Norbert A'Campo
"Courbes de reduction de la monodromie et cycles quadratiques evanescents"
Résumé:
On construit pour la fibre de Milnor d'une singularité isolée de courbe
plane un model (F,\Sigma,\Delta), qui montre sur la surface
à bord F le système
de reduction \Sigma de la monodromie géométrique et un système
distingué de cycles évanescents quadratiques \Delta. Le système
\Sigma est une famille de courbes simples
sur F, deux à deux disjointes, qui est invariante par la monodromie géométrique. Une
application concernant les groupes de monodromie homologique ou géométrique
est donnée.
Edward Bierstone
"Algorithmes de desingularisation"
Résumé:
On compare quelques algorithmes récents de
désingularisation canonique, en terme de la
dépendance des invariants locaux en fonction
des diviseurs exceptionnels. Exemple: désingularisation
des variétés toriques (ou localement binomiales).
Jean-Paul Brasselet
"Nombres et classes de Milnor"
Résumé:
Il existe plusieurs notions d'indices de champs de vecteurs
ayant des singularités (isolées) en les
singularités isolées d'hypersurfaces. En particulier, la différence
entre le "GSV"-indice et l'indice "classique" (de Poincaré-Hopf)
s'exprime en fonction du nombre de Milnor.
La généralisation de ce resultat est une expression
géométrique de la différence entre classe de Schwartz-MacPherson
et classe de Fulton-Johnson, dans le cas d'intersections completes.
Helmut Hamm
"Théorèmes de Lefschetz : théorie de Morse et pinceaux de Lefschetz"
Résumé:
Pour la démonstration du théorème classique de Lefschetz sur les
sections hyperplanes, on peut ou bien utiliser la théorie de Morse
(Andreotti-Frankel) ou bien des pinceaux de Lefschetz. On va comparer
les deux méthodes et leur comportement par rapport aux
généralisations.
Claus Hertling
"Primitive forms, Frobenius manifolds, and moduli of
singularities"
Abstract:
The construction of Frobenius manifolds in singularity
theory is based on K. Saito's primitive forms and
results of M. Saito. As an application I can show the
existence of global moduli spaces for the singularities
in one mu-homotopy class. The construction and this
application will both be sketched. I also intend to give
an idea of the relation to topics as Gauss-Manin connections,
isomonodromic deformations, mixed Hodge structures,
versal Lagrange maps, free divisors.
Anatoly Libgober
"Elliptic genus of toric varieties."
Abstract:
I will discuss two variable elliptic genus for
toric varieties or hypersurfaces in toric varieties
with Gorenstein singularities as well as a relationship
between elliptic genera of hypersurfaces which are
mirrors of each other (joint work with L.Borisov).
Mutsuo Oka
"Elliptic curves from sextics."
Abstract:
Let N be the moduli space of sextics
with 3 (3,4)-cusps. The quotient moduli space N/G is
one-dimensional and consists of two components,
N_{torus}/G and N_{gen}/G. By quadratic transformations,
they are transformed into one-parameter families C_s and
D_s of cubic curves respectively.
First we study the geometry of N_\eps/G, where \eps = torus or gen, and
their structure of elliptic fibration. Then we
study the Mordell-Weil torsion
groups of cubic curves C_s over Q and D_s over Q(\sqrt{-3})
respectively.
We show that C_s has the torsion group
Z/3Z for a generic s in Q and it also contains
subfamilies which coincide with the universal families given by
Kubert with the torsion groups
Z/6Z, Z/6Z + Z/2Z,
Z/9Z or Z/12Z. The cubic
curves D_s have torsion Z/3Z + Z/3Z generically but also
Z/3Z + Z/6Z for a subfamily which is parametrized by
Q(\sqrt{-3}).
Dirk Siersma
"Embeddings of smooth lines into hypersurfaces."
Abstract:
We use singularity theory in order to study the positions
of smooth lines in singular hypersurfaces. Sometimes many positions
are possible, but in some case they don't exist. We define
invariants, related to the relative position, study Morsification
and monodromy.
David Trotman
"Stratified transversality via isotopy."
Abstract:
For X a (w)-regular or (c)-regular stratification, hence for any Whitney
stratification and, via regular embedding, for any abstract stratified set,
we give an extension theorem for diffeomorphisms defined on strata of a
given dimension.
Then we show that after isotopy a stratified subspace W of X can be made
transverse to a stratified map $ g: Y \to X$, and study cases where the
isotopy
preserves regularity of W. There are applications to the homology theory
of stratified sets
extending work of Goresky and Murolo.