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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.

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Research fields

  • geometry, in particular algebraic, differential, and noncommutative geometry, also with applications to quantum field and string theory
  • mathematical analysis, in particular calculus of variations, control theory, partial and ordinary differential equations
  • mathematical modelling, in particular mechanics of solids and fluids, modelling of complex and biological systems, multiscale analysis
  • mathematical physics, in particular integrable systems and their applications, nonlinear partial differential equations, mathematical aspects of quantum physics
  • numerical analysis and scientific computing, applied to partial differential equations and to control problems

PhD and MSC courses:

Laboratories:

  • SISSA MathLab: a laboratory for mathematical modeling and scientific computing
  • SAMBA a laboratory in collaboration with the Cognitive Neuroscience Group

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Markov diffusion semigroups and functional inequalities

The course aims to provide a brief introduction to analytic and geometric aspects of Markov diffusion semigroups and their infinitesimal generators. A particular attention will be given to functional inequalities that can be studied in this framework, including but not limited to Poincaré, Sobolev and log Sobolev inequalities, and to relations with Ricci curvature bounds.

The main reference is the book of Bakry, Gentil, Ledoux “Analysis And Geometry Of Markov Diffusion Operators”.

Topics in mathematical epidemiology

In this course, we revisit the classical compartmental models of mathematical epidemiology and present some of their recent advances. The topics can be organized in three main modules.In the first module, we trace the history of epidemiological models starting from the pioneering work of Bernoulli (1766) up to the well-known Susceptible-Infected-Removed (SIR) model by Kermack and McKendrick (1927) and its subsequent variants.

Numerical Solution of PDEs Using the Finite Element Method

Advanced course dedicated to the Numerical Solution of Partial Differential Equations through the deal.II Finite Element Library.

Topics:

Optimal Transport

The course intends to be a broad introduction to Optimal Transport theory and some of its applications in geometry and analysis. 

Periodic Orbits of Hamiltonian systems through Variational Methods

Hamiltonian systems give a very good description of those physical phenomena where the energy is (approximately) conserved: from planetary orbits to the motion of particles.

Typically, however, the dynamics is highly sensitive to the initial conditions and therefore it is difficult to find specific orbits in the systems such as those connecting two subsets of phase space or those which are periodic.

Some aspects of mean curvature flow

Topics covered in the course: Main properties of classical mean curvature flow and anisotropic mean curvature flow. The minimizing movements method. Mean curvature flow as a limit of the Allen-Cahn equations.

Geometric Control Theory

"Geometric Control Theory” by Ugo Boscain (CNRS, LJLL, Sorbonne Université, and INRIA), Paris)

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