Norbert A'Campo	Fibre de Milnor et recollement des espaces d'arcs.
Résumé :
Lê Dung Trang	Topologie des courbes planes et des surfaces complexes
Abstract:	Dans cet exposé nous allons rappeler les résultats classiques concernant la topologie locale des singularités de courbes planes et nous en donnons une présentation plus "moderne" qui nous montrera comment peut-on "voir" les singularités de surfaces.
David Massey	Milnor fibres and monodromies
Abstract:	For an affine hypersurface singularity with a one-dimensional critical locus, S, the sheaf of vanishing cycles forms a local system, L, on S, outside of a discrete set of points, P. The set P consists of points where there is a non-empty relative polar curve, and the local system L is characterized by the monodromy representations around the points of P. The Milnor monodromy is a natural automorphism of the vanishing cycle functor, and so yields monodromy representations on the Milnor fibres at each point in the critical locus; moreover, these Milnor monodromies must be compatible with the monodromies of the local system L. By analyzing the relations between these Milnor fibres and monodromies, and using general results on polar curves, we obtain restrictions on the cohomology of the Milnor fibres.
Jan Stevens	Surfaces with many triple points
Abstract:	In [Endra\ss, Persson and Stevens, {\it Surfaces with triple points\/}] upper bounds for the number of ordinary triple points on a hypersurface in ${\bf P}^3$ are given and in particular it is shown for sextics that the maximum number of 10 is obtained. We study these surfaces in more detail. The clue to the classification of sextics with many triple points is the study of exceptional curves of the first kind on the minimal resolution. It turns out that only a few different cases can occur. With nine triple points one has three families of blown-up $K3$ surfaces and two families of properly elliptic surfaces. Each $K3$ occurs in a pencil of sextics of the form $\alpha g+\beta q^3$, where $q$ defines the (unique) quadric through the nine points, and $g$ is a reducible sextic. In the other two cases we find even a net of sextics, again containing $q^3$ and reducible sextics. Regarding the existence of a sextic $\{g=0\}$ with 10 triple points we first observe that the pencil $\alpha g+\beta q^3$, where $\{q=0\}$ passes through nine of them, falls into one of our five families of sextics with nine triple points. Assuming that an element of such a family has a tenth triple point gives conditions on the coefficients. In [E--P--S] the resulting equations were only solved in one case by imposing extra symmetry. We show that there are four different three-dimensional families, only defined over ${\bf Q}(\sqrt{-3})$. They are related by Cremona transformations.
S.M.Gusein-Zade	Poincaré series of multi-index filtrations and integrals with respect to the Euler characteristic
Abstract:	There was elaborated a method of computing (generalised) Poincare series of multi-index filtrations on ring of functions which reduces it to some integrals with respect to the Euler characteristic. This includes integration over (infinite dimensional) projectivisations of the spaces of functions (defined in the spirit of motivic integration and, in some sense, dual to the integration over spaces of arcs). The method permits, in particular, to get a short proof of the result that the Poincaré series of the multi-index filtration on the ring of functions of two variables defined by a reducible plane curve singularity coincides with the Alexander polynomial in several variables of the corresponding algebraic link, to compute the Poincaré series of filtrations on the ring of functions on certain surface singularities defined by components of the exceptional divisor of a resolution. The talk describes joint results with A.Campillo and F.Delgado.
David Mond	Intersection indices of vanishing cycles, the Morse-Witten complex and an ansatz of Hori, Iqbal and Vafa
Abstract:	This a seminar on joint work (in progress) with Duco van Straten. Our aim was to find a geometrical justification for an assumption explicitly made by the physicists Hori, Iqbal and Vafa when counting soliton solutions running between vacua in Landau-Ginzburg theory (read: symplectically horizontal lifts of paths between critical values of a holomorphic function). What we have understood so far points to links with symplectic geometry and perhaps with Gromov-Witten invariants. My intention in speaking on this unfinished work is to describe a rather beautiful example in some detail, and perhaps to get some help.
Victor Goryunov	Vanishing topology of singularities of simple symmetric matrix families
Abstract:	We are showing that simple families of symmetric matrices in two variables are classified by certain subdiagrams of marked affine Dynkin diagrams of Weyl groups $A_\mu, D_\mu, E_\mu$. The subdiagrams are obtained by omitting vertex subsets of total marking 2: forgetting two 1-vertices gives rise to a simple $2 \times 2$ matrix family, while forgetting one 2-vertex corresponds to a $3\times 3$ family.
Jörg Schürmann	Milnor classes and vanishing cycles
Abstract:	The classical Milnor number of an isolated hypersurface singularity has been generalized in recent years to the Milnor classes of a complete intersection X in a complex manifold M. They are the difference of two kinds of Chern classes for the singular space $X$, i.e. Fulton- and Schwartz-MacPherson Chern-classes. After recalling the history of this subject, we explain our algebraic geometric approach to these Milnor classes in terms of generalized vanishing cycles. Here we use the famous deformation to the normal cone. Our formula gives a far reaching generalization of Parusinski-Pragacz's formula for the hypersurface case. Moreover, our approach works more generally for any regular embedding of singular spaces, and implies a corresponding Verdier-Riemann-Roch theorem for Chern-Schwartz-MacPherson classes.
Christian Sevenheck	Deformation theory for lagrangian singularities
Abstract:	We study flat deformation of lagrangian singularities. We introduce a lagrangian de Rham complex, related to the de Rham complex of a D-module as well as to the Chevalley-Eilenberg complex of a Lie algebra. We describe the relationship of this complex to the deformation theory of the lagrangian singularity. Finally, we will state and prove a result on the finiteness of the cohomology yielding a versal deformation theorem.
Dirk Siersma	Deformation of polynomial functions
Abstract:	We study the topology of polynomial functions by deforming them generically. We explain how the non-conservation of the total ``quantity'' of singularity in the neighbourhood of infinity is related to the variation of topology in certain families of boundary singularities along the hyperplane at infinity.
Alberto Verjovsky	A nonsingular Peano-Plateau problem
Abstract:
Mario Escario	Monodromy group and discriminant of the polar mapping
Abstract:	We construct an effective algorithmic method to compute the homological monodromy of a complex polynomial in two variables which is good at infinity. Applications are given to compute the intersection matrix of Iomdine surfaces in distinguished basis of vanishing cycles and to prove the existence of conjugate polynomials in some number field which are not topologically equivalents.
Françoise Michel	Topology of holomorphic germs of surfaces
Abstract:	We describe the boundary of the Milnor fiber in the non-isolated cases. We compare the topology of the boundary of the Milnor fiber with the boundary of the normalization. This is a joint work with Anne Pichon.