Mathematics

This course examines group theory and ring theory, with a view towards commutative algebra, algebraic number theory and representation theory.

This course offers students a grounding in the language of modern machine learning, with a focus on particular topics in linear algebra, differential calculus, probability, and statistics. Rather than focusing on theorems and their proofs, the course covers the key tools (and theorems) within the topic areas, and to illustrate these with exemplars drawn from machine learning. The course is delivered through a mixture of lectures and classes, and involves a mix of traditional lecture delivery, interactive notebooks, and problem sets.

This course provides an introduction to programming within the statistical package R. Various computer-intensive statistical algorithms are discussed and their implementation in R is investigated. Topics to include basic commands of R (including plotting graphics); data structures and data manipulation; writing functions and scripts; optimizing functions in R; and programming statistical techniques and interpreting the results (including bootstrap algorithms).

The course covers sufficient statistics, factorization criteria, exponential families, Rao-Blackwells theorem, ancillary statistics, Cramér-Rao's bound, Neyman-Pearson's lemma, permutation test, and connection between hypothesis testing and confidence intervals. Asymptotic methods: maximum likelihood estimation, profile, conditional and penalized likelihood as well as hypothesis testing with likelihood ratio-, Wald- and score-method. Bayesian inference: estimation, hypothesis testing, and confidence interval and the difference compared to frequentist interpretation.

This course examines analytical functions; cauchy-riemann equations; complex mappings; cauchy's integral formulas; morera's, liouville's & rouche's theorems; taylor & laurent series; analytic continuation, residues & applications to integration; and boundary-value problems.

Dynamic programming is a neat way of solving sequential decision optimization problems. Integer Programming provides a general method of solving problems with logical constraints. Game theory is concerned with mathematical modelling of behavior in competitive strategic situations in which the success of strategic choices of one individual (person, company, server, ...) depends on the choices of others. By the end of this course, students have gained: ability to formulate and solve a sequential decision optimization problem; ability to formulate and solve optimization problems with logical constraints; ability to find optimal and equilibrium strategies for zero- and nonzero-sum 2x2 matrix games; and mastery of the theory underlying the solution methods.

This course covers basic notions of information theory. Entropy as measure of uncertainty. Constrained optimization with Lagrange multipliers. Maximum entropy inference with constraints. Partition function, free energy as generating function. Collective behavior in spin systems: from independent voters to the tight-knit model (or Curie-Weiss ferromagnet); phase transitions and spontaneous symmetry breaking.  Distributions of functions of random variables using Kronecker delta.  Laplace's approximation for integrals. Bolzmann distribution and 1d Ising chain: exact calculation for free energy. Variational approximations and trial (factorized) distributions. Time permitting: multi-party voters, stochastic dynamics and Markov Chains, models on social networks, traffic flow and epidemic models.

What is a reasonable value for a derivative on the financial market? The course consists of two related parts. The first part looks at option theory in discrete time. The purpose is to introduce fundamental concepts of financial markets such as free of arbitrage and completeness as well as martingales and martingale measures. Tree structures to model time dynamics of stock prices and information flows are used. The second part studies models formulated in continuous time. The models used are formulated as stochastic differential equations (SDE:s). The theories behind Brownian motion, stochastic integrals, Ito-'s formula, measures changes, and numeraires are presented and applied to option theory both for the stock and the interest rate markets. Students derive e.g. the Black-Scholes formula and how to create a replicating portfolio for a derivative contract.

Linear algebra is the branch of mathematics that is primarily concerned with problems involving linearity of one kind or another. This is reflected by the three main themes around which this introductory course is centered. The first theme concerns how to solve a system of linear equations. For this problem, a complete solution procedure is developed which provides a way to deal with such problems systematically, regardless of the number of equations or the number of unknowns. The second theme addresses linear functions and mappings, which can be studied naturally from a geometric point of view. This involves geometric ‘primitives’ such as points, lines, and planes, and geometric ‘actions’ such as rotation, reflection, projection, and translation. One of the main tools of linear algebra is offered by matrices and vectors, for which a basic theory of matrix-vector computation is developed. This allows one to bring these two themes together in a common, exceptionally fruitful, framework. By introducing the notions of vector spaces, inner products, and orthogonality, a deeper understanding of the scope of these techniques is developed, opening up a large array of rather diverse application areas. The third theme shifts from the geometric point of view to the dynamic perspective, where the focus is on the effects of iteration (i.e., the repeated application of a linear mapping). This involves a basic theory of eigenvalues and eigenvectors. Examples and exercises are provided to clarify the issues and to develop practical computational skills. They also serve to demonstrate practical applications where the results of this course can be successfully employed. Prerequisites include Basic Mathematical Tools or substantial high school experience in Mathematics.