# What is the degree of a smooth hypersurface?

The degree of a real algebraic object provides a strong upper bound for its geometrical and topological complexity.  By approximating smooth objects with algebraic ones in a controlled way (controlling the degree) then, it is possible to extend such bounds to hold in a smooth setting.
Following this philosophy, in this talk I will define an alternative parameter that controls the geometry of certain smooth objects (the singular loci of smooth maps) up to isotopy, and provides, in particular, a bound for their Betti numbers. This parameter is obtained as a function of the distance of the object from the set of singular ones in the appropriate functional space, in analogy with the concept of  "condition number".
The objects under consideration are the so called type-$W$ singularities, where $W$ is a semialgebraic subset of the space of $r$-jets. In this sense, the type-$W$ singularity defined by a smooth map $f$ is the preimage of $W$ via the $r$-jet prolongation map of $f$. Basic examples are: the set of critical points or the zero set.
I will focus on the example of a smooth closed hypersurface in R^n to illustrate the main ideas and results. Here in particular, I will present a quantitative version of Seifert’s Theorem, stating that the hypersurface is isotopic to an algebraic one (inside a given ball) having degree controlled by an explicit function of the reach of the original one.

This talk is based on a recent paper by Antonio Lerario and me, with the same title (see arXiv:2010.14553).

Le séminaire aura lieu en ligne, et sera accessible à cette adresse:

https://webconf.lal.cloud.math.cnrs.fr/b/ann-dtn-6uw

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