Mathematical Sciences
Doctoral programme (PhD)
A.Y. 2022/2023
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: Dario Paolo Bambusi
[email protected] - Degree course website
http://www.mat.unimi.it/dottorati
| Title | Professor(s) |
|---|---|
| Study of stochastic partial differential equations. Field theory (stochastic quantization)
Requirements: Stochastic calculus, mathematical analysis of partial differential equations |
Sergio Albeverio
|
| Normal forms and growth of Sobolev norms in Hamiltonian PDEs in more than higher space dimension
Requirements: Perturbation theory for classical Hamiltonian systems, Birkhoff normal form, elements of functional analysis |
|
| Energy sharing in multiparticle quantum systems, quantum Nekhoroshev’s theorem
Requirements: Perturbation theory for classical Hamiltonian systems, Birkhoff normal form, elements of functional analysis |
|
| Normal form theory for quantum systems and validity in time of effective non-linear equations (connected to the ERC Starting Grants 2021 „FermiMath“ and „HamDyWWa“
Requirements: Hamiltonian mechanics, basic quantum theory |
|
| Mathematical Methods in Many-Body Quantum Mechanics: emergent behavior in Fermi and Bose gases (connected to the ERC Starting Grant 2021 „FermiMath“)
Requirements: Fundamentals of functional analysis (Hilbert space theory) |
|
| Bosonization in 1+1 and 3+1 dimensions for interacting fermionic systems in the scaling limit, ground state energy and relation with Adler-Bardeen anomalies (connected to the ERC Starting Grant 2021 „FermiMath“)
Requirements: Knowledge of mathematical physics, analytical skills |
|
| Group Theory and Representation Theory
Requirements: Basic Algebra and Group Theory |
|
| 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 |
|
| Integrodifferential equations and nonlocal minimal surfaces
Requirements: Good knowledge of partial differential equations and of the basics of geometry |
|
| Mean Field Games and applications in Mathematical Finance
Requirements: Stochastic optimal control, Probability, Stochastic Calculus |
M. Burzoni
|
| 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 |
|
| Stochastic differential games and mean field games with applications
Requirements: Stochastic processes, stochastic calculus |
|
| Stochastic optimal control, backward stochastic differential equations and control of systems of McKean-Vlasov type.
Requirements: Stochastic processes; 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. |
|
| 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. |
|
| Mathematical modelling in applications: Research line connected with project SCICULT- Bando SEEDS-SOE Unimi.
Requirements: Good knowledge of basic mathematical analysis. Elements of real and functional analysis. |
|
| 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. |
|
| 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 |
|
| Optimal Transport, Martingales and financial mathematics
Requirements: Functional Analysis, convex analysis, measure theory and stochastic process |
|
| Financial Mathematics
Requirements: Functional analysis, probability and stochastic process |
|
| Stochastic differential equations
Requirements: Stochastic processes; stochastic calculus |
|
| Varieties with trivial canonical bundle: quotients, fibrations and Hodge structures- part of the Prin project "Curves, Ricci flat varieties and their interactions"
Requirements: Basic notions in algebraic and complex geometry |
|
| Algebraic logic and duality theory, model-checking and decision procedures, nonstandard analysis and Ramsey theory
Requirements: Good general mathematical background |
|
| Algebraic Geometry; Derived categories of sheaves
Requirements: Algebraic geometry, scheme theory, complex manifolds |
|
| 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 |
|
| Ambiguity in financial mathematics and economics
Requirements: Functional analysis, measure theory and stochastic process |
M. Burzoni
|
| Categorical Algebra
Requirements: Basic knowledge of Category Theory, Universal and Homological Algebra |
|
| Analysis via Renormalization Group methods of Statistical Mecanics models, like spin dimer or vertex models and their non integrable extensione and proof of their universality properties. Many Body quantum systems with interaction, in particular Quantum Hall effect, Weyl semimetals and Graphene. Rigorous Renormalization for QFT. Prin Project Mathematical Quantum Matter (PI) and collaborations with two ERC projects, Roma3 e Sissa.
Requirements: Basic knowledge of mathematical physics, analysis and probability. |
|
| 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 |
|
| Normal form methods for singular perturbation problems - ERC project : Hamiltonian Dynamics, Normal Forms and Water Waves.
Requirements: Hamiltonian systems and basic normal form theory. Elementary knowledge of Fourier Analysis and partial differential equations |
|
| Stability of periodic multi-solitons and perturbations of nonlinear integrable systems - ERC project : Hamiltonian Dynamics, Normal Forms and Water Waves.
