Craig Westerland's Homepage

I wrote a thesis at the University of Michigan under the direction of Igor Kriz, and my field of specialty is algebraic topology, particularly homotopy theory. I am generally interested in applications of topology to geometry and mathematical physics, and someday hope to promote my status as a part-time number theory poser to full-time poser.

Contact information Office: 521 Van Vleck Phone Number: 608.263.3168 Fax: 608.263.8891 Email Address: westerla AT math DOT wisc DOT edu Mailing Address:  University of Wisconsin-Madison Mathematics Department 480 Lincoln Dr. Madison, WI 53706-1388, USA Teaching Spring '08: Math 752 (Introductory Topology II). Spring '07: Math 752 (Introductory Topology II). Fall '06: Math 221 (Calculus and Analytic Geometry). Fall '05: Math 221 (Calculus and Analytic Geometry) and Math 751 (Introductory Topology I).

The professional me

Here are this year's (9/15/07) versions of my CV and research statement.

Papers

• Dyer-Lashof operations in the string topology of spheres and projective spaces, in Mathematische Zeitschrift 7/05.  The original publication is available at www.springerlink.com.  Abstract: We compute the 2-primary Dyer-Lashof operations in the string topology of several families of manifolds, specifically spheres and a variety of projective spaces. These operations, while well known in the context of iterated loop spaces, give a collection of homotopy invariants of manifolds new to string topology. The computations presented here begin a first exploration of these invariants.  Apology to Kudo and Araki: Browder has pointed out that since these operations are computed at the prime 2, they are Kudo-Araki operations, not Dyer-Lashof operations.  Unfortunately, the paper was published before I could revise the language.
  

• Stable splittings of surface mapping spaces, in Topology and its Applications, 9/06.  Abstract: We study the homotopy type of mapping spaces from Riemann surfaces to spheres.   Our main result is a stable splitting of these spaces into a bouquet of new finite spectra. From this and classical results one may deduce splittings of the configuration spaces of surfaces.

• String homology of spheres and projective spaces in Algebraic and Geometric Topology, '07.  Abstract: We study a spectral sequence that computes the S¹–equivariant homology of the free loop space LM of a manifold M (the string homology of M). Using it and knowledge of the BV operations on HH^*(H^*(M), H^*(M)), we compute the (mod 2) string homology of M when M is a sphere or a projective space.

• Equivariant operads, string topology, and Tate cohomology, in Mathematische Annalen.   Abstract: From an operad C with an action of a group G, we construct new operads using the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized G-Tate cohomology of C is trivial. Applying this theory to the little disk operad C₂ (which is an S¹ operad) we obtain variations on Getzler's gravity operad, which we show governs the Chas-Sullivan string bracket. Other examples from the higher dimensional little disk operads are studied. 

Preprints

• String topology prospectra and Hochschild cohomology, with K. Gruher. Abstract: We study string topology for classifying spaces of connected compact Lie groups, drawing connections with Hochschild cohomology and equivariant homotopy theory. First, for a compact Lie group G, we show that the string topology prospectrum LBG^-TBG is equivalent to the homotopy fixed-point prospectrum for the conjugation action of G on itself, G^hG. Dually, we identify LBG^-ad with the homotopy orbit spectrum (DG)_hG, and study ring and co-ring structures on these spectra. Finally, we show that in homology, these products may be identified with the Gerstenhaber cup product in the Hochschild cohomology of C^*(BG) and C_*(G), respectively. These, in turn, are isomorphic via Koszul duality.
  

• Koszul duality and the framed disk operads (in progress).

Me, in yellow

If I were a Springer-Verlag Graduate Text in Mathematics, I would be Frank Warner's Foundations of Differentiable Manifolds and Lie Groups. I give a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. I include differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition provide a proof of the de Rham theorem via sheaf cohomology theory, and develop the local theory of elliptic operators culminating in a proof of the Hodge theorem. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find me extremely useful. Which Springer GTM would you be? The Springer GTM Test