פעילויות השבוע

Singular spaces appear everywhere. And the singularity is often non-isolated, i.e. the singular locus is of positive dimension. These non-isolated singularities are more complicated and less studied.

Let X be a variety with singular locus Z, the simplest example being the surface {y^2=x^2z}. Generically along Z the singularity ”factorizes“, i.e. X is locally at each point the product: (the germ of Z)\times (the germ of space with an isolated singularity).

But at some special points of Z the picture degenerates and the family of sections of X, transversal to Z, becomes not equi-singular (in whichever sense). These points form the discriminant of transversal singularity type. We study this discriminant, assuming X,Z are locally complete intersections and X is of ”ordinary type“ generically along Z.

First I will define the discriminant, as a subscheme of Z, and formulate its properties. This discriminant is a (effective) Cartier divisor in Z, nef but not necessarily ample, with nice pullback/pushforward properties under some maps. The discriminant deforms flatly under some deformations of X.

Then I will give the formula for the class of this discriminant in the cohomology/Chow group/Picard group of Z. This class ”counts the number of points“ where the transversal type degenerates as one travels along the singular locus. In most cases this class is not zero (when Z is complete or projective). This places a ”topological“ obstruction to the naive expectation (from differential geometry): ”In the generic case the transversal type does not degenerate“.

The talk is based on arXiv:1705.11013 and arXiv:1308.6045.

It is a well-known fact that 1D systems and non-selfadjoint operators are closely related via the notion of operator colligation. The study of the characteristic function of a colligation is related to the study of de Branges spaces of analytic functions on an open set in the Riemann sphere. It allows us, for instance, to give an alternative proof for the Beurling‘s Theorem using Livšic Colligations and de Branges spaces.

In this talk, I will characterize de Branges spaces, i.e. reproducing kernel Hilbert spaces of analytic sections defined on a real compact Riemann surface, rather than on the Riemann sphere. This is done through the vessel theory, a generalization of the colligation theory to the case of $n$-tuple commuting non-selfadjoint operators. The characteristic function of a vessel is then a bundle mapping defined on a compact Riemann surfaces and which also carries the input-output relation of a 2D system. As a consequence, I will introduce a Beurling-type Theorem on finite bordered Riemann surfaces.

Problems pertaining to approximation and their applications have been extensively studied in the theory of convex bodies. In this talk we discuss several such problems, and focus on their extension to the realm of measures. In particular, we discuss variations of problems concerning the approximation of convex bodies by polytopes with a given number of vertices. This is done by introducing a natural construction of convex sets from Borel measures. We provide several estimates concerning these problems, and describe an application to bounding certain average norms.

Based on joint work with Han Huang