The objective is to give an introductory presentation of mathematical
constructions and tools that can be used to study problems where the
modeling, optimization, or control variable is no longer a set of
parameters, vectors, or functions but the shape or the structure of a
geometric object.
In that context, a good analytical framework and good modeling techniques
must be able to handle the occurrence of singular behaviors whenever they
are compatible with the mechanics or the physics of the problems at hand.
In some optimization problems the natural intuitive notion of a geometric
domain undergoes mutations into relaxed entities such as microstructures.
So the objects under consideration need not be smooth open domains, or even
sets, as long as they still makes sense mathematically.
The talk will cover basic mathematical ideas and methods that often come
from very different areas of applications and mathematical activities that
have traditionally evolved in parallel directions. The field of research is
extremely broad because it touches on areas that include classical
geometry, modern partial differential equations, geometric measure theory,
topological groups, constrained optimization, with applications to
classical mechanics of continuous media such as fluid mechanics, elasticity
theory, fracture theory, modern theories of optimal design, optimal
location and shape of geometric objects, free and moving boundary problems,
image processing? Good analytical formulations are also essential to reduce
the size and complexity of computations.
New issues raised in some applications force a new view of the fundamental
aspects of mathematical areas such as boundary value problems to find
suitable relaxation of solutions of PDEs, or, of geometry to relax the
basic notions of volume, perimeter, and curvature. In that context Henri
Lebesgue was a pioneer when in 1907 he relaxed the intuitive notion of
volume to the one of measure on an equivalence class of sets that can be
"measured". He was followed in that spirit in the early 1950s by the
celebrated work of E. De Giorgi who used the relaxed notion of finite
perimeter defined on the class of Caccioppoli sets to solve Plateau's
problem of minimal surfaces.
Reference. M.C. Delfour and J.-P. Zolésio, Shapes and
Geometries: Analysis, Differential Calculus and Optimization, SIAM series
on Advances in Design and Control, US, 2001
