Mathematics

The main objective of this course is the acquisition of language and basic probabilistic and statistical concepts, and their uses in computer science. The student must apply mathematical skills, logic, and reasoning. This course deals with notions of probability, statistics, and combinatorics. The teaching focuses on discrete probabilities (event concept, probabilistic calculation rules, conditional probabilities, random variables, expectation, variance, probability laws: uniform, binomial, geometric and Poisson). Practical manipulations (simulation and implementation) deepen the theoretical notions of the course. The link with certain continuous laws is approached (normal law, exponential). On the combinatorial aspects, the notions of elementary counts (i.e., sums, products, combinations), recurrences and generating function are approached and applied to the counting of words, sequences and discrete structures usual in computer science. Descriptive statistics (i.e., series, histograms, mean, standard deviation) and an introduction to hypothesis testing are also discussed.

This course introduces fundamental concepts and analytical tools in mathematics. This course contains basic ideas and knowledge in number theory, calculus, analysis, linear algebra, differential equation and probability. Part of the results are given a rigorous proof to show what real mathematics is. To take this course, students are only assumed to have a good command of knowledge of elementary mathematics included in high school and minimum knowledge of undergraduate mathematics.

This course explores topics in linear algebra and multi-variable calculus, including linear transformations, quadratic forms, double and triple integrals, and constrained optimization. The course is a preparation for advanced courses in mathematics, statistics, physics, and other applied sciences.

This course explores continuity, differentiation, and integration of complex valued functions, with applications to real integration.

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.