Requirements: Hamiltonian systems and basic normal form theory. Elementary knowledge of Fourier Analysis and partial differential equations. Elementary knowledge on the theory of integrable systems |
|
| Kam and normal form methods for pdeS in fluid dynamics - ERC project : Hamiltonian Dynamics, Normal Forms and Water Waves
Requirements: Hamiltonian systems and basic normal form theory. Elementary knowledge of Fourier Analysis and partial differential equations |
|
| Computational statistical and stochastic analysis in the framework of the modelling of phenomena such as the degradation of cultural heritage, energy markets and so on. The research line is of modellistic and applicative nature, connected to both the SEED-UNIMI SCICULT project and to a PON research project on Green themes, under the supervision of the proposers and to current research lines. Possible collaboration with researchers of the CNR, the university of Pisa and Karlsdadt (Sweden)
Requirements: Stochastic calculus, statistics, numerical methods for PDE |
|
| Analysis of stochastic and deterministic differential equations, boundary problems and interacting particle system for the modelling connected to problems of climate and environmental changes and the degradation of cultural heritage. The research line is of modellistic and analitic nature, connected to both the SEED-UNIMI SCICULT project and to a PON research project on Green themes, under the supervision of the proposers and to current research lines. Possible collaboration with researcher of the CNR, the university of Pisa and Karlsdadt (Sweden)
Requirements: Analutical skills, knowledge of deterministic differential equations, stochastic calculus |
|
| Motivic homotopy theory, motivic cohomology, motives, K-theory
Requirements: Algebraic geometry and topology, homotopical algebra, infinity categories |
|
| (Perturbation theory for) finite dimensional Hamiltonian systems and application to the dynamics of non linear lattices
Requirements: Knowledge of mathematical physics and basic elements of Hamiltonian dynamical systems |
|
| Mathematical Methods in Quantum Mechanics and in General Relativity; Evolution equations (especially, in fluid dynamics) . The proponent is financially supported by: 1) MUR, 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 |
|
| Epistemology of Mathematics
Requirements: Good knowledge of geometry, analysis,...and of the philosophical aspects of the theory of knowledge |
|
| Differential Geometry and Global Analysis
Requirements: Riemannian Geometry and PDE's |
|
| Study of p-adic L-functions and p-adic regulators with applications to the arithmetic of automorphic forms
Requirements: Knowledge of basic number theory and modular forms |
|
| Algebraic Geometry and Homological Algebra
Requirements: Solid background in algebraic geometry |
|
| Geometric Measure Theory and regularity of solutions to geometric variational problems
Requirements: Solid background in Mathematical Analysis and Measure Theory. Knowledge of the main techniques in elliptic PDE and Calculus of Variations. Geometric intuition |
|
| Fine properties and regularity of weak solutions to Mean Curvature Flows
Requirements: Solid background in Mathematical Analysis and Measure Theory. Basic knowledge of the geometry of hypersurfaces. Knowledge of the main techniques in parabolic PDE |
|
| Birational geometry of the moduli space of curves: MMP and Hassett-Keel program
Requirements: Good knowledge of algebraic geometry. Basic notions of moduli spaces. |
|
| Stochastic methods in quantum mechanics: stochastic description of the quantum phenomenon of Bose-Einstein condensation. Scaling, convergence and an approach via stochastic optimal control. Research line is carried out in collaboration with the University of Bonn (HCM)
Requirements: Stochastic calculus, mathematical analysis |
Sergio Albeverio
|
| Study of invariance and symmetry properties of stochastic dynamical systems. The research is carried out in collaboration with the University of Bonn (HCM)
Requirements: Stochastic calculus, mathematical analysis. |
Sergio Albeverio
|
| Non-local Problems and Free boundary problems
Requirements: Advance skills in mathematical analysis |
Enrico Valdinoci
|
| Nonlocal minimal surfaces
Requirements: Knowledge of the basics of analysis and geometry. Geometric intuition and knowledge of partial differential equations. |
Enrico Valdinoci
|
| Phase coexistence problems
Requirements: Knowledge of the basics of analysis and mathematical physics, with emphasis in partial differential equations. |
Enrico Valdinoci
|
| Algebraic geometry and Hodge theory (PRIN)
Requirements: Basic knowledge of algebraic and complex geometry |
|
| Foundations of adaptive methods for the numerical solution of differential equations. Involved projects: PRIN 2017, Numerical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of complex systems governed by Partial Differential Equations
Requirements: Sound knowledge of conforming and non-conforming Galerkin methods, in particular finite element methods; basic knowledge of nonlinear approximation. |
|
| 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
November 2022
| Courses or activities | Professor(s) | ECTS | Total hours | Language |
|---|---|---|---|---|
| Optional | ||||
| An Introduction to Kam and Normal Form Methods for Pdes | 4 | 20 | English | |
| Some Contributions of Epistemology and Logic to Research in Didactics of Mathematics | 3 | 18 | English | |
| The Geometry of Einstein-Type Structures and Phi-Curvatures On Riemannian Manifolds (module I/II) | 6 | 30 | Italian, English | |
December 2022
| Courses or activities | Professor(s) | ECTS | Total hours | Language |
|---|---|---|---|---|
| Optional | ||||
| Mckean-Vlasov Stochastic Differential Equations | 3 | 15 | Italian | |
January 2023
| Courses or activities | Professor(s) | ECTS | Total hours | Language |
|---|---|---|---|---|
| Optional | ||||
| Particle Systems and Scaling Limits in Kinetic Theory | 3 | 18 | Italian, English | |
February 2023
| Courses or activities | Professor(s) | ECTS | Total hours | Language |
|---|---|---|---|---|
| Optional | ||||
| An Introduction to Mean Curvature Flow and Its Regularity Theory | 4 | 20 | English | |
May 2023
| Courses or activities | Professor(s) | ECTS | Total hours | Language |
|---|---|---|---|---|
| Optional | ||||
| Mathematics for Statistical Physics and Quantum Field Theory | 5 | 25 | Italian | |
Following the programme of study
Contacts
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