MATH101 Calculus I
Further information about this module can be found on the associated VITAL pages.
I study the topology of stratified and stratifiable spaces using various techniques. These include essentially algebraic viewpoints (constructible derived categories, perverse sheaves etc) but also homotopy-theoretic ones (partially-ordered and transversal homotopy). Recent work includes
- relating natural stratifications of spaces of Bridgeland stability conditions on a triangulated category to t-structures in the category, and using this relation to prove contractibility of stability spaces in some cases;
- studying metric properties of Bridgeland stability conditions;
- (joint with Joerg Schuermann) describing the structure of Witt groups of perverse sheaves with applications to signature formulae for singular spaces;
- (joint with Conor Smyth) proving a version of the Tangle Hypothesis for Whitney categories, a kind of `n-category with duals' defined as a presheaf with glueing properties on a certain category of n-dimensional Whitney stratified spaces.
- Some metric properties of spaces of stability conditions. J. Woolf. Bulletin of the London Mathematical Society, Vol 44, no. 6, pp. 1274-1284, 2012. (arXiv:1108.2668)
- Transversal homotopy theory. J. Woolf. Theory and applications of categories, Vol 24, Issue 7, pp 148-178, 2010. (arXiv:0910.3322)
- Stability conditions, torsion theories and tilting. J. Woolf. Journal of the London Mathematical Society, Vol 82, Issue 3, pp 663-682, 2010. (arXiv:0909.0552)
- The fundamental category of a stratified space. J. Woolf. Journal of Homotopy and Related Structures, Vol 4, Issue 1 pp 359-387, 2009. (arXiv:0811.2580)
- Woolf J (2008) Witt groups of sheaves on topological spaces. J. Woolf. Commentarii Mathematici Helvetici, Vol 83, Issue 2, pp 439-494, 2008.(arXiv:math/0510196)
- Intersection pairings on singular moduli spaces of bundles over a Riemann surface and their partial desingularisations. L. Jeffrey, Y-H Kiem, F. Kirwan, and J. Woolf. Transformation Groups, Vol 11, Issue 3, pp 439-49, 2006.(arXiv:Math/0505362)
- The Kirwan map for singular symplectic quotients. Y-H. Kiem and J. Woolf. J. London Math. Soc. 73 no.1:209-230, 2006.
- The Kirwan map, equivariant Kirwan maps, and their kernels. L. Jeffrey and A.-L. Mare and J. Woolf. J. Reine Angew. Math. 589:105-128, 2005(arXiv:math/0211297)
- Intersection cohomology of symplectic quotients by circle actions. Y-H. Kiem and J. Woolf. J. London Math. Soc., 71, no. 2:531–544, 2005.
- Cohomology pairings on singular quotients in geometric invariant theory. L. Jeffrey, Y-H. Kiem, F. Kirwan and J. Woolf. Transform. Groups , 8 no. 3:217–259, 2003. (math.AG/0101079)
- The decomposition theorem and the intersection cohomology of quotients in algebraic geometry. J. Woolf. Journal of Pure and Applied Algebra, 182:317–328, 2003. (math.AG/0110137)
- Witt groups of perverse sheaves. J. Schuermann and J. Woolf. 2011.
- Whitney categories and the Tangle Hypothesis. C. Smyth and J. Woolf. Submitted to Geometry and Topology, 2011. (arXiv:1108.3724)
- Notes on Euler Calculus in an o-minimal structure. C. Henderson-Moggach and J. Woolf. 2013. (Euler Calculus)
- Algebraic stability conditions and contractible stability spaces. J. Woolf. 2014 (arXiv:1407.5986)
Slides from talks
- Witt groups and Witt spaces, given at Topology and Analysis in Interaction, Oberwolfach, March 2006.
- Introduction to derived and triangulated categories, given at the Peripatetic Seminar on Sheaves and Logic, Glasgow, May 2006.
- Transversal Homotopy Theory, given at Queens' University, Belfast, Pure Mathematics Colloquium, March 2010.
- Introduction to topology for Social Scientists, given at University of Manchester, February 2010.
- Transversal Homotopy Theory and the Tangle Hypothesis, given at Transpennine Topology Triangle meeting, University of Leicester, November 2010.
- Intersection Cohomology and Perverse Sheaves, given at Stratified Spaces meeting, Oberwolfach, December 2011.
This is a (fairly primitive) webtool for investigating the potential renewable energy contribution to a building in the UK. It was developed whilst I was at the Martin Centre in Cambridge but I haven't done any work on it for a long time and probably never will now.