The activity in mathematical analysis is mainly focussed on ordinary and partial differential equations, on dynamical systems, on the calculus of variations, and on control theory. Connections of these topics with differential geometry are also developed.The activity in mathematical modelling is oriented to subjects for which the main technical tools come from mathematical analysis. The present themes are multiscale analysis, mechanics of materials, micromagnetics, modelling of biological systems, and problems related to control theory.The applications of mathematics developed in this course are related to the numerical analysis of partial differential equations and of control problems. This activity is organized in collaboration with MathLab for the study of problems coming from the real world, from industrial applications, and from complex systems.

## Calculus of Variations and Multiscale Analysis

## Research topics

- Homogenization of variational problems
- Gamma-Convergence and relaxation
- Variational methods in continuum mechanics
- Variational methods in rate independent evolution prooblems
- Variational methods in phase transitions
- Variational methods in micromagnetics
- Applications of geometric measure theory
- Existence problems in the calculus of variations
- Hamilton-Jacobi equations

## Geometry and Control

## Research topics

- Optimal Control and Optimal Synthesis
- Sub-Riemannian geometry
- Feedback Equivalence and Feedback Invariants
- Switching Systems
- Quantum Control
- Control of Fluid Mechanics Systems
- Optimal Transportation
- Aplications to Vision and Robotics
- Applications to Differential Geometry and Dynamical Systems
- Stochastic Geometry
- Real algebraic Geometry

## Ordinary Differential Equations

**Research topics**

- Periodic solutions with oscillatory and superlinear nonlinearities
- Sturmian theory for multipoints and for matrix equations
- Functional boundary value problems a la Conti-Lasota
- A general spectral theory for nonsymmetric BVP

**Main External Collaborators**

- G. Vidossich
- S. Ahmad
- G. A. Degla
- M. Gaudenzi
- F. Zanolin
- A. Fonda

## 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

## Dynamical Systems and PDEs

## Research topics

- KAM for PDEs
- Water waves, KdV, Schrodinger and Klein-Gordon equations
- Hamiltonian systems
- Birkhoff normal form for PDEs and long time existence
- Chaotic Dynamics and Arnold Diffusion
- Variational Methods
- Perturbative Methods in Critical Point Theory
- Elliptic Equations on $\mathbb R^n$ and Nonlinear Schrodinger Equation

## Research Group

## Mathematical Analysis, Modelling, and Applications

A research group in mathematical analysis has been active at SISSA since 1978, the year SISSA was founded. Starting from 2000, the activity of this group has been extended to mathematical modelling and applications.