My current research interests are in the emerging area of computational topology. Recent projects include the analysis of human speech data via persistent homology techniques, and refining various theoretical results about multi-dimensional persistence. 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.

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 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. My full publication list, broken down by area, is shown below.

Publications

Homology of Linear Groups and K-theory

The homology of \( SL_2(F[t,t^{-1}]) \), J. Algebra 180 (1996), 87-101.

The homology of special linear groups over polynomial rings, Ann. Sci. Ecole Norm. Sup. (4) 30 (1997), 385-416.

Congruence subgroups and twisted cohomology of \( SL_n(F[t]) \), J. Algebra 207 (1998), 695-721.

Integral homology of \( PGL_2 \) over elliptic curves, Algebraic K-theory, Seattle, WA, 1997, Proc. Symp. Pure Math. 67 (1999), 175-180.

On the K-theory of elliptic curves, J. reine angew. Math. 507 (1999), 81-91.

Unstable homotopy invariance and the homology of \( SL_2(Z[t]) \), J. Pure Appl. Algebra. 148 (2000), 255--266.

Congruence subgroups and twisted cohomology of \( SL_n(F[t]) \) II. Finite fields and number fields, Comm. Alg. 29 (2001), 5465--5475.

Unstable homotopy invariance for finite fields, Fund. Math. 175 (2002), 155--162.

(with M. Walker) Homology of linear groups via cycles in \( BG x X \), J. Pure Appl. Algebra 192 (2004), 187--202.

Homology and finiteness properties for \( SL_2(Z[t,t^{-1}]) \), Algebr. Geom. Topol. 8 (2008), 2253--2261.

Relative Completions

Relative completions of linear groups over \( Z[t] \) and \( Z[t,t^{-1}] \), Trans. Amer. Math. Soc. 352 (2000), 2205--2216.

Correction to "Relative Completions...", Trans. Amer. Math. Soc. 353 (2001), 3833--3834.

Relative completions and the cohomology of linear groups over local rings, J. London Math. Soc. (2), 65 (2002), 183--203.

Relative completions of linear groups over coordinate rings of curves, Comm. Alg. 35 (2007), 3904--3908.

Relative completions of linear groups L'Enseign. Math., 54 (2008), 119--121.

Computational Topology and Discrete Morse Theory

(with H. King and N. Mramor) Generating discrete Morse functions from point data, Experimental Math. 14 (2005), 435--444.

(with H. King and N. Mramor) Birth and death in discrete Morse theory, J. Symbolic Comput. 78 (2017), 41--60.

A refinement of multi-dimensional persistent homology, Homology, Homotopy Appl., 10 (2008), 259--281.

(with K. Brown) Nonlinear statistics of human speech data, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), 2307--2319.

(with L. Sjoberg) Theoretical geometry, critical theory, and concept spaces in IR, submitted (2015).

(with K. Brown)Persistent homology for speech recognition, preprint (2008).

Persistent homology as discrete Morse theory, in preparation.

Miscellaneous Topology Papers

On the kernel of the Gassner representation, Arch. Math. (Basel), 85 (2005), 108--117.

The homology of invariant group chains, J. Algebra 298 (2006), 15--33.

Invariant chains and the homology of quotient spaces, preprint(2004).

Books

Homology of Linear Groups, Progress in Mathematics, vol. 193, Birkhauser Boston, 2000.

Morse Theory: Smooth and Discrete, World Scientific, Singapore, 2015.

POPULARIZATIONS

Topology looks for the patterns inside big data, The Conversation, May 18, 2015.

John Nash: A beautiful mind and its exquisite mathematics, The Conversation, May 26, 2015.

Don't freak if you can't solve a math problem that's gone viral, The Conversation, June 25, 2015.

Franz and Georg: Cantor's Mathematics of the Infinite in the work of Kafka, Journal of Humanistic Mathematics 7 (2017), 147--154.