Mathematics

1. The Ph.D. program in Mathematics at the University Roma Tre started 16 years ago in 2000. We had 15 starting classes (in Italy we call a class a cycle) involving altogether about 80 students. At the present time 13 cycles have been completed and we graduated 50 Ph.D. students. About 70% of our Ph.D. graduates have, at the moment, a research position in research institutes or universities in Italy or abroad (about 50% abroad) 2. The Roma Tre Ph. D. program in Mathematics aims at training students towards a research activity in pure or applied Mathematics at high international standards. Therefore, a Ph.D. graduate in Mathematics from Roma Tre is in a very good position to find high level employments in either the academic world (Universities and research center either domestic or foreign) or companies of the private sector carrying out advanced research projects. 3. Main Research Fields: 3.1 Commutative and non-commutative Algebra: Rings of Kronecker functions, Nagata rings, multiplicative system of ideals in commutative rings, Gabriel Popescu localizing system and associated (semi)star operations; representation theory, infinite-dimensional Lie algebras, conformal and vertex algebras. 3.2 Logic: mathematical logic, proof theory, computational content of mathematical proofs, linear logic. 3.3 Algebraic geometry and differential geometry: moduli spaces of curves and algebraic varieties, deformations theory, higher dimensional algebraic varieties, classification of algebraic varieties. Self-dual 4 manifolds, twistor theory; hermitian geometry of complex surfaces. 3.4 Analytic number theory: Artin L-series, distribution of primitive roots, elliptic curves. 3.5 Mathematical Analysis and Dynamical Systems: Differential equations with Hamiltonian structure and small divisors problems (classical Hamiltonian systems and Celestial Mechanics, partial differential equations with Hamiltonian structure, extension of Aubry-Mather theory). Nonlinear eigenvalue problems with singular nonlinearities: existences, unicity and compactness; asymptotic analysis and construction of blow¬up solutions for two dimensional elliptic equations arising from Gauge theory; asymptotic analysis for singular perturbation problems or Sobolev with critical growth. Special functions and inequalities. 3.6 Probability: Stochastic evolution for system of interaction particles with emphasis on relaxation time. Optimization problem in a random environment, Markov chain Monte Carlo algorithm for computational complex problem, metastability and estimates for large deviations, random walks on random graphs. 3.7 Mathematical Physics: Random Walks in time - fluctuating random environments; Anderson's parabolic model for almost stationary environments. 3.8 Numerical Analysis: Level set methods: schemes for mean curvature motion, convergence, fast marching implementation (non iterative). Large time-steps schemes, semi-Lagrangian and Lagrange-Galerkin schemes, linear diffusion-transport problems and conservation laws with viscosity terms. 3.9 Applied Mathematics and Scientific Computing: Probabilistic method (and Probabilistic domain decompositions as well) for numerically solving: boundary problems for elliptic equations, initial boundary problem for parabolic equations and furthermore application to certain class of nonlinear partial differential equations (such as KPP, Navier-Stokes, Vlasov-Poisson) . 3.10 Theoretical Computer Science: Cryptography, Computer Security, Distributed Computation, Computational Number Theory, Pisot Numbers and Applications, Computational Methods in Systems Biology. 4. Organization of the Ph.D. program: During the first and a half year, Ph.D. students are asked to attend four Ph.D. courses. The courses can be chosen not only in the Department of Mathematics and Physics of Roma Tre but also in the other universities of Rome, or even in other universities upon approval of the Director of graduate studies. The students are also encouraged to follow joint activities co-organized by the three universities of Rome (Roma Tre, La Sapienza and Tor Vergata Universities). After the second year, students have to choose an advisor (which can be freely chosen among the faculties of the three universities of Rome) and are expected to finish the program within the third year. Participation to schools , workshops and conferences are encouraged (starting from the second year) and partly financed.