Computational Methods and Mathematical Models for Sciences and Finance

*The course is offered in the framework of the project ''Departments of Excellence''*

Research activity

Scientific computing and mathematical modeling (both deterministic and stochastic) are fundamental tools for the solution of problems arising in the study of complex systems, whether originating from the physical, chemical, or biological sciences, or of an economic and social nature. This new PhD program aims at providing a thorough computational  training with a strong mathematical content while giving  students the opportunity to specialize their studies in a specific scientifc area, to be chosen among the following ones:

• Applied and computational mathematics.
• Operations research.
• Quantitative finance.
• Computational chemistry.
• Computational physics.
• Computational biology.
• Computational mechanics and mechanobiology.

The research activities in computational methods and mathematical modeling at the Scuola Normale Superiore include the following fields:

• Fundamental research in Numerical Analysis and Scientific Computing, in particular, numerical linear algebra and numerical methods for the solution of partial differential equations (PDEs) and PDE-constrained optimization problems. Applications in fluid mechanics, biology, network science and data sciences.
• Mathematical analysis of optimization problems (optimal transport, shape optimization), imaging, mechanics, biomathematics, etc.
• Stochastic Processes and Mathematical Statistics, with applications to fluids, complex systems, and finance.
• Mathematical Models for the Sciences. Applications to biology and mechanics.
• Quantitative Finance. Portfolio optimization, mathematical finance, microstructure of financial markets and financial econometrics.
• Computational methods for Quantum Chemistry.
• Quantum Information Theory and Quantum Computing.
• Computational physics. Numerical simulation techniques in condensed matter physics, astrophysics and cosmology.
• Computational Biology. Bioinformatics.

Teaching activity

All teaching will be in the English language. Each first year PhD student is required to take at least 160 hours of course work. Of these, 120 hours will be devoted to fundamental notions of numerical analysis, optimization, differential and stochastic modeling, and scientific programming. The remaining 40 hours may be chosen from the course offerings on special topics in mathematics, computational chemistry, physics or biology, or quantitative finance, according to the student’s interests. The details of the proposed course of study will have to be submitted to the Faculty Board for approval.

Students may be asked to follow some course from the Undergraduate program, to fill any 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. At the end of the first year students are expected, in close consultation with the Coordinator and with approval from the Faculty Board, to choose the Thesis supervisor and project. At the end of the second and third years, PhD students should present a written report concerning the research done and the results achieved so far, together with any publications produced. The report will be discussed in an oral presentation in front of a panel of experts appointed by the Faculty Board. Upon successful performance, the student will be admitted to the subsequent year.