Mathematics

This course provides students with tools for handling high-dimensional statistical problems. The course contains models and methods with practical applications, mainly for spatial statistics and image analysis. Of special importance are the Bayesian aspects, since they form the foundation for many modern spatial statistical and image analysis methods. The course emphasizes methods with applications in climate, environmental statistics, and remote sensing. The following topics are covered: Bayesian methods for stochastic modelling, classification, and reconstruction; random fields, Gaussian random fields, Kriging, Markov fields, Gaussian Markov random fields, non-Gaussian observationer; covariance functions, multivariate techniques; simulation methods for stochastic inference (Gibbs sampling); applications in climate, environmental statistics, remote sensing, and spatial statistics.

This course on complex analysis covers the following: theory of functions of complex variables, analytic/holomorphic functions, Cauchy integral theorem and formula, null spaces, singularities, analytic continuity, Riemann mapping theorem.

This course uses mathematics to discuss the global environment. Basic fluid dynamics and simple physics for the atmosphere and oceans are used to discuss some of the mechanisms involved in the dispersion of pollutants along coasts and the four-yearly (on average) El Nino oscillation in the equatorial Pacific, with its attendant Australia drought and blight of the Peruvian anchovy industry. Typical analysis involves the solution of linear partial differential equations for the velocity and density of the flows.

This is a graduate level course that is part of the Laurea Magistrale program in Mathematics. Enrollment is by consent of the instructor. The course is intended for students who have a strong background in math, especially calculus. The course focuses on the fundamental aspects of scientific calculus. Emphasis is placed on numerical solutions to problems of scientific calculus. A section of the course is devoted to teaching scientific calculus as described by the Italian Educational Mathematics Curriculum in Secondary Schools. This section emphasizes how to face and solve mathematical problems within a uniform, integrated computer algebra environment. Students are also required to prepare lesson plans and exercises. A special section of the course is devoted to the presentation of numerical methods for solving, via computer, certain mathematical problems, and the comparison with corresponding computer algebra methods, wherever appropriate. The course covers the use of computer tools in Math courses in the Secondary School system in Italy. Required readings are assigned from the following sources, available in the Departmental library: NUMERICAL COMPUTING WITH IEEE FLOATING POINT ARITHMETIC by M. Overton, ACCURACY AND STABILITY OF NUMERICAL ALGORITHMS by N. Higham, NUMERICAL ANALYSIS: MATHEMATICS OF SCIENTIFIC COMPUTING by D. Kincaid, E. W. Cheney, NUMERICAL LINEAR ALGEBRA, by D. Bau, N. Trefethen, AFTERNOTES ON NUMERICAL ANALYSIS by G. W. Stewart. Assessment is based on a laboratory test and a short written report and discussion of course materials.

This course provides a study of the theory of functions of one variable and focuses on the properties of monotonicity, continuity, derivability, and concavity/convexity of functions.

This course offers a study of sets, functions, mappings, and relations; groups, subgroups, cyclic groups, and cosets; homomorphism and isomorphism; group actions on sets; methods of proof; complex numbers; divisibility theory for integers and modular arithmetic; divisibility theory for polynomials; rings, ideals, and quotient rings; and fields and construction of fields from polynomial rings.

This course provides an introduction to probability theory aimed at third-year mathematics majors. Topics include: probability space; random variables: distribution, distribution functions, discrete random variables, moments of discrete random variables; random vectors; sequences of random variables; law of large numbers and the central limit theorem. Recommended prerequisites: introductory differential calculus, introductory integral calculus, multivariable differential calculus, multivariable integral calculus.