Abstract: En esta platica, que esta pensada especialmente para los estudiantes, presentaremos una introduccion a los procesos Gaussianos, concentrandonos en las propiedades de las trayectorias. Recordaremos los resultados basicos sobre continuidad y diferenciabilidad de trayectorias demostrados por Kolmogorov y veremos como han sido enfocados estos problemas para procesos Gaussianos, tanto en el caso estacionario como en el caso general. En el camino hablaremos sobre algunas de las desigualdades fundamentales para estos procesos y sus aplicaciones.

Abstract: Imagine installing a bunch of cheap, infrared eye beams throughout a
complicated warehouse, office, or shopping center. Just like the safety beam on a motorized garage door, a single bit of information is provided: Is the beam currently obstructed?  Now suppose that there are one or more moving bodies, which could be people, robots, animals, and so on.  What can we infer about the paths taken by the moving bodies?  It turns out that the subject is much more general than the particular scenario just described.  In addition to binary detection beams or regions placed around an environment, a sensor beam abstraction arises in other contexts.  For example, if a robot carries a camera and certain image features critically change, or the robot detects that a pair of landmarks align with its moving position, then these events may be equivalent to crossing a ``virtual'' beam in the environment.
During the talk, methods to recover the possible agents' paths based only on the binary sensor data will be presented.  This is a basic filtering problem encountered in many settings, which may arise from physical sensor beams or virtual beams that are derived from other sensing modalities.  Methods are given for three alternative representations: 1) the possible sequences of regions visited, 2) path descriptions up to homotopy class, and 3) numbers of times winding around obstacles.  The solutions are adapted to the minimal sensing setting; therefore, precise estimation, distances, and coordinates are replaced by topological expressions.

Abstract: This work addresses the problem of surveillance in an environment with obstacles. We consider the problem in which a mobile  pursuer attempts to maintain visual contact with an evader as it moves through an environment containing obstacles. This surveillance problem is a variation of traditional pursuit-evasion games, with the additional condition that the pursuer immediately loses the game if at any time it loses sight of the evader.
We formulate the problem in the framework of Differential Game theory. For the game of kind, we provide complete solutions to the tracking problem in a simple environment containing a single corner. Then we use these results to address the problem in a general polygonal environment containing multiple obstacles. Next, we formulate the problem as a game of degree and obtain strategies in Nash equilibrium. These strategies are integrated back in time to yield trajectories of the players near the termination situations. Finally, I will discuss some future problems.