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Workshop

Self-intersections of surfaces that contain two circles through each point

  • Niels Lubbes (Johannes Kepler University Linz, Austria)
Felix-Klein-Hörsaal Universität Leipzig (Leipzig)

Abstract

Surfaces containing two lines through each point are smooth quadrics as was discovered about 350 years ago by Sir Christopher Wren. The Möbius geometric analogues of such doubly ruled surfaces are "celestial surfaces", namely surfaces containing two circles through each point. Celestial surfaces are of interest in for example architecture (supporting frameworks of buildings), kinematics (configuration spaces of linkages) and computer vision (surface reconstruction from silhouettes and camera calibration). In this talk, we present a classification of the real singular loci of celestial surfaces. Although the simplest curve after a line is perhaps a circle, this classification is surprisingly diverse: there are 5 different possible configurations for the complex components and at least 7 different possible topological types for the real components. Our methods include intersection theory on real del Pezzo surfaces, Möbius geometry and factorization of quaternionic polynomials. Our classification plays a key role in the proof of a recent result which states that almost all celestial surfaces are homeomorphic to one of 4 topological normal forms. We do an effort to make our results and methods accessible to a wide audience.

conference
29.07.24 02.08.24

MEGA 2024

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig) E1 05 (Leibniz-Saal)
Universität Leipzig (Leipzig) Felix-Klein-Hörsaal

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Christian Lehn

Ruhr-Universität Bochum

Irem Portakal

Max Planck Institute for Mathematics in the Sciences

Rainer Sinn

Universität Leipzig

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Simon Telen

Max Planck Institute for Mathematics in the Sciences

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