Mathematics

This course is an introduction to the theory and application of linear algebra. Students learn to draw diagrams that illustrate the basic operations of vector algebra in two and three dimensions, recognize the equations of lines and planes in two and three dimensions, perform the matrix computations outlined in the following, solve a linear system using Gaussian elimination, compute the inverse of an invertible matrix, find the orthogonal projection of a vector onto a hyperplane, compute the determinant of a square matrix, compute the characteristic equation of a matrix, find the eigenvectors corresponding to a given eigenvalue, and to diagonalize a diagonalizable matrix.

The course presents the fundamental statistical methods for extreme value analysis, discusses examples of applications regarding floods, storm damage, human life expectancy, and corrosion, provide practical use of the models, and points to some open problems and possible developments. Extreme value theory concerns mathematical modelling of random extreme events. Recent development has introduced mathematical models for extreme values and statistical methods for them. Extreme values are of interest in economics, safety and reliability, insurance mathematics, hydrology, meteorology, environmental sciences, and oceanography, as well as branches in statistics such as sequential analysis and robust statistics. The theory is used for flood monitoring, construction of oil rigs, and calculation of insurance premiums for re-insurance of storm damage. Often extreme values can lead to very large consequences, both financial and in the loss of life and property. At the same time the experience of really extreme events is always very limited. Extreme value statistics is therefore forced to difficult and uncertain extrapolations, but is, none the less, necessary in order to use available experience in order to solve important problems.

This course presents elementary linear analysis in a concrete setting, emphasizing specific techniques important to analysis and its applications. Topics include normed linear spaces; Banach spaces; Hilbert spaces; convexity; orthogonal expansions and their applications; and Riesz representation theorem for functionals.

Topics in this course include: bilinear mappings; complex differential functions; power series; complex contour integrals; Cauchy's theorem and integral formula; Taylor's theorem; zeros of analytic functions and Rouche's theorem; maximum modulus principle; singularities and Laurent series; poles and residues; and residue calculus.

This course provides an introduction to calculus. Emphasis is on an understanding of the basic concepts and techniques, and on developing the practical, computational skills to solve problems from a wide range of application areas. This course illustrates the methods learned by looking at real problems from different fields where these techniques can be applied and through this applied lens all students explore new facets of calculus and deepen their knowledge. The course discusses: functions, limits, and continuity; derivatives; rules of differentiation; maxima and minima; implicit differentiation and rates Integration; definite integrals; applications of integration; improper integrals; differential equations. Prerequisites for this course include substantial high school experience in Mathematics.

This course discusses different concepts of numerical analysis including algorithms, stability, accuracy, and efficiency. Topics include: errors, algorithms, and estimates; nonlinear equations and nonlinear systems; methods for linear systems of equations; polynomial interpolation-- Lagrange, Hermite, piecewise, and splines; numerical quadrature and differentiation.

Prerequisites: Linear Algebra, Differential Calculus, Integral Calculus, and Programming.  

This course is an introduction the theory of metric spaces. The real line with its natural notion of distance is a metric space, from which the metric space definition and theory evolves. Familiar concepts such as convergence and continuity are explored in this new broader context while new concepts and properties, such as closed sets and compactness, illuminate key facts about functions.

In this course, students learn how to understand, analyze, and resolve complex problems using a range of quantitative techniques. Emphasis is placed on the importance of understanding the science and techniqies behind these ideas in engaging with the modern world. Students tackle estimation problems, learn coding with Python, and explore statistics and game theory.