Final Report Summary - NMST (New methods and interacions in Singularity Theory and beyond) Singularity Theory is a highly interdisplinar topic, that combines ideas and techniques from various parts of mathematics. NMST was a research project designed to advance in Singularity Theory several directions. We have succeeded to generate new interesting knowledge in most of them, and in some cases the answers are very complete. Some of these complete developments are: Concerning Poincare Duanlity Theories for pseudomanifolds and algebraic varieties we have produced a complete theory of Intersection Space Complex, which should be regarded as the analogue to Intersection Homology Complexes in the realm of Banagl’s Intersection Space Theory. Such a theory includes an obstrction theory scheme to determine when the construction is possible. We have proved that the theoru applies to large classes of algebraic varieties including toric ones.We have completely developed A’Campo conjectural theory of tête à tête graphs and twists for the curve case. The final result is that tête à tête graphs twists coincide with monodromies of functions in normal surface singularities. Such a beautiful characterization was one of our most optimistic previsions.We have developed a theory that classifies Maximal Cohen Macaulay Modules in gorenstein surface singularities, and studies their deformations and moduli. The final result was very unexpected: indecomposable special MCM modules are in bijection with discrepancy=0 divisorial valuations over the singularity, and generalizes beautifully previous work of McKay, Gonzalez-Sprinberg, Artin, Verdier, Esnault, Wunram, Kahn, Greuel, Drodz, Kashuba... As a consequence of our results we prove the Gorenstein case of a conjecture of Drodz, Greuel and Kashuba.We have proved a conjecture of Brasselet, Schurmann and Yokura on L-type characteristic classes of algebraic varieties which are rational homology manifolds. We have completely resolved the abstract Lipschitz analogue of Zariski multiplicity conjecture. We have developed a complete metric algebraic topology theory, which provides a rich set of invariants for the emerging field of Lipschitz geometry of singularities. We have provided progress in disentanglement theory, with several advances towards the undestanding of Mond’s conjecture.We have provided significant progress towards the understanding of the Seiberg-Witten invariant of the link of a normal surface singularity, and its relation with Poincare series and zeta functions associated with the singularities.We made progress in several directions focussing at the interplay between singularity theory and algebraic geometry: toric resolutions and spaces of valuations, degeneration of jacobians, cohomology ring of moduli spaces, cohomology jump loci ...
