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Homotopy methods is the name for a set of algorithms designed to find approximate solutions in a broad class of very important problems such as polynomial system solving or the eigenvalue problem. Roughly speaking, these methods consider continuous deformations of problems with a known solution to problems we want to solve. Despite homotopy methods exhibiting an excelent experimental performance, we still don't know optimal bounds for its complexity. The works of Shub and Smale, among others, suggesst to study the behaviour of geodesics in the so-called "condition metric”.
In this talk we will introduce homotopy methods together with some of the main known results. We will also have a look at some situations where homotopy methods have been used and we will see the relation between geodesics in the condition metric and the geometry of the problem.
Juan G. Criado del Rey is a PhD student at the University of Cantabria, Spain. He has been working on some geometric aspects of problems in Numerical Analysis such as S. Smale's problems 7 (elliptic Fekete points) and 17 (solving systems of polynomial equations). His main interests are Differential Geometry and Analysis, and their interplay with Computational Mathematics. Juan received his degree in Mathematics from the Complutense University of Madrid and his master's degree from the University of Cantabria. He currently lives in Santander, Spain.