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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.

AJS - Analysis Junior Seminars 2021-2022

Date Speaker Seminar
October 8 Daniele Tiberio Geodesic distances on Banach manifolds
October 15 Martina Zizza Properties of mixing BV vector fields
October 22 Simone Carano The relaxed area with respect to the strict convergence in BV
October 29 Moaad Khamlich Model order reduction for bifurcating phenomena in fluid-structure interaction problems
November 05 Daniela Di Donato C^1 submanifolds and Lipschitz graphs in Carnot groups
November 12 Giulio Ortali Subgrid Closure for the Shell Model of Turbulence using Deep Learning
November 19 Emanuele Caputo Parallel transport on ncRCD(K,N) spaces
November 26 Francesco Nobili Progress on the independence on p of p weak gradients
December 3 Niccolò Tonicello High-order spectral element methods for the simulation of compressible turbulent flows
December 10 Moreno Pintore Progress on the independence on p of p weak gradients
December 17 Mattia Manucci Contour Integral Methods and Reduced Basis for parametric dynamical problems
January 14 Davide Manini Localization via disintegration and isoperimetric inequality in non-compact MCP spaces
January 21 Jose Raul Bravo Martinez KratosMultiphysics and Local ROM
January 28 Luigi De Masi Min-max theory for minimal surfaces
February 4 Dario Andrini Optimal design of planar shapes with active materials
February 11 Alessandro Scagliotti Deep Learning approximation of diffeomorphisms via linear-control systems
February 18 Maria Strazzullo Full Order Model and Reduced Order Model Consistency for Evolve-Filter-Relax Regularization
March 11 Meneghetti Laura Reduced Convolutional Neural Networks for image recognition and object detection
March 18 Carlo Gasparetto You can't always touch a minimal surface
April 22 Davide Torlo Arbitrary High-Order Positivity-Preserving Time Integration with Application on Shallow-Water equations
April 29 Nicolò De Ponti From a quantitative Feller property to functional and geometrical inequalities
May 6 Camillo Brena Weakly non-collapsed RCD spaces are non-collapsed
June 10 Giorgio Stefani Bakry-Èmery curvature condition and entropic inequalities on metric-measure groups
June 24 Alberto Della Noce Numerical simulation of in-flight ice accretion: present and future challenges

Advanced Programming

The course aims to provide advanced knowledge of both theoretical and practical programming in C++14 and Python3, particularly the principles of object-oriented programming and best practices of software development.

Syllabus:

A diagrammatic approach to perturbation theory

In this course I will present a diagrammatic approach which can be conveniently used to study perturbative series. After a general introduction, I will start by focusing on the classical KAM theorem in the "mechanical case'', and then discuss various generalizations, such as the case of more general hamiltonians, non-maximal tori, harmonic oscillators, depending on the audience preferences.

Bifurcation theory and perturbation of linear operators

The first part of the course will be devoted to a self-contained presentation of perturbation theory for eigenvalues of linear operators via complex analysis. The required preliminary properties of analytic functions in Banach spaces will be given. In the second part of the course, we will develop local bifurcation theory with main applications in fluid dynamics. 

Introduction to Elliptic Equations

1. Laplace equation:

  • harmonic functions, mean value properties,
  • maximum principle,
  • Green's function,
  • Poisson kernel,
  • Harnack inequality,
  • subharmonic functions,
  • Perron-Wiener-Brelot method for the Dirichlet problem,
  • regular boundary points.

2. Variational theory of elliptic equations:

Nonsmooth Differential Geometry

1. First order calculus on metric measure spaces:

Topics in Computational Fluid Dynamics

  • Introduction to CFD, examples.
  • Constitutive laws
  • Incompressible flows.
  • Numerical methods for potential and thermal flows
  • Boundary layer theory

Computational Mechanics by Reduced Order Methods

Lectures Prof Gianluigi Rozza, Tutorials coordinated by Dr Giovanni Stabile,  Dr Francesco Ballarin, Dr Maria Strazzullo and Dr Federico Pichi

Learning outcomes and objectives

The course aims to provide the basic aspects of numerical approximation and efficient solution of parametrized PDEs for computational mechanics problem (heat and mass transfer, linear elasticity, viscous and potential flows) using reduced order methods.

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