Annual report The PhD students of our program in Mathematic have an intense research activity on a large spectrum of themes which range from more abstract mathematical questions to topics in applied mathematics. Most of the topics treated by our students (and the results obtained) are of great interest for international mathematical community. Some Key Words to have an idea of the variety of topics studied by the students last year: Biological Evolution, Mutation Processes, Simulations, Replicator Dynamics Topological Insulators, Wannier Functions, Schroedinger operators with magnetic field Analysis and simulations of: theory of travelling waves and applications to biomedicine and geophysics Nonlinear PDE with non regular coefficients, Lorentz regularity Elliptic boundary valued problems Feynman-Kac formula, PDEs on networks Harmonic analysis, Functional analysis and Volterra equations. Percolation, conductivity in disordered systems at low temperatures Quantum field theory, quantum electrodynamics, semiclassical analysis Quantum vertex algebra, quantum group, R-matrix Anyons, Schrodinger Operators, Self-Adjoint Extensions, Spectral Theory, Von-Neuman Theory, quadratic Forms 3-manifolds, minimal entropy, growth Variational poisson cohomology, W-algebras Pre-Lie Algebras, deformation theory W-algebras, Vertex Algebras Numbers theory, Weber Class Number One Problem, Class Field Theory Operator algebras, conformal field theory, representation theory Liquid crystals, harmonic maps, calculus of variations, geometric analysis Irregular manifolds, abelian manifolds Differential Geometry Logic, Desctiptive Set Theory, Game Theory Gromov hyperbolic, Tits alternative, doubling, entropy, group of isometries Some more data on the activity of the students during 2023 Number of partecipation to schools or conferences: 77 Number of talks or posters at conferences: 13 Seminars at other Institutions: 23 Days spent abroad: 849 Number of publications/preprints of the current students: 21 For more details please visit the pages of the students here.