Mathematics

This course introduces mathematical and programming skills that are employed by researchers in the Molecular and Biophysical Life Sciences to analyze and integrate data and to understand the physics of living systems. The course is divided into two parts that run in parallel. The mathematics part of the course consists of nine lectures that cover: basic algebra, goniometry, differentiation and integration (including functions of multiple variables), limits, (partial) differential equations (first order and second order), Taylor expansion, basic probability theory and statistics and vectors (including dot product and cross product). Each lecture is followed by a supervised practical session. The programming part consists of six lectures that introduce the basics of programming by discussing the modulare structure of programs (modules, functions, loops), different data types and variables, as well as good practices. For some calculations of the mathematics part of the course it is explained how to perform those calculations using Python. After each lecture, students work individually on a series of practical coding assignments that familiarize them with the basics of programming in Python during supervised tutorials, where regular instruction and feedback is provided.

The course focuses on key algebraic structures including ring and polynomial theory, with a strong emphasis on mathematical proofs and applications of algorithms including Euclid's, Lagrange interpolation, RSA cryptography, and the Fast Fourier Transform.

This course covers the following topics and subtopics: reduction of endomorphisms, determinants, eigenvectors, and eigenvalues; characteristic polynomials and minimal polynomials; Cayley-Hamilton Theorem; diagonalization and trigonalization; Dunford and Gauss-Jordan Reductions; Hermitian and Euclidean spaces; bilinear forms; quadratic forms; self-adjoint; and orthogonal groups in 2 or 3 dimensions.

This course features a study of selected topics in Linear Algebra in continuation of Linear Algebra II: eigenvalues, eigenvectors, diagonalization and Jordan normal form. 

This is a special studies course involving an internship with a corporate, public, governmental, or private organization, arranged with the Study Center Director or Liaison Officer. Specific internships vary each term and are described on a special study project form for each student. A substantial paper or series of reports is required. Units vary depending on the contact hours and method of assessment. The internship may be taken during one or more terms but the units cannot exceed a total of 12.0 for the year.

Mathematics is at the same time a conceptual framework, a collection of proven theorems, and a toolbox. In this course, students encounter all three of these aspects by studying one of the central mathematical issues for applications in science and engineering. The general topic of the course is the solution of linear partial differential equations using the separation of variables, Fourier series, and Fourier transforms. The study involves both computational and rigorous mathematical aspects. While the actual computation of solutions is the main objective, students also learn the mathematical theorems establishing the validity and limitation of the different methods. Interested students are also offered the possibility to experiment with numerical approaches.  In addition to the contact hours, each student is expected to work nine hours a week on the course. This time should be devoted to reviewing the material of the preceding lecture; finishing the exercises started in the preceding problem session; preparing exercises to hand in; studying the corrections of the previously returned hand-in problems and making sure everything is clear. Entry Requirements: Calculus and Linear Algebra.

This course introduces exciting new developments in advanced mathematics. The barriers between fields are being broken, many new unexpected applications are continually found, and out of this cross-fertilization, new kinds of mathematics are born. Topics are subject to change but may include various new advances of pure mathematics and logic, computational science and numerical analysis, fluid mechanics and geophysics, wavelets and signal processing, cryptology, quantum computation, mathematical biology (including bioinformatics, proteomics and neuroscience), intelligence science, financial mathematics and mathematical economics, and probability theory with various applications.  

This course introduces concepts and theories of mathematical analysis. Topics include limits of continuous functions and differentiable series of functions, uniform convergence of series of functions, Arzela-Ascoli theorem, Weierstrass theorem, power series, analytic functions, trigonometric series, Fourier series, etc.