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. This page summarizes my research work for non-experts. If you would like more detail, see my pages on motivic homotopy theory and other research interests.
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.
The chart below 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.
Sometimes people (including mathematicians) ask how we are confident that our complicated, lengthy, and detailed computations are correct. This is an excellent question that deserves careful consideration. First of all, we rely on computer calculations whenever possible. However, eventually we want to go beyond what can be verified by computer.
The different parts of the structure that we are calculating are linked in multiple complicated ways with each other. As a result, when we make a mistake, inconsistencies typically arise quickly with other parts of the calculation. This keeps us on the right track because the method itself catches many errors.
One possible analogy is to building a model of a ship. Imagine having all of the parts of a kit for building a ship, but no instructions. With enough patience and attention to detail, you might be able to figure out how to build the ship. You might make mistakes along the way, which you can correct later when you realize that other parts won’t fit. In the end, if you’ve used all the parts, and the result floats in water, then you’ve probably built the model correctly.