Ursula Ludwig Home Page
The main focus of my research lies in the interaction of geometry, topology and analysis on singular spaces. Singular spaces arrive naturally in geometry, the most prominent examples being quotients of manifolds under group actions, or - in algebraic geometry - zero sets of polynomia.
I have studied singular spaces from different viewpoints: After my DEA and Diploma-thesis in algebraic geometry, I switched for my PhD project to studying singular spaces with methods from differential topology, differential geometry and dynamical systems.
In the last few years I started to use intersection homology tools on singular spaces on the one hand and analytic techniques (L2-techniques) as well as methods from global analysis on the other.
In particular I worked successfully on the generalisation of the Witten deformation and the Cheeger-Müler Theorem to singular spaces.
- topological and analytic invariants of singular spaces
- (stratified) Morse-Novikov theory
- Witten deformation
- global analysis, index theory, analytic and topological torsion
submitted
- An Extension of a Theorem by Cheeger and Müller to Spaces with Isolated Conical Singularities,
submitted, 74 pages, 2018. The results of this article have been announced in a note aux CRAS (see below).
published or accepted for publication
- An Index Formula for the Intersection Euler Characteristic of an Infinite Cone,
to appear Math. Z., DOI:10.1007/s00209-019-02423-5, 28 pages.
- Comparison between two complexes on a singular space,
J. Reine Angew. Math. (Crelle) 724, 1-52, 2017.
- A complex in Morse theory computing intersection homology,
Ann. Inst. Fourier 67 (1), 197-236, 2017.
- An analytic approach to the stratified Morse inequalities for complex cones,
Int. J. Math., Vol. 24, No. 12, 2013, arXiv:1107.1636.
- The Witten deformation for even dimensional conformally conic manifolds,
Trans. Amer. Math. Soc. 365, no. 2, 885-909, 2013, arXiv:1011.5357.
- A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation,
Ann. Inst. Fourier 61, No. 5, 1749-1777, 2011.
- The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions,
Ann. Inst. Fourier 60 (5), 1533-1560, 2010.
- The Witten complex for singular spaces of dimension 2 with cone-like singularities,
Math. Nachrichten 284 (No 5-6), 717-738, 2011.
- Morse-Smale-Witten complex for gradient-like vector fields on stratified spaces,
Proceedings of the 2005 Marseille singularity school and conference, CIRM, Marseille, France, January 24--February 25, 2005. World Scientific, 683-713, 2007.
Notes published in CRAS:
- An Extension of a Theorem by Cheeger and Müller to Spaces with Isolated Conical Singularities,
C. R., Math., Acad. Sci. Paris 356, No. 3, 327-332, 2018.
- An Index Formula for the Intersection Euler Characteristic of an Infinite Cone,
C. R., Math., Acad. Sci. Paris, 355, No. 1, 94-98, 2017.
- Comparison between two complexes on a singular space,
C. R., Math., Acad. Sci. Paris 350, No. 9-10, 525-528, 2012.
- The Witten deformation for even dimensional spaces with cone-like singularities and admissible Morse functions,
C. R., Math., Acad. Sci. Paris 348, No. 15-16, 915-918, 2010.
- The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions,
C. R., Math., Acad. Sci. Paris 347, No. 13-14, 801-804, 2009.
- The Witten complex for algebraic curves with cone-like singularities,
C. R., Math., Acad. Sci. Paris 347, No. 11-12, 651-654, 2009.
Contribution to OWF-Reports:
- Comparison between two complexes on a singular space,
in OWF-Report 56/2011, for the Workshop Stratified Spaces: Joining Analysis, Topology and Geometry, Dec 11 - Dec 17, 2011.
- The Witten deformation for singular spaces with cone-like singularities,
in Oberwolfachreport 07/2010, for the Workshop Analysis and Geometric Singularities, June 27th - July 3rd, 2010.