# The School's curricula

The PhD Programme in Mathematics lasts three years and is articulated into five curricula:

**a. General Mathematics**

This area focuses on one or more of the following research themes:

**Calculus of Variations**: analysis in metric spaces, geometric measure theory, variational convergences (Gamma-convergence), optimal transport.**Geometric Analysis, Riemannian Geometry, Geometrical Flows.****Nonlinear Partial Differential Equations (PDE)**: free boundary problems, models of hysteresis, asymptotic behavior and PDE homogenization, variational and topological methods, nonlinear equations of Ginzburg-Landau and Schrödinger.**Analytic Geometry and Algebraic Geometry**: Algebraic curves and moduli spaces. Surfaces of general type and moduli spaces. Manifolds of high dimensions: Mori theory, Fano varieties. Real algebraic geometry, complex and hypercomplex analysis. History of algebraic geometry. Mathematical visualization.**Mathematical physics**: foundational, analytical and geometrical aspects of classical, quantum and relativistic theories**Dynamical Systems and Control Theory**: existence, multiplicity, stability of periodic solutions of differential equations, Lagrangian and Hamiltonian systems, differential games and optimal control problems, viscosity solutions of Hamilton-Jacobi equations; hybrid system optimizations.**Stochastic processes**: stochastic partial differential equations, functional integration and applications.**Mathematical Statistics and Data Science:**it will cover classical techniques of statistical inference, both under a frequentist and Bayesian paradigm, as well as modern techniques for complex and high-dimensional data. In particular, it will include topics from multivariate statistics, such as graphical models and their link to network science, robust statistics and statistical data depth.**Mathematical Logic and Theoretical Computer Science**: applications of non standard techniques (à la A. Robinson) in functional analysis, non classical logics, programming languages theory, type systems, static analysis, general and philosophical aspects, foundations, constructive mathematics and Hilbert’s program.**Group theory**, in particular permutation groups and finite p-groups, Lie groups and algebras, computational methods and applications in theoretical physics. Commutative and computational algebra, monomial algebras and associated combinatorial structures. Algorithms for algebraic and combinatorial invariants. Coding theory and cryptography. Tensor decomposition, secant varieties, algorithms and applications to complexity theory, quantum information, and data analysis. Representations of algebras, homological algebra.

**b. Mathematical Modelling and Scientific Computing (MOMACS)**

This area crosses the following research themes:

**Stochastic Processes**: integral-differential equations and stochastic partial differential equations for the modeling of physical, biological, and financial phenomena.**Numerical Methods for Partial Differential Equations**: modeling of electromagnetic phenomena and (classic and quantum) fluid dynamics, approximation methods based on finite elements, boundary elements, differences or finite volumes.**Approximation/numerical interpolation of multivariate functions**: efficient methods and applications.**Discrete Mathematics**: modeling in operations research, graph theory, combinatorial optimization, and applications to computational biology.**Optimal Control, Optimization**: applications to decision science, image processing, cultural heritage.**Mathematical and computational models in medicine**: simulation of physiological and pathological mechanisms in the human organism, with special focus on the circulatory and lymphatic system and their interactions with the central nervous system.

**c. Cryptography and coding theory**

This area focuses on several mathematical methods used in cryptography and in the theory of error correcting codes. More specifically, research is performed in the following topics:

**Algebraic methods**: linear algebra, commutative algebra, algebraic combinatorics, computational algebra, Gröbner bases, number fields, group theory.**Geometric methods**: algebraic geometry, elliptic curves.**Cryptographic protocols**: design and formal proofs.

The proposed research problems range from purely theoretical classifications to problems close to industrial research. Industrial research can also be integrated with internships at leading companies within the field.

**d. Mathematical and computational biology**

This area focuses on the vast field of applications of **mathematical and computational models in biology**.

**Mathematical and computational models in medicine**: simulation of physiological and pathological mechanisms in the human organism, with special focus on the circulatory and lymphatic system and their interactions with the central nervous system.

**e. Mathematical applications to Quantum Science and Technologies**

This area focuses on theoretical and applied research on topics related to quantum physics in a broad sense (**quantum mechanics, quantum information, quantum field theory**) both from foundational-theoretical and applied points of view, involving advanced mathematical techniques in **algebra**, **analysis**, **geometry**, **mathematical physics**, **computer science**, and **probability**. Doctoral students can collaborate with the cross-disciplinary doctoral project **“Quantum Science and Technology”** within the Q@TN consortium.