Mathematics

This course covers the following topics: Calculus of variations: functional, variation, extremals, Euler-Lagrange equation, Beltrami identity, brachistochrone, catenary, natural boundary conditions, isoperimetric constraints; Conservation laws: transport equation, Burgers' equation, method of characteristics, weak solutions, Rankine-Hugoniot jump condition, Oleinik entropy condition; and Wave equation: Spherical means, Euler-Poisson-Darboux equation, Kirchhoff's formula in 3 dimensions, Poisson's formula in 2 dimensions, energy method, finite speed of propagation, domain of dependence.

This course introduces approximation techniques in numerical analysis and scientific computing. It examines numerical methods for solving nonlinear equations, systems of linear and nonlinear equations, and related problems. Techniques for interpolation, approximation, and numerical differentiation and integration are studied.

This course begins with an understanding of the concept of distance in mathematics, and introduces the fundamental concepts and various properties of topological structures. It provides a theoretical foundation for topics previously used in calculus, including the real number system, limits, continuous functions, the extreme value theorem, the mean value theorem, the existence of definite integrals, the fundamental theorem of calculus, and the intermediate value theorem. In addition, the course explores the geometric properties of topological concepts. Topics include Set Theory and Logic, Topological Spaces and Continuous Functions, Connectedness and Compactness, Countability and Separation Axioms, The Tychonoff Theorem, Metrization Theorems and Paracompactness, and Complete Metric Spaces and Function Spaces.

This course examines ways of solving the (usually partial) differential equations that arise in physical, biological, and engineering applications. Many of the methods covered, such as Fourier Transforms, also have applications beyond the solution of differential equations.

This course provides an introductory overview of probability theory, presented in a mathematically rigorous manner. Starting from the definitions of events, random variables, independence, and expectation, we also cover some basic applications such as weak convergence, the law of large numbers, characteristic functions, the central limit theorem, etc. 

Prerequisites: Elementary level of calculus (required), analysis (required), and linear algebra (optional). 

This course introduces the basic results and techniques of linear programming (LP) and its related topics in operations research. There is an equal emphasis on all three aspects of understanding, algorithms and applications. The course serves, together with a course on network models, as essential concept and background for more advanced studies in operations research. The topics include the simplex method, the dual simplex method, parametric programming, decomposition methods and interior point methods.

This course provides essential mathematical tools for students of Biology, reinforcing key concepts and introducing methods for modeling biological systems. Topics include matrix algebra, systems of equations, real functions, calculus (limits, derivatives, and integrals), and an introduction to ordinary differential equations. Emphasis is placed on practical applications relevant to biological phenomena and experimental sciences.

This course provides a solid foundation in essential mathematical concepts for students in economics and management. It ensures a consistent level of mathematical proficiency to prepare for further study of advanced quantitative techniques. The course covers linear functions, absolute values, square roots and inverses, second-degree polynomials, and derivatives.

Starting from the concept of limit of a sequence, learn how it is possible to give a precise meaning to the concepts of "infinite", "infinitely small", and "infinitely large". Students also learn how to work with series, and understand how these tools can be applied to define limits of functions. Students see some of the concepts that they have covered in school in a new light, and analyze them in great depth. Students learn how to give rigorous proofs of mathematical statements, and how to sketch the graph of a function.