| TITLE | REQUIREMENTS | PROFESSOR/S |
| Space-time stochastic processes, Stochastic geometry and statistical shape analysis: point processes, random sets, random measures | Measure theory; Probability and Mathematical Statistics | E. Villa; A. Micheletti |
| Biomathematics and Biostatistics | Probability, Mathematical Statistics. Partial differential equations, analytical and numerical aspects. Differential Modelling. | C. Cavaterra; A. Micheletti |
| Categorical algebra | Basic knowledge of Category Theory, Universal and Homological Algebra | S. Mantovani; A. Montoli |
| Stochastic methods in quantum mechanics | Stochastic calculus and analytical skills | S. Albeverio; S. Ugolini |
| Invariance properties in stochastic dynamics | Stochastic calculus and analytical skills | S. Albeverio; S. Ugolini |
| Stochastic partial differential equations and quantum field theory | Stochastic calculus and analytical skills | S. Albeverio |
| Foundations of adaptive methods for the solution of differential equations | Sound knowledge of Galerkin methods with conforming and nonconforming spaces, basic knowledge of nonlinear approximation | A. Veeser |
| Numerical Galerkin methods for partial differential equations | Theory and practice of finite element methods, numerical linear algebra | A. Veeser; C. Lovadina |
| Algebraic Geometry and Homological Algebra | Slid background in algebraic geometry | P. Stellari |
| p-adic modular forms and L-functions, algebraic cycles, motives and their realizations | Theory of schemes, number theory and homological algebra | F. Andreatta |
| Mathematical logic, algebraic logic, duality theory, model-checking and decision procedures. | Good general mathematical background | S. Ghilardi; V. Marra |
| Isogeometric Analysis and Virtual Element Method; Numerical methods for partial differential equations; Biomathematics | Numerical Methods for PDEs | C. Lovadina; S. Scacchi |
| Non-local Problems and Free boundary problems | Advance skills in mathematical analysis | E. Valdinoci |
| Nonlocal minimal surfaces | Knowledge of the basics of analysis and geometry. Geometric intuition and knowledge of partial differential equations | E. Valdinoci |
| Phase coexistence problems | Knowledge of the basics of analysis and mathematical physics, with emphasis in partial differential equations. | E. Valdinoci |
| Evolution systems of PDE | Real analysis, functional analysis | E. Bonetti; C. Cavaterra |
| Mathematical models for applications | Real analysis, functional analysis | E. Bonetti; C. Cavaterra |
| Inverse problems | Real analysis, functional analysis | E. Bonetti; C. Cavaterra |
| Differential Geometry and Global Analysis | Riemannian Geometry and PDE’s | M. Rigoli; P. Mastrolia |
| Epistemology of Mathematics | Good knowledge of geometry, analysis,…and of the philosophical aspects of the theory of knowledge | M. Rigoli |
| Mathematical Physics for quantum and classical statistical mechanics and quantum field theory | Knowledge of mathematical physics, analytical skills | V. Mastropietro |
| Mathematical Methods in Quantum Mechanics and in General Relativity; Evolution equations (especially, in fluid dynamics) | Basic knowledge of functional analysis and quantum mechanics; Basic knowledge of differential geometry and general relativity | L. Pizzocchero |
| Non linear Analysis, nonlinear partial differential equations | Basic knowledge of Functional analysis, PDEs and Sobolev spaces | B. Ruf |
| Algebraic geometry and Hodge theory, Moduli spaces of curves and Geometry of Calabi-Yau varieties | Basic knowledge of algebraic and complex geometry | L. van Geemen |
| Non linear Dynamics | Elementary techniques of dynamic systems | D. Bambusi |
| KAM and normal form theory for PDEs | Basic elements of Hamiltonian systems | D. Bambusi |
| Group Theory and Representation Theory | Basics in Algebra and Group Theory | M. Bianchi |
| Geometric properties of solutions to partial differential equation | Knowledge of the basics of analysis and geometry, with emphasis in partial differential equations and basics of functional analysis. | G. Ciraolo |
| Finite dimensional Hamiltonian dynamics: from nonlinear chains to celestial mechanics | Knowledge of mathematical physics and basic elements of Hamiltonian dynamical systems | M. Sansottera; T. Penati; S. Paleari |
| Financial Mathematics | Functional analysis, probability and stochastic processes | M. Frittelli |
| Ambiguity modelling in mathematical finance | Functional analysis, measure theory, stochastic calculus | M. Maggis; M. Burzoni |
| Martingale Optimal Transport and Financial mathematics | Functional analysis, convex analysis, measure theory, stochastic calculus | M. Frittelli |
| Functional analysis and infinite-dimensional convexity | Real analysis, Elements of Functional analysis | L. Vesely |
| kam and normal form methods for pdeS in fluid dynamics | Hamiltonian systems and basic normal form theory. Elementary knowledge of Fourier Analysis and partial differential equations | R. Montalto |
| normal form methods for singular perturbation problems | Hamiltonian systems and basic normal form theory. Elementary knowledge of Fourier Analysis and partial differential equations | R. Montalto |
| stability of periodic multi-solitons and perturbations of nonlinear integrable systems | Hamiltonian systems and basic normal form theory. Elementary knowledge of Fourier Analysis and partial differential equations. Elementary knowledge on the theory of integrable systems | R. Montalto |
| Stochastic optimal control, backward stochastic differential equations and control of systems of McKean-Vlasov type. | Stochastic processes; stochastic calculus. | L. Campi; M. Fuhrman |
| Stochastic differential equations. | Stochastic processes; stochastic calculus. | M. Fuhrman |
| Algebraic geometry: geometry, automorphisms and constructions of varieties with trivial canonical bundles and with elliptic fibrations | Basic knowledge of algebraic and complex geometry | A. Garbagnati |
| Computational topology for machine learning | Real and functional analysis; topology; Statistics; neural networks | A. Micheletti |
| Stochastic differential games and mean field games with applications | Stochastic processes, stochastic calculus | L. Campi |
| Algebraic Geometry: projective models, automorphism groups and moduli spaces of Hyperkähler manifolds and irreducible symplectic varieties. | Good knowledge of algebraic geometry ad of complex geometry | C. Camere |