Mathematics is the oldest science, and still one of the most important and widely applicable. The mathematical tradition at the Scuola Normale is quite strong, and our PhD program prepares high-level researchers, who routinely go on to teach and do research in very prestigious institutions, in Italy and abroad. Although our program is mostly in pure mathematics, the preparation that one obtains can easily be put to use in a vast range of fields with high scientific and technological content.
One of the characteristics of the Scuola Normale is the constant flow of visitors and post-docs from all over the world; this, together with the financial support offered to our students for visits to research institutes and universities of very high standing, greatly enlarges their research horizons. Also, its limited size favors interaction among students in different disciplines, resulting in a remarkable cultural and personal enrichment.
The mathematical research at the Scuola Normale is concentrated in some of the most significant areas of mathematics at the international level.
• In calculus of variations, our main activity is in geometric measure theory, even in metric measure spaces, elliptic and parabolic partial differential equations, optimal mass transport and its applications.
• In harmonic analysis the main themes are invariant operators on Lie Groups, singular integrals, Fourier multipliers, and Radon transforms. Methods of commutative and noncommutative harmonic analysis are also applied to complex analysis.
• In differential geometry and global analysis we treat non-linear problems in riemannian geometry and geometric evolutions, riemannian manifolds with special structures, such as solitons of the Ricci flow or quasi-Einstein manifolds, calibrated geometries and varieties with special holonomy, and the geometry of submanifolds.
• In probability and stochastic analysis we investigate stochastic partial differential equations, particle systems and scaling limits, Kolmogorov and Fokker-Planck equations, applications in turbulence and geophysics.
• In algebraic geometry we do research on algebraic stacks, essential dimension, actions of algebraic groups and their invariants, derived categories and toric varieties.
• In number theory the most studied aspects are the applications of geometric methods to the problems of integral points, with particular stress on the effectivity of solutions, heights, and intersection problems in algebraic groups.
• In dynamical systems we concentrate on problems of stability of quasi-periodic orbits, on KAM theory, on holomorphic dynamics, on the ergodic theory of continued fractions and on Teichmüller flows.
• In numerical analysis and scientific computing our main research interests are in numerical linear algebra, especially sparse matrix algorithms, iterative methods, preconditioning, and functions of matrices. Our activities range from theoretical foundations to algorithm development with applications to fluid mechanics, network science and quantum chemistry.
Teaching is divided into lecture and seminar courses (all held in English). Graduate students in Mathematics will annually agree with the PhD Coordinator a study plan to be presented to the Faculty Board. Such a document will specify the planned research and education activities for the relevant academic year. The courses will be chosen to enlarge the student background and deepen specific aspects related to the PhD Thesis project. PhD students are expected to take at least three courses and to pass the corresponding exams. Students may be asked to follow some course from the Undergraduate program, to fill some gaps in their preparation and these may or may not be in addition to the above three compulsory courses, on a case by case basis.
Students must pass the exams at the end of the first, second and third year, before a commission appointed by the Faculty Council.
At the end of the first year students are expected, in close consultation with the Coordinator and with approval from the Faculty Council, to choose the Thesis supervisor and a preliminay project.
At the end of the second year students are expected to have a precise work plan, while at the end of the third year the results must have taken on an almost definitive form. During the fourth year the research must be completed and condensed into a thesis.
According to research needs, students are encouraged to spend periods of study and research at Italian and foreign institutions.