[Todos] Coloquios del Departamento de Matemática

[Daniel Carando]

Próxima charla: Jueves 15 de mayo a las 16:00
Aula E 24

Mike Shub
Universidad de Toronto .

Bezout's Theorem and Complexity: Geometry, Topology and the Condition Number.

The theory of the solution of non-linear problems which can only be
solved approximately is not very well developed from a formal
complexity point of view. It is suspected that the condition of the
problem instance, that is the sensitivity of the solution to
perturbation of the data, will play a role, not only in the numerical
precision needed to solve the problem to desired accuracy, but also in
the number of arithmetic operations themselves which need to be
performed. We illustrate this phenomena with the study of homotopy
methods to solve systems of polynomial equations. The object of study
is paths in the solution variety, that is the set of
(problem,solution) pairs. The geometry and topology of the solution
variety intervenes.The complexity of a homotopy path is measured by
its length in a particular Riemannian structure related to the
condition number, called the condition metric, and the topology of the
variety itself is studied via the Morse function defined by a smooth
variant of the condition number function. Geodesics in the condition
metric are surprisingly short and the topology is surprisingly simple.

Might polynomial systems be "easy" to solve approximately? Carlos
Beltran will discuss this question in a forthcoming talk.


Están todos cordialmente invitados.

Daniel