- Hyperbolic Systems of Conservation Laws in One Space Dimension
- Fundamental theory: existence, uniqueness and continuous dependence of weak entropy admissible solutions, characterization of semigroup trajectories
- Problems with large BV data, blow-up of BV norm, local existence and uniqueness
- Structure of solutions, local behavior, structural stability, generalized shift-differentiability w.r.t. parameters
- Initial-boundary problems, inhomogeneous balance laws, asymptotic blow-up patterns, global existence
- Convergence rates for approximation schemes: wave-front tracking, Glimm, finite element
- Vanishing viscosity approximations, a-priori estimates, convergence

## Introduction to Optimal Transport

The theory of Optimal Transport finds its roots in the work of Gaspard Monge in 1781, whose interest was in finding the cheapest possible way to transfer some resources from producers to consumers.

After some major contributions due to Leonid Kantorovich (who was later awarded the Nobel Prize in Economics, partly for his work on the subject) in the 1940s, Optimal Transport has been blooming in the last thirty years, with applications covering partial differential equations, geometric analysis, mathematical finance, and machine learning, among the others.

## Problems of Moving Sets

The course is divided into two parts, each related to the control of expanding sets either with a barrier or by removing a fixed amount of area per unit time.

## Linear wave propagations for half space problems

In this series of talks, we are aimed at constructing the solution formula of the Green's functions for various hyperbolic and hyperbolic-parabolic partial differential equations in a half multi-D space domain. We will use the transform variables to derive a master relationships of the Dirichlet-Neumann data of the PDE, and use it to obtain the full Dirichlet-Neumann data in the transform variables; and obtain the Rayleigh surface.

## Conservation Laws and Transport Problems

## Research topics

- Hyperbolic Systems of Conservation Laws in One Space Dimension
- Fundamental theory: existence, uniqueness and continuous dependence of weak entropy admissible solutions, characterization of semigroup trajectories
- Problems with large BV data, blow-up of BV norm, local existence and uniqueness
- Structure of solutions, local behavior, structural stability, generalized shift-differentiability w.r.t. parameters
- Initial-boundary problems, inhomogeneous balance laws, asymptotic blow-up patterns, global existence
- Convergence rates for approximation schemes: wave-front tracking, Glimm, finite element
- Vanishing viscosity approximations, a-priori estimates, convergence
- Flow of weakly differentiable vector fields
- Linear transport problems