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There are many objects in geometry that are called "singularities",
depending on the context. The most basic examples are the zero set (i.e.
hypersurface) or the set of critical points of a function, the set of
points where two hypersurfaces are tangent to each other, etc. In this
talk we will investigate the topology of different types of singular
loci from a broad perspective. The topology of the singular set of a polynomial imposes a lower bound on the degree, due to the Thom-Milnor bound and similar results. I will discuss some quantitative version of this concept, for smooth maps. Such topic is relevant in the context of smooth rigidity and Whitney extension problem, but it also offer an alternative approach to the polynomial case. By using polynomial approximations in a quantitative way, one obtains a Thom-Milnor bound valid for all smooth maps. However, the standard way of controlling the topology in the approximation: maintaining a transversality condition, produces a non-sharp inequality. I will present a general result about the behavior of the Betti numbers under C^0 approximations that allows to improve the above method. |