## 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. The methods have been integrated into complex multidisciplinary systems.

## Research topics

• CONTROL AND OPTIMIZATION
• The efficient solution of optimal control or shape optimization problems involving partial differential equations (PDEs) is a problem of interest in computational science and engineering. The goal of an optimal control problem is the minimization/maximization of a given output of interest (expressed by suitable cost functionals) under some constraints, controlling either suitable variables (such as sources, model coefficients or boundary values) or the shape of the domain itself. In the latter case, we deal with shape optimization or optimal shape design problems.
• REDUCED ORDER MODELLING
• Model order reduction techniques provide an efficient, accurate and reliable way of solving (systems of) parametrized partial differential equations in the many-query or real-time context thanks to offline-online computational splittings, such as (shape) optimization, flow control, characterization, parameter estimation, uncertainty quantification. Our research is mostly based, but not limited to, on certified reduced basis methods and proper orthogonal decomposition for parametrized PDEs.
• FREE SURFACES
• Techniques to study the position of an interface as a part of the problem itself, when studying the dynamics of a boat, for example.
• FLUID-STRUCTURE INTERACTION
• Development of efficient algorithms and methods for the coupling between the fluid and structure dynamics finds applications in a large variety of fields dealing with internal or external flows, also at the reduced order level (cardiovascular applications, naval engineering).
• PARALLEL and HIGH PERFORMANCE COMPUTING
• OPEN SOURCE SOFWARE DEVELOPMENT
• Several open source software libraries are developed and maintained

## Collaborating Institutes

• Politecnico di Milano, MOX, Modeling and Scientific Computing Center
• EPFL, Lausanne, Switzerland
• Massachusetts Institute of Technology, Cambridge, US
• Università di Pavia, Italy
• University of Houston, US
• Laboratoire Jacques Louis Lions, Paris VI, France
• Duke University, Durham, US
• Imperial College, London, UK
• Politecnico di Torino, Italy
• Virginia Tech, Blacksburg, Virginia, US
• Scuola Superiore S.Anna, Pisa, Italy
• University of Cambridge, UK
• University of Sevilla, Spain
• University of Santiago de Compostela, Spain
• RWTH Aachen, Germany
• University of Ghent, Belgium

## Grants

ERC CoG 2015 AROMA-CFD grant 681447: Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics (PI Prof. Gianluigi Rozza)

## Purpose of the PhD Course

The aim of the PhD Course in Mathematical Analysis, Modelling, and Applications is to educate graduate students in the fields of mathematical analysis and mathematical modelling, and in the applications of mathematical and numerical analysis to science and technology. The goal is to enable PhDs to work as high level researchers in these fields in universities, research institutes, and private companies.

This PhD Course continues in a unified way the PhD Courses in Mathematical Analysis and in Applied Mathematics, active until the academic year 2012-2013.

## Research Topics

The activity in mathematical analysis is mainly focused on Dynamical Systems and PDEs, on the Calculus of Variations, on Hyperbolic Conservation Laws and Transport Problems, on Geometric Analysis and on Geometry and Control theory. Connections of these topics with differential geometry and reduced order modelling 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, computational fluid and solid dynamics, numerical analysis and scientific computing, 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 developed in collaboration with MathLab  for the study of problems coming from the real world, industrial and medical applications, and complex systems.

Students are selected yearly by means of an entrance examination; there are two admission sessions: spring and autumn. The latter will be activated only if there should still be places available after the former. Deadlines for the academic year 2023-2024 will be available after the publication of the call.
[PhD announcements are available here ]

The entrance examination procedure will be based on:

• Evaluation of academic and scientific qualifications
• Written exam (in presence)
• Oral exam (in presence)

Candidates domiciled in extra-EU countries will be allowed to attend remotely contemporaneously to the other candidates. Non-EU students' preselection will not take place anymore due to the new entrance examination procedure.

The admission process starts by registering at the page https://pica.cineca.it/sissa/ and filling out the requested form.

## Important Dates and Info

Spring entrance session:
• deadline for application: 3 February 2023
• exams: 14-15 February 2023
Autumn entrance session :
• deadline for application: 22 August 2023
• exams: 5-6 September 2023

Progress exam from 1st to 2nd and from 2nd to 3rd year:
• date: 15 September 2023
• exam commission: proff. A. Braides (president), S. Bianchini, G. Rozza, G. Noselli (secretary), A. Maspero and dr. M. Girfoglio

Progress exam from 3rd to 4th year:
• date: 7 September 2023
• exam commission: proff. G. Dal Maso (president), N. Gigli, A. Cangiani (secretary), L. Rizzi, L. Heltai and G. Noselli

Final AMMA PhD exams:
• date: 29 September 2023

Should you have any queries or require any further information please do not hesitate to contact us by email: phd@sissa.it

## Former Faculty Members

Former Professors

## Previous PhD Theses

Click here to see the previous PhD Theses.

## Regulation

Click here to see the regulation of this PhD course (in Italian).

## 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
• Dynamic 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 problems
• Variational methods in phase transitions
• Variational methods in micromagnetics
• Applications of geometric measure theory
• Existence problems in the calculus of variations
• Hamilton-Jacobi equations

## Grants

• National Research Project (PRIN 2015) “Calculus of Variations”, funded by the Italian Ministry of Education, University, and Research, February 5, 2017 – February 5, 2020, National Coordinator: Luigi Ambrosio.

## Research topics

• Stochastic Geometry
• Real algebraic Geometry
• Sub-Riemannian geometry
• Optimal Control and Optimal Synthesis
• Feedback Equivalence and Feedback Invariants
• Switching Systems
• Ensemble Control
• Control of Fluid Mechanics Systems
• Optimal Transportation
• Applications to Hamiltonian Dynamics