Mathematical Sciences | Università degli Studi di Milano Statale

Doctoral programme (PhD)

A.Y. 2023/2024

Study area

Science and Technology

Doctoral programme (PhD)

Years

Dip. Matematica 'Federigo Enriques' - Via Saldini, 50 - Milano

Italian

PhD Coordinator

Bambusi Dario Paolo

Course objectives

The doctoral programme in Mathematical Sciences aims to teach students the research techniques and methods typical of the sectors of contemporary Mathematics and their applications, both qualitative and quantitative, so as to obtain the wide ranging scientific and cultural autonomy required to produce original and significant results. The programme will produce graduates with expertise in exploiting the full potential of mathematical tools and methods and statistics to tackle the intrinsic complexity of problems posed by the applied sciences and industry. The first year syllabus includes advanced theory and workshops held by international scholars chosen by the Board of Lecturers to offer students the opportunity to establish direct contacts with the international scientific community. Doctoral students will have personalised courses under the guidance of a tutor. Once mandatory attendance of courses and examinations are completed, students can concentrate on their chosen research project. Students are assessed on the basis of their doctoral thesis, to which the three-year doctoral programme dedicates considerable attention.

Entry requirements

Tutte le classi di laurea magistrale - All classes of master's degree

Course location

Dip. Matematica 'Federigo Enriques' - Via Saldini, 50 - Milano

Contacts

Main offices
Dip. Matematica 'Federigo Enriques' - Via Saldini, 50 - Milano
Degree course coordinator: Dario Paolo Bambusi
[email protected]
Degree course website
http://www.mat.unimi.it/dottorati

