a conference at Reed College in Portland, Oregon, USA.

I use a tool called the Adams spectral sequence to carry out these computations.  It is delicate, intricate, subtle, and complicated, but it is also understandable with enough insight and patience.  Each time we understand a new part of the machinery, another layer of complexities becomes accessible for further study.  By grappling with the difficulties of the Adams spectral sequence, we are widening human knowledge and developing new skills for thinking about abstract mathematics. 

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