Intersections of ellipsoids, intersections of hyperboloids

Plática dada por Santiago López de Medrano (Instituto de Matemáticas, UNAM) en las Conferencias Samuel Gitler 2016 en el Centro de Investigación y de Estudios Avanzado del Instituto Politécnico Nacional (CINVESTAV) el viernes 30 de septiembre del 2016.

Any compact differentiable manifold with stably trivial normal bundle
can be realized as a transverse intersection of quadrics in $R^n$. 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 $R^n$ have been studied for many years. They appear naturally in several branches of Mathematics and are a source of important examples in several more. The difficulties for describing their topology in the smooth case grow quite fast with the number of quadrics involved and can be solved completely only when that number is small. For more quadrics we can only describe the topology of large families of them.
This is the work of many people, Sam Gitler included. In the talk I will give a new and unified overview (in terms of the difference between ellipsoids and hyperboloids) of the main results and of the difficulties involved.

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