Qayum Khan

Research Assistant Professor

Kenna Instructor in Mathematics

B.S., University of Illinois at Chicago, 1998
Ph.D., Indiana University, Bloomington, 2006

Research Interests

I work in the Topology of High-dimensional Manifolds.  Loosely speaking, a manifold is a smooth shape, without any edges or singularities. Its dimension is the number of local degrees of freedom in that abstract space.  The ultimate goal is to classify the homeomorphism types of manifolds within a fixed homotopy type.

In other words, one hopes to discover numerical invariants that catalog the diversity of geometric topologies within the same algebraic topology of manifold.  The main mechanism is surgery theory.  It works completely in dimensions at least five, but it only holds partially in dimension four.   On the other hand, the methods of dimension three depend solely on the first homotopy group.

I focus on both the application of Cappell's hypersurface method and the calculation of its obstruction groups.  Currently, my topics of research interest are:  4-dimensional manifolds (topological surgery sequence), high-dimensional geometric topology (equivariant rigidity & fibering problems), and algebraic topology (exotic Nil-groups in K- and L-theory).

Selected Publications

• Manifolds homotopy equivalent to P^n # P^n (with J. Brookman and J.F. Davis), Mathematische Annalen, 338 (4): 947-962, 2007
• On smoothable surgery for 4-manifolds, Algebraic & Geometric Topology, 7: 2117-2140, 2007
• On fibering and splitting of 5-manifolds over the circle, Topology and its Applications, 156(2): 284-299, 2008
• Reduction of UNil for finite groups with normal abelian Sylow 2-subgroup, Journal of Pure and Applied Algebra, 3(3): 279-298, 2009
• Calculation of UNil for the cyclic group of order two, Forum Mathematicum, accepted (19 May 2008), in press (19 pages)
• Dihedral manifold approximate fibrations over the circle (with C.B. Hughes), Geometriae Dedicata, published online (22 Sep 2009), online first (39 pages)

Please direct questions and comments to: Q K H A N at N D dot E D U