Mathematical Sciences | Università degli Studi di Milano Statale

Doctoral programme (PhD)

A.Y. 2024/2025

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 C. Boccato
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 C. Boccato Montalto Riccardo
Homotopical methods in arithmetic geometry Requirements: Commutative algebra, scheme theory and some background in homotopica/homological algebra	Binda Federico Vezzani Alberto
Rigid geometry, logarithmic geometry and perfectoid spaces Requirements: 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 PRIN2022 "Symplectic varieties: their interplay with Fano manifolds and derived categories". 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: Stochastic processes and stochastic calculus	Campi Luciano Cosso Andrea Fuhrman Marco Alessandro
Stochastic differential games and mean field games with applications Requirements: 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: Probability, Mathematical Statistics. Partial differential equations, analytical and numerical aspects. Differential Modelling	Causin Paola Cavaterra Cecilia Micheletti Alessandra Scacchi Simone
Optimal Transport in Lorentzian manifolds. Structure of Lorentzian length spaces and synthetic curvature bounds Requirements: Basic knowledge of optimal transport and of differential geometry	Cavalletti Fabio
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. 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: Stochastic processes and stochastic calculus	Cosso Andrea
Foundational problems in Financial Mathematics: Fundamental Theorem of Asset Pricing with cooperating agents; martingale optimal transport; time consistency in the theory of preferences. Requirements: Good knowledge of functional analysis and of probability theory, in addition to the classical aspects of financial mathemtatics.	Frittelli Marco Maggis Marco
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: 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: Numerical methods for partial differential equations	Lovadina Carlo Scacchi Simone
Existence, non-existence and rigidity results for submanifolds with prescribed curvature in Riemannian and Lorentzian ambient spaces. Requirements: Riemannian Geometry and PDE's	Mari Luciano
Categorical Algebra Requirements: Basic knowledge of Category Theory, Universal and Homological Algebra	Mantovani Sandra Montoli Andrea
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: 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: Measure theory; Probability and Mathematical Statistics	Micheletti Alessandra Villa Elena
Analytic Number Theory, with a special focus on explicit bounds and applications to algorithms computing algebraic invariants Requirements: Good knowledge of bases of Number theory, bot in analytic and algebraic flavours.	Molteni Giuseppe
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: 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: 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: 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
Differential Geometry and Global Analysis Requirements: Riemannian Geometry and PDE's	Rigoli Marco Mastrolia Paolo Mari Luciano
p-adic methods in arithmetic Requirements: Number Theory, Algebraic Geometry and Commutative Algebra	Seveso Marco Adamo Venerucci Rodolfo
Rational points on elliptic curves Requirements: 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
Geometric Analysis, Geometric Measure Theory and regularity of solutions to geometric variational problems (within project PRIN 2022PJ9EFL "Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations") Requirements: Solid background in Mathematical Analysis and Measure Theory. Good knowledge of the main techniques in elliptic PDE theory and Calculus of Variations. Knowledge of Riemannian Geometry. Geometric intuition.	Stuvard Salvatore
Fine properties and regularity of weak solutions to Mean Curvature Flows (within project PRIN 2022PJ9EFL "Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations") Requirements: Solid background in Mathematical Analysis and Measure Theory. Good knowledge of the main techniques in parabolic PDE theory. Knowledge of Riemannian Geometry.	Stuvard Salvatore
Birational geometry of algebraic foliations Requirements: 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: 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: 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: 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: Theory and practice of Finite Element Methods; fundamentals of Numerical Linear Algebra	Veeser Andreas Lovadina Carlo