Research lines

Title	Professor(s)
Inverse boundary value problems for partial differential equations: uniqueness, stability issues and reconstruction algorithms based on artificial intelligence techniques Requirements: Good knowledge of functional analysis and the theory of weak solutions for partial differential equations and systems. Good knowledge of basic numerical analysis and computer programming	Aspri Andrea Cavaterra Cecilia Causin Paola
KAM and normal form methods for modulation equations and for the deduction of effective equations (non relativistic limit, equations for wave packets, many body equations in classical and quantum mechanics). PRIN project “Hamiltonian and Dispersive PDEs” Requirements: Elementary properties of Hamiltonian systems and partial differential equations	Bambusi Dario Paolo
Mathematical methods in quantum many body theory: emergent properties in Fermi and Bose gases (related to the ERC Starting Grant 2021 “FermiMath”) Requirements: Basic knowledge of functional analysis (Hilbert space theory)	Benedikter Niels Patriz Boccato Chiara
Bosonization in dimensions 1+1 and 3+1 for interacting fermionic systems in scaling limits, ground state energy and relation with anomalies of Adler-Bardeen (related to the ERC Starting Grant 2021 “FermiMath” and PRIN 2017 MAQUMA) Requirements: Knowledge of mathematical physics, analytic capacity	Benedikter Niels Patriz Mastropietro Vieri
Semiclassical limits in quantum mechanics: quantitative analysis via Bogolioubov theory and normal forms Requirements: Basic knowledges of quantum mechanics, perturbation theory, partial differential equations and functional analysis	Benedikter Niels Patriz Boccato Chiara Montalto Riccardo
Homotopical methods in arithmetic geometry Requirements: Knowledge of commutative algebra, scheme theory and some background in homotopica/homological algebra	Binda Federico Vezzani Alberto
Rigid geometry, logarithmic geometry and perfectoid spaces Requirements: Knowledge of number theory, algebraic geometry, commutative algebra and homological algebra	Binda Federico Vezzani Alberto
Mathematics learning environments in secondary school and integration of digital technologies for teaching Requirements: Knowledge of mathematics at university level relevant for teaching in upper secondary school. Elements of mathematics teaching	Branchetti Laura
Interdisciplinarity in the initial training of mathematics teachers Requirements: Knowledge of mathematics and physics or computer science at university level relevant for teaching in upper secondary school. Didactic elements of mathematics or physics or computer science	Branchetti Laura
Problems of learning and task design in the school-university transition Requirements: Knowledge of mathematics at university level relevant for teaching in upper secondary school. Elements of mathematics teaching	Branchetti Laura
Algebraic Geometry: projective models, automorphism groups and moduli spaces of Hyperkähler manifolds, irreducible symplectic varieties and Enriques varieties. This project is part of the project PRIN2020 "Curves, Ricci flat varieties and their interactions" Requirements: Good knowledge of algebraic geometry ad of complex geometry	Camere Chiara
Stochastic optimal control, backward stochastic differential equations and control of McKean-Vlasov systems. Requirements: Knowledge of stochastic processes and stochastic calculus	Campi Luciano Cosso Andrea Fuhrman Marco Alessandro
Stochastic differential games and mean field games with applications Requirements: Knowledge of stochastic processes and stochastic calculus	Campi Luciano
Biomathematics and Biostatistics - research line connected with: - Project FAITH - Fighting Against Injustice Through Humanities (strategic project of UniMI) - Project Modeling the heart across the scales: from cardiac cells to the whole organ” PRIN 2017, 2019-2022, PI A. Quarteroni (PoliMI) - Project MICROCARD - “Numerical modeling of cardiac electrophysiology at the cellular scale”, EuroHPC2020, 2021-2024, PI M. Potse (Univ. Bordeaux) - Collaborations with researchers of biomedical areas on: * Survival analysis for oncological patients * Planning of ideotypes of cereal plants to resist to climate changes * Mathematical and numerical modelling of cardiac electromechanical activity - Collaborations with industrial partners * Mathematical and computational models for Diffuse Optimal Tomography Requirements: Knowledge in probability, Mathematical Statistics. Partial differential equations, analytical and numerical aspects. Differential Modelling	Causin Paola Cavaterra Cecilia Micheletti Alessandra Scacchi Simone
Mathematical modelling in Cultural Heritage Requirements: Good knowledge of basic mathematical analysis. Elements of real and functional analysis	Cavaterra Cecilia Bonetti Elena
Evolution systems of partial differential equations and applications. Research line connected with project PRIN 2020: Mathematics for industry 4.0 (Math4I4) Requirements: Good knowledge of basic mathematical analysis. Elements of real and functional analysis	Cavaterra Cecilia Bonetti Elena
Inverse problems for systems of partial differential equations: identification of parameters, inclusions and inhomogeneities Requirements: Good knowledge of basic mathematical analysis. Elements of real and functional analysis	Cavaterra Cecilia Aspri Andrea
Geometric properties of solutions to partial differential equation Requirements: Knowledge of the basics of analysis and geometry, with emphasis in partial differential equations and basics of functional analysis	Ciraolo Giulio
Regularity for solutions to elliptic PDEs Requirements: Knowledge of the basics of analysis and geometry, with emphasis in partial differential equations and basics of functional analysis	Ciraolo Giulio
Hamilton-Jacobi-Bellman equations on Wasserstein spaces or function spaces Requirements: Knowledge stochastic processes and stochastic calculus	Cosso Andrea
Varieties with trivial canoncial bundle: quotients, fibrations and Hodge structures Requirements: Basic notions in algebraic and complex geometry	Garbagnati Alice
Algebraic logic , categorical logic and duality theory, model-checking and decision procedures, non standard analysis and Ramsey theory Requirements: Good mathematical background, together with knowledge of fundamental mathematical logic results and techniques	Ghilardi Silvio Marra Vincenzo Luperi Baglini Lorenzo
Classification of semi-discrete Hamiltonian systems Requirements: Basic knowledge of Hamiltonian mechanics in finite dimensions, differential and Riemannian geometry, projective geometry	Gubbiotti Giorgio
Integrable systems in N dimensions and generalisations Requirements: Basic knowledge of Hamiltonian mechanics in finite dimensions, differential and Riemannian geometry, projective geometry	Gubbiotti Giorgio
Geometry, dynamics, and symmetries of Bertini-Moody-Manin involutions Requirements: Basic knowledge of algebraic and complex geometry	Gubbiotti Giorgio Van Geemen Lambertus
Classification on study of discrete-time systems admitting coalgebra symmetry Requirements: Basic knowledge of Lie algebras and dynamical systems	Gubbiotti Giorgio
Complexity and growth of birational maps of projective spaces in dimension higher than 2 Requirements: Basic knowledge of algebraic and complex geometry	Gubbiotti Giorgio
