# Alessio Figalli: Catalogue data in Autumn Semester 2018

 Name Prof. Dr. Alessio Figalli Field Mathematics Address Professur für MathematikETH Zürich, HG G 45.2Rämistrasse 1018092 ZürichSWITZERLAND Telephone +41 44 632 75 30 E-mail alessio.figalli@math.ethz.ch URL https://people.math.ethz.ch/~afigalli/ Department Mathematics Relationship Full Professor

NumberTitleECTSHoursLecturers
401-0353-00LAnalysis III 4 credits2V + 2UA. Figalli
AbstractIn this lecture we treat problems in applied analysis. The focus lies on the solution of quasilinear first order PDEs with the method of characteristics, and on the study of three fundamental types of partial differential equations of second order: the Laplace equation, the heat equation, and the wave equation.
ObjectiveThe aim of this class is to provide students with a general overview of first and second order PDEs, and teach them how to solve some of these equations using characteristics and/or separation of variables.
Content1.) General introduction to PDEs and their classification (linear, quasilinear, semilinear, nonlinear / elliptic, parabolic, hyperbolic)

2.) Quasilinear first order PDEs
- Solution with the method of characteristics
- COnservation laws

3.) Hyperbolic PDEs
- wave equation
- d'Alembert formula in (1+1)-dimensions
- method of separation of variables

4.) Parabolic PDEs
- heat equation
- maximum principle
- method of separation of variables

5.) Elliptic PDEs
- Laplace equation
- maximum principle
- method of separation of variables
- variational method
LiteratureY. Pinchover, J. Rubinstein, "An Introduction to Partial Differential Equations", Cambridge University Press (12. Mai 2005)
Prerequisites / NoticePrerequisites: Analysis I and II, Fourier series (Complex Analysis)
401-3350-68LIntroduction to Optimal Transport
Number of participants limited to 11.
4 credits2SA. Figalli, further speakers
AbstractIntroductory seminar about the theory of optimal transport.
Starting from Monge's and Kantorovich's statements of the optimal transport problem, we will investigate the theory of duality necessary to prove the fundamental Brenier's theorem.
After some applications, we will study the properties of the Wasserstein space and we will conclude introducing the dynamical point of view on the problem.
Objective
ContentGiven two distributions of mass, it is natural to ask ourselves what is the "best way" to transport one into the other. What are mathematically acceptable definitions of "distributions of mass" and "to transport one into the other"?
Measures are perfectly suited to play the role of the distributions of mass, whereas a map that pushes-forward one measure into the other is the equivalent of transporting the distributions. By "best way" we mean that we want to minimize the map in some norm.

The original problem of Monge is to understand whether there is an optimal map and to study its properties. In order to attack the problem we will need to relax the formulation (Kantorovich's statement) and to apply a little bit of duality theory. The main theorem we will prove in this direction is Brenier's theorem that answers positively to the existence problem of optimal maps (under certain conditions).
The Helmotz's decomposition and the isoperimetric inequality will then follow rather easily as applications of the theory.
Finally, we will see how the optimal transport problem gives a natural way to define a distance on the space of probabilities (Wasserstein distance) and we will study some of its properties.
Literature"Optimal Transport, Old and New", C. Villani