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NMST Report Summary

Project ID: 615655
Funded under: FP7-IDEAS-ERC
Country: Spain

Mid-Term Report Summary - NMST (New methods and interacions in Singularity Theory and beyond)

A team is working in the different aspects of the proposal. After half of the life of the project we have made good progress in most of the aspects. Here is a non-exhaustive selection of results:

- We have advanced in detecting obstructions for the existence of Banagl intersectio spaces, we have constructed intersection spaces for toric varieties, and we have advances in the construction of intersection space sheaves for non-isolated singularities.

- We made a substantial contribution to Mond's conjecture.

- We studied equisingularity of non-reduced curves.

- We improved previous results on bilipschitz invariance of multiplicity.

- We provided a formula to compute gradient Lojasiewicz exponents in terms of polar quotients.

- We proved a conjecture of M. Oka on real singularities with Milnor fibration.

- We studied singularities of the Hilbert scheme of effective divisors of surfaces in the 3-dimensional projective space, contributed to the study of determinant maps from moduli of semistable shaves to the Picard variety, and studied Kollar’s index under specialization.

- We provided an algorith to determine the link of certain normal surface singularities from the defining equation.

- We have studied to what extent A’Campo tête à tête graphs model surface automorphisms, and have elaborated on the notion of mixed tête à tête automorphisms.

- We have made substantial progress in extending McKay correspondence for an arbitrary Gorenstein normal surface singularity.

- We obtained several relevant results in the theory of motivic Milnor fibres, aiming to the general description of the motivic Milnor fibre of the sum of two functions, generalizing motivic Thom-Sebastiani Theorem. We also obtained an explicit description of the motivic Milnor fibre of a plane curve in terms of a resolution graph.

- We have obtained new results on generalised Nash problem for smooth germs of surfaces. In particular we proved that is a topological problem, extracted some consequences, and applied our results to the classical problem of adyacencies.

- We have generalized, to the general curve case the existence of an embedding such that a single toric modification of the ambient produces a resolution of the curve.

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