Where and when: Room A2 at ENS-Lyon, on Thursday from 1:30pm to 4:30pm
Prerequisites: Nothing strictly required. Basic notions of differential geometry will help.
Objective: The course Effective Methods in Geometry is rather unusual in aiming to show how abstract mathematics can be translated into concrete computations. A wide set of methods will be presented, ranging from combinatorial ones to optimal transport. Many examples will be given: isometric embeddings, word problems and computational topology, optical lenses... A first step in this translation requires to discretize the studied problem that may comes from mathematics itself or from other fields such as physics. We shall see on various examples how to perform this discretization and how to solve the corresponding formulation.
One part will focus on numerical aspects. We shall concentrate on some equations of the Monge-Ampčre type that intervene in many fields such as geometry, optimal transport, or optics. We will show how the geometrical discretization of the theory of optimal transport leads to solving optimization problems. In turn, this approach may lead to the realization of concrete objects, such as optical lenses.Tentative plan for this part:
Another part will focus on combinatorial aspects issued from topology and geometry, where the underlying mathematical problems have a discrete nature. We will start with the isometric embedding of the flat torus in order to explore PL isometric embeddings as well as other questions mixing topology and geometry.Tentative plan for this part:
| PL isometric embeddings|
| Embedding n-complexes in R2n|
| Deciding linear embeddability|
Validation: Oral presentation on a research article followed by questions on the part of the course not covered by the article.