Kevin Houston Equisingularity of Mappings.

A classical theorem of Briançon, Speder and Teissier states that a family of isolated hypersurface singularities is Whitney equisingular if and only if the \mu^*-sequence for a hypersurface is constant in the family. Similar results are true for families of finitely A-determined map-germs from C^n to C^p, where n and p are general. In this case one defines an analogue of the \mu^*-sequence using the disentanglement of a map rather than the Milnor fibre. The constancy of this sequence is equivalent to Whitney equisingularity in some cases.

Adam Parusinski Blow-analytic equivalence of two variable real analytic function germs.

We give a complete characterization of blow-analytic equivalence classes of two variable real analytic function germs.
pdf file with abstract

We establish a formula for the total Gauss-Bonnet-Chern curvature of a closed semi-algebraic set
$X \subset \mathbb{R}^n$ in terms of its Euler-Poincaré characteristic and its behaviour at infinity.

Satoshi Koike Finiteness theorems on Blow-Nash and Blow-semialgebraic triviality for a family of Nash sets.

Given a family of of Nash mappings $f_t : N \to \R^k$ defined over a Nash manifold $N$ with a semialgebraic parameter space $J$, we ask whether a finiteness theorem holds on some triviality for the family of zero-sets $\{ (N,f_t^{-1}(0)) \}$. More precisely, we ask whether there is a finite partition of $J$ into Nash open simplices $Q_i$'s such that the family of zero-sets is trivial over each $Q_i$ on the triviality we consider. We do not assume that the source manifold $N$ is compact. Then it follows from Hardt's theorem that a finiteness theorem holds on semialgebraic triviality for the family of zero-sets. Here we consider the notions of Blow-Nash triviality and Blow-semialgebraic triviality for a family of Nash sets, which are stronger than semialgebraic triviality. Thanks to Hardt's theorem, when we we consider our finiteness problem on those trivialities, we may assume from the beginning that $\dim f_t^{-1}(0)$ is constant over $J$. Under this setup we explain the following finiteness results for the family of Nash sets $\{ (N,f_t^{-1}(0)) \}$: (0) In any case a finiteness theorem holds on the existence of Nash trivial simultaneous resolution. (I) In the regular case a finiteness theorem holds on Nash triviality. (II) In the case of isolated singularities, a finiteness theorem holds on Blow-Nash triviality. (III) In the case of non-isolated singularities, a finiteness theorem holds on ``Blow-semialgebraic triviality" and ``Blow-semialgebraic triviality consistent with a compatible filtration" for a family of 2-dimensional Nash sets and a family of 3-dimensional Nash sets, respectively. Addendum: T. Fukui constructed a family of 4-dimensional real algebraic varieties in $\R^6$ and its Nash trivial simultaneous resolution such that the restriction of the resolution mapping to the intersection of the strict transform of the variety and the exceptional divisor at some point represents the Nakai family in which local topological moduli appear. This implies that our programme to show finiteness on Blow-semialgebraic triviality for 2 or 3-dimensional Nash sets is not applicable to the 4-dimensional case.

Tadeusz Krasinski The Lojasiewicz exponent of isolated singularities

      This is a survey lecture on the Lojasiewicz exponent of isolated singularities. The main topics are:
    1. characterization of rational numbers which are Lojasiewicz exponents of isolated singularities.
    2. the Lojasiewicz exponent of isolated nondegenerate singularities.

 Michel Merle Singularités et intégration motivique.

Johannes Nicaise Rigid geometry and complex singularities.

I will try to convince you that rigid geometry over the field of Laurent series C((t)) is a natural and powerful tool in the study of complex algebraic singularities. The main example will be the construction of the analytic Milnor fiber, associated to the germ of a morphism f from a smooth algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over C((t)), and it fully determines the analytic type of the singularity of the germ f. Its (étale) cohomology coincides with the singular cohomology of the classical topological Milnor fiber of f; the monodromy transformation is given by the Galois action. Moreover, the points on the analytic Milnor fiber are closely related to the motivic zeta function of f, and the arc space of X. The general observation is that the arithmetic properties of the analytic Milnor fiber reflect the structure of the singularity of the germ f.

Anne Pichon Principal analytic link theory in surface singularity links.

(joint work with Walter D. Neumann) An important corpus of works in singularity theory concentrate on the following class of questions : given a singularity - e.g. a germ of analytical space or morphism - which analytical properties of the singularity are reflected by its topology ? We focus on the following situation : let $Z$ be a normal complex surface and let $p \in Z$. One denotes by $M_{(Z,p)}$ the link of the singularity $(Z,p)$, i.e. the 3-dimensional manifold obtained as the boundary of a small regular neighbourhood of $p$ in $Z$. Then, $Z$ is locally homeomorphic to the cone $C(M_{(Z,p)})$ on $M_{(Z,p)}$. Given a surface singularity link $M$, there may exists many different analytical structures on the cone $C(M)$, i.e. normal surfaces singularities $(Z,p)$ whose $M$ is (homeomorphic to) the link. A natural problem - particular formulation of the above question - is to understand the different analytic structures which arise on $C(M)$. In this talk, we will present an approach which consists of studying the principal analytic link-theory on $M$. Our aim is to present this point of view, to outline this study through examples and to encourage people to pursue this area.

Jean-Jacques Risler Deformations de Harnack d'une branche plane réelle.

We define a class of local deformations of a real plane branch C in terms of the embedded resolution of singularities of the branch. We investigate the relations between the embedded topological type of the real branch, viewed in the affine complex plane, and the embedding of its Harnack's deformations, those real local deformations which are smooth, have the maximal number of ovals and have good oscillation conditions with the coordinate axes. These relations involves a generalization to the local case of a theorem of Mikhalkin on Harnack curves in a toric surface and their amoebas. The amoeba of a plane curve is the image of its points $(x, y)$ under the map $(x, y) \mapsto (\log|x|, \log |y|)$.

David Trotman  Local geometry of singular spaces and Whitney's fibering conjecture.

Since the foundational work of Whitney and Thom much progress has been made in taming real and complex algebraic/analytic varieties. Whitney showed the existence of (b)-regular stratifications which Thom proved to be locally topologically trivial. Hardt showed further local semialgebraic triviality of semialgebraic sets, while Mostowski and Parusinski showed local bilipschitz triviality. Recently Valette combined these results by obtaining local semialgebraic bilipschitz triviality of semialgebraic sets, and proved a conjecture of Siebenmann and Sullivan on the countability of metric types of germs of analytic spaces. Whitney's fibering conjecture asks that a neighbourhood of a stratum be foliated by analytic copies of a core stratum such that the tangent spaces to the leaves of the foliation vary continuously. We describe recent progress made with Murolo and du Plessis on a subanalytic Whitney fibering conjecture and its smooth version. There are applications to other conjectures - for example Whitney triangulation of Whitney stratified sets (proved by Shiota in 2005 for semialgebraic sets).