Continuous k-regular maps

Speaker(s)

Jarosław Buczyński

Affiliation

UW/IMPAN

Date

May 16, 2017, 2:30 p.m.

Room

room 4070

Seminar

Seminar Algebraic Topology

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    A continuous map from R^m to R^N or from C^m to C^N is called k-regular if the images of any k points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value N_0 of N for which such maps exist. The methods of algebraic topology provide lower bounds for N_0, however there are very few results on the existence of such maps for particular values m and k. Using the methods of algebraic geometry we construct k-regular maps. We relate the upper bounds on N_0 with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k<10, and we provide explicit examples for k<6. We also provide upper bounds for arbitrary m and k. The problem has an equivalent phrasing in the language of interpolation theory. Its generalisations lead to further open questions, and to an idea of a more general framework.

 

Joint work with Tadeusz Januszkiewicz, Joachim Jelisiejew, Mateusz Michałek.

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