Algebraic geometry, derived categories, birational geometry. Part of the project Progetto PRIN 2020: Curves, Ricci flat varieties and their Interactions Requirements: Knowledge in scheme theory, Riemann surfaces, homological algebra	Lombardi Luigi
Isogeometric analsysis and virtual element methods; numerical methods for partial differential equations. Projects involved: PRIN 2017, Virtual Element Methods: Analysis and Applications; PRIN 2020, Advanced polyhedral discretisations of heterogeneous PDEs for multiphysics problems Requirements: Knowledge of numerical methods for partial differential equations	Lovadina Carlo Scacchi Simone
Categorical Algebra Requirements: Basic knowledge of Category Theory, Universal and Homological Algebra	Mantovani Sandra Montoli Andrea
Differential Geometry and Global Analysis Requirements: Knowledge of riemannian Geometry and PDE's	Rigoli Marco Mastrolia Paolo
Computational geometry and topology for machine learning - research related with the Italian PNRR thematic 'Artificial intelligence: foundational aspects' and to the industrial project "Development of methods of computational topology and explainable machine learning applied to molecular docking" Requirements: Knowledge of real and functional analysis; topology; Statistics; neural networks	Micheletti Alessandra
Space-time stochastic processes, Stochastic Geometry and statistical shape analysis: point processes, random sets, random measures - research related with the ECMI Special Interest group "Shape and size in medicine, biotechnology and materials science" Requirements: Knowledge of Measure theory; Probability and Mathematical Statistics	Micheletti Alessandra Villa Elena
Statistical and stochastic analysis and calibration in modelling the degradation phenomenon in cultural heritages. SEED-UNIMI project and PON project on Green themes. The research is carried on in collaboration with the University of Pisa and University of Karlstad Requirements: Knowledge of statistics, stochastic processes and stochastic calculus	Morale Daniela Ugolini Stefania
Analysis of stochastic equations and interacting particles systems in modelling the degradation phenomenon in cultural heritages. Convergence problems from the nano to macroscale. The research is carried on in collaboration with the University of Pisa, University of Pavia and University of Karlstad. Requirements: Knowledge of stochastic processes and stochastic calculus	Morale Daniela Ugolini Stefania
Nonlinear waves in fluid mechanics and dispersive equations via quasi-linear Kam and normal form methods. ERC project: Hamiltonian Dynamics, Normal Forms and Water Waves Requirements: Basic knowledges of evolution PDEs, perturbation theory, Fourier and functional analysis	Montalto Riccardo
Stability of solitons and periodic and quasi-periodic waves for integrable and quasi-integrable Partial Differential equations - ERC project: Hamiltonian Dynamics, Normal Forms and Water Waves Requirements: Basic knowledges of evolution PDEs, integrable systems, perturbation theory, Fourier and functional analysis	Montalto Riccardo
Normal form methods for singular perturbation problems - ERC project: Hamiltonian Dynamics, Normal Forms and Water Waves Requirements: Basic knowledges of evolution PDEs, perturbation theory, Fourier and functional analysis	Montalto Riccardo
Motivic homotopy theory, motivic cohomology, motives, K-theory Requirements: Knowledge of Algebraic geometry and topology, homotopical algebra, infinity categories	Oestvaer Paul Arne
Stability conditions on triangulated categories and the geometry of moduli spaces Requirements: Solid background in complex algebraic geometry	Pertusi Laura
Mathematical Methods in Quantum Mechanics and in General Relativity; Evolution equations (especially, in fluid dynamics) . The proponent is financially supported by: 1) MIUR, Project PRIN 2020 ''Hamiltonian and dispersive PDEs''; 2) Università degli Studi di Milano, PSR2021, Project ''Classical and quantum dynamical systems, statistical mechanics''; 3) Istituto Nazionale di Fisica Nucleare, Specific Initiative BELL Requirements: Basic knowledge of functional analysis and quantum mechanics; Basic knowledge of differential geometry and general relativity	Pizzocchero Livio
p-adic methods in arithmetic Requirements: Knowledge of Number Theory, Algebraic Geometry and Commutative Algebra	Seveso Marco Adamo Venerucci Rodolfo
Rational points on elliptic curves Requirements: Knowledge of Number Theory, Algebraic Geometry and Commutative Algebra	Seveso Marco Adamo Venerucci Rodolfo
Algebraic Geometry and Homological Algebra: derived, triangulated and dg categories in algebraic geometry Requirements: Solid background in algebraic geometry	Stellari Paolo
Birational geometry of algebraic foliations Requirements: Knowledge of foundations of algebraic geometry and/or theory of holomorphic foliations. It may be beneficial to have at least a rudimentary knowledge of the Minimal Model Program	Svaldi Roberto Tasin Luca
Boundedness problems in algebraic geometry Requirements: Knowledge of foundations of algebraic geometry, particularly of Fano and/or Calabi--Yau varieties. It may be beneficial to have at least a rudimentary knowledge of the Minimal Model Program,	Svaldi Roberto Tasin Luca
Stochastic methods in quantum mechanics. The main research line is the stochastic description of the Bose-Einstein condensation phenomenon. The research is carried on in collaboration with the University of Bonn and the University of Pavia Requirements: Knowledge of stochastic processes and stochastic calculus	Ugolini Stefania S. Albeverio
Study of invariance and symmetry properties of stochastic dynamical systems, generalizing the classical theory of S. Lie. The research is carried on in collaboration with the University of Bonn and the University of Pavia. Requirements: Knowledge of stochastic processes and stochastic calculus	Ugolini Stefania S. Albeverio
Algebraic geometry and Hodge theory (PRIN) Requirements: Basic knowledge of algebraic and complex geometry	Van Geemen Lambertus
Galerkin methods for partial differential equations. Projects involved: PRIN 2017, Virtual Element Methods: Analysis and Applications; PRIN 2017, Numerical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of complex systems governed by Partial Differential Equations; PRIN 2020, Advanced polyhedral discretisations of heterogeneous PDEs for multiphysics problems Requirements: Knowledge of theory and practice of Finite Element Methods; fundamentals of Numerical Linear Algebra	Veeser Andreas Lovadina Carlo
Development of imaging algorithms for biomedical applications (ex DM 117/2023)	Causin Paola Scacchi Simone

Courses list

Educational Plan

Optional programme year

November 2023

Courses or activities	Professor(s)	ECTS	Total hours	Language
Optional
Mathematical Methods for Many-Body Quantum Systems	Benedikter Niels Patriz	3	15	English

December 2023

Courses or activities	Professor(s)	ECTS	Total hours	Language
Optional
Geometric and Numerical Invariants of Derived Categories of Sheaves	Lombardi Luigi	3	15	Italian

January 2024

Courses or activities	ECTS	Total hours	Language
Optional
Analytic Techniques Applied to Derived and Triangulated Categories	3	15	English
Boundary Layers and Applications to the Vanishing Viscosity Limit for Incompressible Fluids	3	16	English