Most of my research concerns geometry and topology, broadly defined. I have been interested in manifolds, their range of possible metrics and their group actions, and in stratified spaces, homology manifolds, and their invariants, the large scale geometry of discrete groups and other natural large scale metric spaces such as spaces of Riemannian metrics. I am currently interested in spaces of discontinuous functions, processes on large networks, and their scientific applications. In short, my focus these days is on quantitative and applied topology.
(This is experimental; talks are often written in a hurry and should be used for information and inspiration, not as completely reliable sources of fact or scholarship.)
Quantitative and Applied Topology: 2012 Clifford Lectures Hopefully more information to come soon.
Disordered solids and the dynamics of manifolds of bounded geometry, 2011 talk at ETH (Applied Algebraic Topology)
Group actions on aspherical manifolds, 2011 talk at Dubrovnik Topology Conference (handwritten)
Persistent homology of data, function spaces, and landscapes, 2010 Benter lecture at the Hong Kong City University
2008 AAAS talk on data
Taming 3-manifolds by Positive Scalar Curvature, 2009 talk at Vanderbilt
2008 talk on the work of Sylvain Cappell (at the Courant celebration of his birthday)
Playing the Novikov game, 2008 Hardy lecture in Edinburgh
Entropy, function spaces, complexity and variational problems, 2008 Hardy lecture in London
Matthew Strom Borman
Algebraic Topology Seminar
Point Set Topology
Second steps in topology
Network course additional readings
Quantitative Topology Readings (2010)
Differential Topology 2010
Readings for "Applied Algebraic Topology". (2009)
Readings for "Topology of Manifolds" (2008)
Readings for "Aspherical Manifolds" (2007)
Some of my favorite geometric arguments/theorems.
Fundamental group and Covering Spaces: the facts.
Degrees and intersections
Introduction to Mathematics (Numbers and Stuff)