Intersection of Ellipsoids, Intersection of Hyperboloids

Plática dada por Santiago López de Medrano (Instituto de Matemáticas, UNAM) en el evento Graded Algebras, Geometry and related Topics para celebrar el 60 aniversario de Adolfo Sánchez Valenzuela el viernes 18 de noviembre del 2016 en la Universidad Autónoma de Yucatán (UADY)

Abstract:
Any compact differentiable manifold with stably trivial normal bundle can be realized as a transverse intersection of quadrics in ℝⁿ.
It can be an intersection of ellipsoids (when it bounds a parallelizable manifold) or an intersection of hyperboloids (if it does not). So there are too many topological types of such intersections to hope for a description of them all.

Intersections of concentric quadrics in ℝⁿ appear in several branches of Mathematics and are a source of important examples in many more. In this case we have now a complete description of the topology of all the generic intersections of up to 3 ellipsoids or up to 2 hyperboloids and their topology is quite simple. The proof involves the normal forms for pairs of quadratic forms, some theory of polytopes and some Algebraic and Geometric Topology.

For intersections of higher numbers of quadrics problems become severe: their algebraic classification is a wild problem, their combinatorics are out of reach and their algebraic topology includes often non-trivial secondary cohomology operations. Even so, work by Bosio-Meersseman and Gitler-GómezGutiérrez-LdM lets us describe a good part of the diagonalizable cases with any number of quadrics and some mildly non-diagonalizable ones.

The hope is to clarify some other zones of the dark regions: New families that can be described? Some general understanding of non-diagonalizability? Of the relations between algebraic, combinatorial and topological classifications?

I will present an overview of what is known and of the difficulties that appear, hoping that someone has dealt with similar problems in other questions.

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