The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.[-]

The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.[-]

Robotic design involves modeling the behavior of a robot mechanism when it moves along potential paths set by the users. In this lecture, we will first give an overview of different approaches to design a set of kinematic equations associated with a robot mechanism. In particular, these equations can be used to solve the forward and the backward kinematics problems associated with a robot mechanism or to model its singularity locus. Then we will review methods to solve those equations, and notably methods to draw with guarantees the real solutions of an under-constrained system of equations modeling the singularities of a robot.[-]

Robotic design involves modeling the behavior of a robot mechanism when it moves along potential paths set by the users. In this lecture, we will first give an overview of different approaches to design a set of kinematic equations associated with a robot mechanism. In particular, these equations can be used to solve the forward and the backward kinematics problems associated with a robot mechanism or to model its singularity locus. Then we will review methods to solve those equations, and notably methods to draw with guarantees the real solutions of an under-constrained system of equations modeling the singularities of a robot.[-]