Mathematics

The use of computers is increasingly pervasive in all areas of mathematics. This course introduces the foundational concepts of programming and some of the many computational tools in common use by mathematicians.

This course provides an overview of the theory of and practically useful methods in computer vision, with applications within e.g. vision systems, non-invasive measurements, and augmented reality. Participants develop problem solving skills, with and without a computer, using mathematical tools taken from many areas of the mathematical sciences, in particular geometry, matrix theory, algebraic geometry, optimization, mathematical statistics, invariant theory and transform theory. Course topics include projective geometry, geometric transformations, modelling of cameras. camera calibration. epipolar geometry. stereo vision, photogrammetry. model fitting., robust metrics, minimal solvers, 3D-modelling. geometry of surfaces and their silhouettes. deformable models, and visualization. Assumed prior knowledge: FMAF05 Systems and Transforms, or equivalent (for example FMAF10).

Linear Algebra is a field of mathematics that focuses on vectors and matrices, covering fundamental concepts and theories of linear spaces such as systems of linear equations, vector spaces, matrix operations, eigenvalues, and eigenvectors. It serves as an essential tool in economics and applied statistics. Specifically, linear algebra provides the foundation for optimization theory, which is widely used in economics. In statistics, it plays a key role in estimating regression coefficients. Furthermore, matrix factorization is frequently applied in big data analysis and machine learning algorithms. The objective of this course is to develop a solid understanding of the core concepts and theories of vectors and matrices, and to cultivate the ability to apply them to a variety of problems.

This course explores both theoretical and practical aspects of cryptography, authentication, and information security. Students learn the relevant mathematical techniques associated with cryptography, the principles of cryptographic techniques and how to perform implementations of selected algorithms in this area, and explore the application of security techniques in solving real-life security problems in practical systems.

This interactive course revolves around problem-solving and both playing and analyzing mathematical games. Throughout the course, students are given a sequence of carefully selected problems to solve individually. Students are given ample time to think about the questions - with hints from the instructors as needed - and they receive follow-up problems after each problem solved. Later in the course, the solutions and key learnings are presented by the instructors. Topics include mathematical games, graph theory, chapters from combinatorics, information theory, and invariants.

This is a first course in combinatorics and graph theory and covers graphs, Euler's V-E+F=2 Theorem, Kuratowski's Theorem, counting sets, generating functions, matching, Hall's Marriage Theorem, Polya counting, and counting paths in graphs.