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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 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.

Advanced FEM Techniques

This an advanced monographic course on the numerical analysis of finite element techniques. Each year, the state-of-the-art of a research level topic is selected and presented with strong interaction with the students.

Numerical Methods for PDEs

This is a Joint course, between SISSA PhD in Mathematical Analysis, Modeling, and Applications, Laurea Magistrale in Matematica, and the Laurea Magistrale in Data Science and Scientific Computing.

Course materials are available on the course github repository.

Syllabus

Metric mathematical relativity and optimal transport

Abstract. Lorentzian geometry is the mathematical foundation of Einstein's theory of general relativity, which explains gravity as a manifestation of spacetime curvature (unlike Newton’s theory, which treats gravity as a force). This course has two objectives. First, we give an overview of classical concepts from Lorentzian geometry, encompassing e.g.

Optimal Transport

Location and selected dates:

Lectures will take place in Room 133-Ambrosetti.

  • October: 2, 3, 9, 10, 23, 24, 30, 31
  • November: 6, 13, 20, 21, 27, 28
  • December: 4, 5, 12, 18, 19
  • January: 8, 9, 15, 16, 22, 23, 29, 30
  • February: 5, 6

The lecture of Friday, November 14 is suspended to allow students to participate in the SISSA Orientation Day 2025.

Topis in advanced analysis II

Content:

The course is focused on Nonlinear Partial Differential Equations. It will start from classical theorems up to some nonlinear PDEs under active research.
1) First order PDEs
2) Transport equations and weakly regular vector fields
3) Hyperbolic conservation laws
4) Euler Equations and convex integration techinques
5) Hamilton-Jacobi equation

Schedule:

23/9/2025 - 15/12/2025

Tuesday 9-11, room 133

Wednesday 14-16, room 133

 

Topics in Hamiltonian PDEs and critical point theory

Abstract: 
In this course I will present existence and multiplicity results of periodic solutions of Hamiltonian systems, finite and infinite dimensional, like wave equations, which can be obtained by variational methods and critical point theory. I will also present other results regarding the long time dynamics of Hamiltonian PDEs on tori, addressing both stable and unstable dynamics.

 

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