My current research interests are in the emerging area of computational topology. Recent projects include the analysis of various data sets via persistent homology techniques. For example, in joint work with Laura Sjoberg (UF Political Science) I have studied the structure of point clouds constructed from empirical measures of democracy, and I am currently investigating college student education data with the hope of finding predictors of academic success. On the theoretical side, I have refined various results about multi-dimensional persistence; this ongoing project aims to find more complete invariants of these structures. Also, in collaboration with Henry King and Neza Mramor, I have done work in discrete Morse theory. We have developed algorithms to generate discrete Morse functions from point cloud data and to study birth-death phenomena in families of discrete Morse functions. Joint work with Ulrich Bauer connects discrete Morse theory with persistence.
My original research area was the study of the homology of linear groups. I did a great deal of work in this field, beginning with the calculation of the homology of the special linear group over various polynomial rings, including coordinate rings of elliptic curves. The latter result led to an interesting theorem about the second K-group of elliptic curves. However, my primary focus was on attempts to prove the Friedlander-Milnor conjecture, which relates the cohomology of the discrete group of k-rational points of a reductive algebraic group to the etale cohomology of the simplicial classifying scheme of the group. Joint work with Mark Walker establishes a link between this problem and algebraic cycles, thereby putting the problem in a motivic context. I published a research monograph, Homology of Linear Groups, in 2000, which has become the standard reference in the field.
I have also done some work on miscellaneous problems in topology. I have studied Deligne’s notion of relative completion for linear groups over several types of rings. I have also studied the Gassner representation of the pure braid group, obtaining information about the size of the kernel.
In addition to my original research, I have a keen interest in public outreach and mathematical communication. I am a contributor to Forbes, where I write regularly about mathematics and its applications. I am also a columnist for The Conversation, writing about math for a general audience.