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Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

Damien Gayet, Jean-Yves Welschinger (2016)

Journal of the European Mathematical Society

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from...

BGG resolutions via configuration spaces

Michael Falk, Vadim Schechtman, Alexander Varchenko (2014)

Journal de l’École polytechnique — Mathématiques

We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik–Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the 𝔰𝔩 2 Bernstein–Gelfand–Gelfand resolution as an Aomoto complex.

Biholomorphic maps determined on the boundary

Nozomu Mochizuki (1977)

Annales de l'institut Fourier

Let D be a bounded domain in C n such that the boundary b D is topologically S 2 n - 1 in R 2 n with a regular point; let f : D ˜ C n be a holomorphic map where D ˜ is a neighborhood of D . If f is one-to-one when restricted to b D , then f : D f ( D ) is biholomorphic.

Bi-Lipschitz trivialization of the distance function to a stratum of a stratification

Adam Parusiński (2005)

Annales Polonici Mathematici

Given a Lipschitz stratification 𝒳 that additionally satisfies condition (δ) of Bekka-Trotman (for instance any Lipschitz stratification of a subanalytic set), we show that for every stratum N of 𝒳 the distance function to N is locally bi-Lipschitz trivial along N. The trivialization is obtained by integration of a Lipschitz vector field.

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