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Participants

Arnaud Bodin (Lille) Nero Budur  (Leuven) Octave Curmi (Lille) James Damon  (Chapel Hill) Anne Frühbis-Krüger (Hannover) Laurentiu Maxim (Bonn) Daniel Matei  (Bucuresti) A.J. Parameswaran (Mumbai) Youssef Hantout  (Lille)

Patrick Popescu-Pampu (Lille) Dirk Siersma (Utrecht) Jan Stevens (Göteborg) Yongqiang Liu (Leuven) Mihai Tibar (Lille) Matthias Zach (Hannover)

 

Talks

Laurentiu Maxim Characteristic classes of singular complex hypersurfaces

Abstract.

An old problem in geometry and topology is the computation of topological and analytical invariants of complex hypersurfaces, e.g. Betti numbers, Euler characteristic, signature, Hodge-Deligne numbers, etc...
While the non-singular case is easier to deal with, the singular setting requires a subtle analysis of the intricate relation between the local and global topological and/or analytical structure of singularities.
In this talk I will explain how to compute characteristic classes of complex hypersurfaces in terms of local invariants of singularities.
This is joint work with S. Cappell, M. Saito, J. Schuermann and J. Shaneson.

James Damon   Topology of Exceptional Orbit Hypersurfaces of Prehomogeneous Spaces

Abstract.

We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as “exceptional orbit varieties” of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. 
These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors.
Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. 
We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse-type arguments on the Milnor fibers.
This includes replacing the local Milnor fiber by a global Milnor fiber which has a “complex geometry” resulting from a transitive action of an appropriate algebraic group, yielding a compact “model submanifold”  for the homotopy type of the Milnor fiber. 
Unlike isolated singularities, the cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial.
  We also deduce from Bott’s periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the “stable range”.

Dirk Siersma  Milnor fibre homology via deformation

Abstract

We study hypersurfaces of dimension n with a 1-dimensional singular set and consider admissible deformations for the study of the Milnor fibre.
It’s homology depends very much on the types of special singularities in the deformation.
In dimension n-1 there are strong bounds related to the minimum of the n-1 Betti numbers of the special fibres.
In several cases we can even show that the homology is concentrated in dimension n only.
The same technique has been used by us to compute the vanishing homology of projective hypersurfaces with a1-dimensional singular set. This is joint work with Mihai Tibar (two titles on arXiv).

Daniel Matei Real Arrangements and Contact Structures

Abstract

We consider contact structures that appear in connection with singularities of polynomial mappings through the notion of open bo! ok decompositions.
For real singularities that arise from arrangements of real subspaces, we analyze the range of possible contact structures, from the perspective of the dichotomy tight versus overtwisted, highlighting the tightness restrictions of the holomorphic case. This is joint work with M. Tibar.

Matthias Zach The Topology of Isolated Cohen-Macaualy codimension 2 Singularities

Abstract

A matrix singularity $(X,0)\subset (C^N,0)$ is a complex space germ whose ideal $I$ is generated by minors of a matrix $M$ with polynomial (resp. analytic) entries.
Those singularities in the title, abbreviated as ICMCd2-singularities, fall into this category. As in the case of complete intersection singularities deformations
of $(X,0)$ can easily be described and if $(X,0)$ admits a smoothing, one can ask for the topology of the smooth fiber.

We have recently described the Tjurina modification for ICMCd2-singularities, which in case of Cohen-Macaulay-type 2 translates
the problem to a local complete intersections scheme. However, this comes at the cost of possibly having a one-dimensional singular locus.

I will discuss how to apply old and recent results about the latter singularities to tackle the topology of matrix singularities.

Nero Budur  Cohomology support loci

Abstract

Cohomology support loci are homotopy invariants of topological spaces.
For complex varieties, they encode a rich part of the geometry of the variety.
These loci are notoriously difficult to compute.
We will present two recent advances in this direction. One is due to joint work with Liu-Saumell-Wang and provides an A'Campo type formula.
The other is due to Maisonobe who proved a big part of some of our conjectures on Bernstein-Sato ideals.

A.J.Parameswaran On mixed functions of type $f\bar g$

Abstract.

Let $f$, $g$ be two holomorphic function germs. The study of the class of functions $f\bar g$ produced many results.
Here we study the Thom regularity of  $f\bar g$
This is a joint work with Mihai Tibar.

 

Jan Stevens Improvements of non-isolated surface singularities

Abstract

 

Anne Frühbis-Krüger   An Algorithm for Computing the Discriminant of Families of Essentially Isolated Determinantal Singularities (EIDS)

Abstract

The compuation of the discriminant of a family of singularities is a computationally expensive task, because it requires a Groebner basis
computation of an ideal of minors with respect to an elimination ordering. In many relevant cases, the computation is not feasible due
to the complexity of the result.
For families of EIDS, however, the special structure of the equations can be used to apply a divide and conquer approach to the problem. As a byproduct, this approach
provides a meaningful decomposition of the discriminant.
In this talk, I shall explain this approach in detail.