The singular side, 16-17 mai

Singularités et Applications

Salle Duhem
Workshop - Ursula Ludwig, Nicolas Dutertre, Cezar Joita, Piotr Migus
Mercredi, 16 Mai, 2018 - 17:30 - 18:30


Ursula Ludwig     An Extension of a Theorem by Cheeger and Müller to Spaces with Isolated Conical Singularities

Abstract. An important comparison theorem in global analysis is the comparison of analytic and topological torsion for smooth compact manifolds equipped with a unitary flat vector bundle. It has been conjectured by Ray and Singer and has been independently proved by Cheeger and Müller in the 70ies. Bismut and Zhang combined the Witten deformation and local index techniques to generalise the result of Cheeger and Müller to arbitrary flat vector bundles with arbitrary Hermitian metrics. In this talk we present an extension of the Cheeger-Müller theorem to spaces with isolated conical singularities by generalising the proof of Bismut and Zhang to the singular setting.

Nicolas Dutertre   Généralisations de l'obstruction d'Euler globale et points critiques

Abstract. L'obstruction d'Euler locale a été introduite par MacPherson dans les années 1974 pour définir les classes de Chern des espaces singuliers. Seade, Tibar et Verjovsky (Math. Annalen, 2005) ont défini l'obstruction d'Euler globale d'un ensemble algébrique affine complexe et ils ont donné une formule de multiplicité polaire pour cette obstruction. Dans cet exposé, on définit plusieurs généralisations de l'obstruction d'Euler globale et on donne plusieurs généralisations de la formule de multiplicité polaire. C'est un travail avec Nivaldo Grulha.

Cezar Joita   Bifurcation set of multi-parameter families of complex curves

Abstract. We characterize the regular bifurcation values of polynomial maps $C^{n+1} \to C^n$, $n>1$. One of the main tools is a theorem by Ilyashenko on coverings of complex manifolds foliated by complex curves. We use specific results about Stein spaces and the work by Meigniez on the topology of non-proper fibrations. Jointwork with Mihai Tibar.


Piotr Migus Sufficient end necessary conditions for local equivalence of functions

Abstract.  Let $f,g: (R^n, 0) \to (R,0)$ be Ck -functions. We say that $f$ and $g$ are CkCr-right-equivalent if there exits a Ck -diffeomorphism $\varphi$ such that $f= g\circ \varphi$. We will recall classical Kuiper-Kuo, Bochnak-Łojasiewicz Theorem which deal with right- -equivalence of functions in the case of isolated singularities. Using similar technique, namely integration of vector field, we obtained a couple generalizations of this theorem without assumption about isolated singularities: a) sufficient condition for right-equivalence of functions, b) non-isolated version of Bochnak-Łojasiewicz Theorem, c) Kuiper-Kuo Theorem for mappings. Moreover, we will give simple connection between equivalence of functions and Łojasiewicz exponent in the gradient inequality.


Mihai Tibar   Polar degree conjectures, after Dolgachev, Dimca-Papadima, Huh etc.

Abstract. Dedicated to the memory of Stefan Papadima

Dolgachev (Michigan Math J, 2000)  has initiated the study of Cremona polar transformations
i.e. the birational maps $grad f : \mathbb P^{n} \dashrightarrow \mathbb P^{n}$ defined by the gradient
map of a homogeneous polynomial. He conjectured that the topological degree of $grad f$ depends
only on the projective zero locus $V$ of $f$, so that it can be called polar degree  of $V$, denoted $pol(V)$.

I'll discuss the proof of Dolgachev's conjecture found by Dimca and Papadima (Annals of Math 2003),
their conjecture on the classification of homaloidal hypersurfaces (i.e.
$pol(V)=1$ ) with isolated singularities proved by 
Huh (Duke Math J, 2014), and Huh's  conjecture on the classification of hypersurfaces with isolated singularities and $pol(V) =2$, with its recent proof.