# 2006

**14 June 2006****José Antonio Vallejo**, Department of Applied Mathematics IV, UPC, Barcelona**Understanding supermanifolds**- Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, aula 005; 12.00 h
**Abstract**Due to the needs in Physics, mathematicians have created several generalizations of the notion of manifold whose goal is to encompass both commuting and anti-commuting "variables" as those appearing in Quantum Physics. Unfortunately, the generalization preferreed by physicists (Rogers-DeWitt supermanifolds) has some serious drawbacks (and even inconsistencies) from the mathematical point of view, and the more accurate version of supermanifolds given by Kostant-Leites-Manin is way too technical, involving a heavy use of algebraic geometry and sheaf theory.

The aim of this talk is to present a simple description of what a supermanifold in this last sense is, following some ideas by Koszul, and to give a intuitive model for the study of supervector fields and ordinary differential superequations.

**9, 10, 11, 12, 15 and 16 May 2006**Ph.D. course**Andrew D. Lewis**, Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada**A course on Pontryagin's maximum principle**- Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, aula 006; 15:30-18:30 h
**Contents**The maximum principle of Pontryagin and coworkers is one of the triumphs in the applications of mathematics. The principle gives an enormous generalisation of the calculus of variations and provides a basis for solving a number of problems of great importance in modern control theory. As well, the ideas behind the theory are themselves of fundamental importance in control theory in general.

In this course, the maximum principle will be motivated from the point of view of the classical calculus of variations through the so-called Skinner-Rusk formulation of variational problems. This leads one fairly quickly to reasonable (but incorrect) conjectures about the form of the maximum principle.

The maximum principle is then stated in its precise form, and some more or less immediate consequences of this statement are explored. The proof of the maximum principle is difficult, although it is possible to understand the main ideas fairly easily. These main ideas will be presented, and the technical points in the proof will be identified and discussed.

The final part of the course will consist of two important applications of the maximum principle to linear control theory. The first will be the theory of quadratic optimal control for linear systems. After some work, this theory leads, somewhat surprisingly, to the stabilising linear quadratic regulator feedback law. This is an important idea in so-called modern control theory. The final topic will be time-optimal control for linear systems. Here the Maximum Principle allows for an elegant description of time-optimal trajectories.- Control systems and optimal control problems.
- The calculus of variations.
- The statement of the maximum principle.
- Tools used in the proof of the maximum principle.
- An outline of the proof of the maximum principle.
- Linear quadratic control theory.
- Linear time-optimal control theory.

**Bibliography**- Athans, M. and Falb, P.L. [1966],
*Optimal control. An introduction to the theory and its applications*. McGraw-Hill, New York. - Lee, E.B. and Markus, L. [1967],
*Foundations of optimal control theory*. John Wiley and Sons, New York. - Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. [1961],
*Matematicheskaya teoriya optimal'nykh protsessov*, Gosudarstvennoe izdatelstvo fizikomatematicheskoi literatury, Moskva.

translated in 1962 from the Russian by K.N. Trirogoff, and reprinted in 1986 as: - Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. [1986],
*The mathematical theory of optimal processes*, Classics of Soviet Mathematics, Gordon & Breach Science Publishers, New York.

**7 and 9 February 2006**mini course**Mariano Santander**, Department of Theoretical Physics, Atomic Physics and Optics, University of Valladolid**Classical mechanics in spaces of constant curvature**- Dep. de Matemàtica Aplicada IV, Campus Nord UPC, edifici C3, 204a (biblioteca de Matemàtica); Tuesday 15:30-17:30, Thursday 11-13
**Abstract**In this course we present a framework that allows to work in the explicit description of both geometrical and mechanical characteristics of the classical motion in spaces of constant curvature with arbitrary signature.

In the euclidian space, Kepler's potential and the harmonic oscillator turn out to be salient systems for their mathematical properties (related to the superintegrability) as well as for their physical relevance. In both cases the orbits are conics and the motion problem admits a complete analytical solution. Such remarkable feature remains unaltered when one replaces euclidian space by any space of constant curvature. Both "curved" Kepler's potential and "curved" harmonic oscillator are superintegrable systems whose orbits are "conics" (in the sphere, the hyperbolic plane, ...).

A complete description of the conics in these spaces of constant curvature will be given, emphasising their visual representations. This information will be applied to describe completelys the kind of orbits for both systems in spaces with curvature.

Finally we will comment on the well-known relationship between the euclidian Kepler motion and the geodesic flow in (velocity) spaces of constant curvature, and we will show that the same relationship applies, with minimal changes, when the motion occurs in a space of constant curvature.