Scientific research at the Department of Mathematics Guido Castelnuovo covers all major fields of mathematics. The Eccellenza project aims to develop three main areas, in synergy with all the scientific activity of the department.
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Hyperkähler manifolds (HK), which, in dimension 2, are K3 surfaces, play a fundamental role in Geometry. In dimensions greater than 2, some classes of deformations of HK manifolds are known, and there is a theory that reproduces and extends the theory developed for K3 surfaces, but the current state of knowledge is very lacking. This project focuses on several research directions which include the study of the Chow ring of HK manifolds, in particular the conjectures formulated by Beauville and Voisin, the study of Lagrangian submanifolds of HK manifolds, and the use of deformation theory techniques to study sheaves on HK manifolds and non-commutative K3 surfaces.
The analysis of the regularity and of the singularities in geometric variational problems is a central topic in mathematical analysis, as already seen in the study of minimal surfaces, harmonic maps and free boundary problems. The aim of the project is to develop analytical tools for understanding the corresponding partial differential equations, which are naturally related to fundamental problems in geometry and mathematical physics. Particular attention in the context of the project is given to the following directions: minimal surfaces and geometric measure theory, harmonic maps theory for liquid crystals, isoperimetric problems and shape optimization, prescribed curvature problems and related geometric flows, evolution and formation of interfaces in phase separation models.
Random effects in statistical physics have become experimentally accessible and a central point of mathematical modelling. For many models, conjectures and predictions can be formulated through numerical simulations and approximations, but a mathematically satisfactory analysis is not yet available. This project focuses on several research directions, including fluctuations in kinetic models, random growth models, quantum-mechanical transport, non dissipative transport in topological materials.
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