STAG

Objective:

"The coordinated reseach proposal contains a very comprehensive set of objectives and developments in Singularity Theory and its applications from a modern viewpoint. At BCAM node we will focus at: (1) Lipschitz Geometry (joint with Madrid team): developing further the recently introduced theory of MD-Homology and apply it to advance in the comprehension of Lipschitz Geometry of singularities. Connections with equisingularity theory and resolution of singularities will be explored. We expect to find connections and applications non-arquimedean geometry. (2) Floer and Contact Homology and their connections with arc spaces (joint with Madrid, Zaragoza and our parters in Leuven and Budapest). Recent discoveries of McLean, Budur, Bobadilla, Lê, Nguyen and Nemethi hint a new connection between Floer theories of symplectic and contact manifolds associated with singularities and algebro-geometric objets like arc spaces, line bundles and their cohomology, and intersection lattices. Precise an appealing conjectures are formulated, and many more are waiting for their discovery. Applications to long-standing problems like Le-Ramanujam and Zariski conjectures could be derived. (3) Characteristic classes on singular varieties. Recently Bobadilla and Pallares have designed a new method based on cubical hyperresolutions, perverse sheaves and Hodge theory and proved with it a conjecture of Brasselet-Schurman-Yokura on L-classes of singular varieties. The method can be further applied to get much finer information on the geometric significance of the various L-classes, and other charactristic classes of singular varieties. (4) Palka and Pelka have succeded to make significant progress in a set of long-standing conjectures on affine algebraic geometry using the Minimal Model Program. Such study is by no means complete, and Pelka will go further on it. The knowledge of Pelka on MMP can be combined with the expertise of Bobadilla and Pallares in perverse sheaves to study the sheaf of nearby cycles and approach the Lê- Ramanujam conjecture. (5) Bobadilla and Romano have recently classified special Maximal Cohen-Macaulay modules on Gorenstein normal surface singularities. The technique can be further applied to the non-Gorenstein/non-special classification, which would have impact in the study of derived categories of coherent sheaves at singularities. The technique also matches with the study of the Zaragoza team, Laszlo and Nemethi of embedded curves in surfaces via associated reflexive modules, and we plan to join forces to deepen in this relation. "