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## Mathematical Analysis, Modelling, and Applications

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 analysismechanics of materialsmicromagneticsmodelling 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.

## Numerical Analysis and Scientific Computing

The research deals with the analysis, development, application of mathematical models for the integration of complex systems. The analysis is conducted using mathematical methods in several fields such as linear algebra, approximation theory, partial differential equations, optimization and control. Solution methods are developed and applied to domains as diverse as (potential and viscous) flow dynamics, (linear and nonlinear) structural analysis, mass transport, heat transfer and in general to multiscale and multiphysics applications.

## Admission to PhD program

There are two admission sessions: on 12-13 April 2018 and on 5-6 September 2018.

Entrance examination is based on:

• Evaluation of curriculum vitae
• Written test (first day of exams)
• Oral interview (second day of exams)
Funds are available to cover travel and lodging expenses: to apply for such funds write at phd@sissa.it and submit your CV.

## Research topics

• Nonlinear solid mechanics: finite elasticity and elasto-plasticity
• Soft matter elasticity: polymers, liquid crystals, granular materials
• Electro-magneto-mechanically coupled systems
• Variational methods fracture mechanics
• Scientific computing
• Cell motility and Mechano-biology
• Stokes Equations

## 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

## 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

## 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

## 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