Mathematical Sciences
Doctoral programme (PhD)
A.Y. 2024/2025
Study area
Science and Technology
PhD Coordinator
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.
Tutte le classi di laurea magistrale - All classes of master's degree
Dip. Matematica 'Federigo Enriques' - Via Saldini, 50 - Milano
- Main offices
Dip. Matematica 'Federigo Enriques' - Via Saldini, 50 - Milano - Degree course coordinator: Prof. Ciraolo Giulio
[email protected] - Degree course website
http://www.mat.unimi.it/dottorati
| 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. |
|
| 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 |
|
| 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) |
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 |
V. Mastropietro
|
| 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 |
C. Boccato
|
| Homotopical methods in arithmetic geometry
Requirements: Commutative algebra, scheme theory and some background in homotopica/homological algebra |
|
| Rigid geometry, logarithmic geometry and perfectoid spaces
Requirements: Number theory, algebraic geometry, commutative algebra and homological algebra |
|
| 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 |
|
| 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 |
|
| 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 |
|
| 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 |
|
| Stochastic optimal control, backward stochastic differential equations and control of McKean-Vlasov systems.
Requirements: Stochastic processes and stochastic calculus |
|
| Stochastic differential games and mean field games with applications
Requirements: Stochastic processes and stochastic calculus |
|
| 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 |
|
| Optimal Transport in Lorentzian manifolds. Structure of Lorentzian length spaces and synthetic curvature bounds
Requirements: Basic knowledge of optimal transport and of differential geometry |
|
| Mathematical modelling in Cultural Heritage
Requirements: Good knowledge of basic mathematical analysis. Elements of real and functional analysis. |
|
| Evolution systems of partial differential equations and applications.
Requirements: Good knowledge of basic mathematical analysis. Elements of real and functional analysis. |
|
| 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. |
|
| 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 |
|
| 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 |
|
| Hamilton-Jacobi-Bellman equations on Wasserstein spaces or function spaces
Requirements: Stochastic processes and stochastic calculus |
|
| 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. |
|
| Varieties with trivial canoncial bundle: quotients, fibrations and Hodge structures
Requirements: Basic notions in algebraic and complex geometry |
|
| 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 |
|
| Classification of semi-discrete Hamiltonian systems
Requirements: Basic knowledge of Hamiltonian mechanics in finite dimensions, differential and Riemannian geometry, projective geometry |
|
| Integrable systems in N dimensions and generalisations
Requirements: Basic knowledge of Hamiltonian mechanics in finite dimensions, differential and Riemannian geometry, projective geometry |
|
| Geometry, dynamics, and symmetries of Bertini-Moody-Manin involutions
Requirements: Basic knowledge of algebraic and complex geometry |
|
| Classification on study of discrete-time systems admitting coalgebra symmetry
Requirements: Basic knowledge of Lie algebras and dynamical systems |
|
| Complexity and growth of birational maps of projective spaces in dimension higher than 2
Requirements: Basic knowledge of algebraic and complex geometry |
|
| 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 |
|
| 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 |
|
| Existence, non-existence and rigidity results for submanifolds with prescribed curvature in Riemannian and Lorentzian ambient spaces.
Requirements: Riemannian Geometry and PDE's |
|
| Categorical Algebra
Requirements: Basic knowledge of Category Theory, Universal and Homological Algebra |
|
| 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 |
|
| 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 |
|
| 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. |
|
| 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 |
|
| 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 |
|
| 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 |
|
| 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 |
|
| 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 |
|
| Motivic homotopy theory, motivic cohomology, motives, K-theory
Requirements: Algebraic geometry and topology, homotopical algebra, infinity categories |
|
| Stability conditions on triangulated categories and the geometry of moduli spaces
Requirements: Solid background in complex algebraic geometry |
|
| 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 |
|
| Differential Geometry and Global Analysis
Requirements: Riemannian Geometry and PDE's |
|
| p-adic methods in arithmetic
Requirements: Number Theory, Algebraic Geometry and Commutative Algebra |
|
| Rational points on elliptic curves
Requirements: Number Theory, Algebraic Geometry and Commutative Algebra |
|
| Algebraic Geometry and Homological Algebra: derived, triangulated and dg categories in algebraic geometry
Requirements: Solid background in algebraic geometry |
|
| 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. |
|
| 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. |
|
| 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 |
|
| 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, |
|
| 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 |
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 |
S. Albeverio
|
| Algebraic geometry and Hodge theory (PRIN)
Requirements: Basic knowledge of algebraic and complex geometry |
|
| 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 |
Courses list
January 2025
| Courses or activities | Professor(s) | ECTS | Total hours | Language |
|---|---|---|---|---|
| Optional | ||||
| Stochastic Control and Games with Different Information Structures | 3 | 15 | English | |
February 2025
| Courses or activities | Professor(s) | ECTS | Total hours | Language |
|---|---|---|---|---|
| Optional | ||||
| Geometric and Numerical Invariants of Derived Categories of Sheaves | 3 | 15 | English | |
April 2025
| Courses or activities | Professor(s) | ECTS | Total hours | Language |
|---|---|---|---|---|
| Optional | ||||
| Boolean Harmonic Analysis with Application to Learning of Classical and Quantum Objects | 3 | 15 | English | |
| Stability Conditions On Derived Categories and Kuznetsov Components | 3 | 15 | English | |
| The Bernstein's Problem | 3 | 15 | English | |
May 2025
| Courses or activities | Professor(s) | ECTS | Total hours | Language |
|---|---|---|---|---|
| Optional | ||||
| Partial Differential Equations Under Curvature Bounds | 3 | 15 | English | |
June 2025
| Courses or activities | Professor(s) | ECTS | Total hours | Language |
|---|---|---|---|---|
| Optional | ||||
| An Introduction to Mathematical Wave Kinetic Theory | 3 | 15 | English | |
Enrolment
Places available: 9
Call for applications
Please refer to the call for admission test dates and contents, and how to register.
Application for admission: from 29/05/2024 to 27/06/2024
Application for matriculation: from 09/08/2024 to 05/10/2024
Attachments and documents
Following the programme of study
Contacts
Office and services for PhD students and companies