Mathematics

The course covers the translation between biology and mathematics; population models and spatial models, simulations: Deterministic versus stochastic simulations of mathematical models; weaknesses, strengths, and applicability; the Gillespie algorithm for stochastic simulations: Naive implementation and possible optimizations for large systems; cost functions; optimization methods including local optimization, thermodynamic methods, particle-swarm optimization, and genetic algorithms; and sensitivity analysis: Estimation of the uncertainty of determined parameter values. Strategies to achieve robustness. Admission to the course requires 90 credits Science studies, including knowledge equivalent to BERN01 Modelling in Computational Science, 7.5 credits or FYTN03 Computational physics, 7.5 credits and English 6/B. Admission to the course also requires knowledge in programming in Python equivalent to NUMA01, 7.5 credits or similar knowledge in Matlab, C++ or the like programming language.

The course is an introduction to vector calculus and a specialization of differential and integral calculus of functions of several variables. The course covers line and surface integrals; Green's formula, Gauss divergence theorem, and Stokes theorem; Basic potential theory. To be eligible for the course, 45 credits in courses in mathematics equivalent to MATA21 Analysis in One Variable (15 credits), MATA22 Linear Algebra 1 (7.5 credits), MATA21 Analysis in Several Variables 1 (7.5 credits), MATB22 Linear Algebra 2 (7.5 credits) and one of the courses NUMA01 Computational Programming with Python (7.5 credits) and MATA23 Foundations of Algebra, (7.5 credits) are required.
 

This course extends the statistical ideas introduced in the first year to more complex settings. Mathematically, the central concept is the linear model, a framework for statistical modelling that accommodates multiple predictor variables, continuous and categorial, in a unified way. There is a focus on fitting models to real data from a variety of problem domains, using R to perform computations. 

This advanced topics course covers reinforcement learning, search, and test-time scaling of large language models that are expected to drive the next generation of AI systems. 

Topics include: Basics of RL (Markov Decision Process and Policy evaluation), Basics RL (Imitation learning, Deep policy gradient methods), Basics of RL (Deep Q-Learning, Rainbow DQN); Symmetric alternating Markov games, Monte Carlo tree search, expert iteration, and AlphaGo; Imperfect information games, Counerfactural regret minimization, and Pluribus; NLP basics (RNN, beam search, tokenizers); NLP basics (Transformers, encoder-decoder architectures); Instruction fine-tuning, Scaling laws of LLM pre-training; Reinforcement learning with human feedback, direct policy optimization, Group Relative Policy Optimization (GRPO); Chain of thought, Process reward models, Prover-verifier games; In-context learning, Scaling LLM Test-Time Compute; DeepSeek-R1. 

This course discusses differential geometry of curves and surfaces in Euclidian Space: curves in 2- and 3-dimensional spaces, local and global theory of surfaces, special classes of surfaces, discrete curves and surfaces.

This course explores the classical theory of games involving concepts of dominance, best response, and equilibria, where it proves Nash’s Theorem on the existence of equilibria in games. Students learn the concept of when a game is termed zero-sum and prove the related Von Neumann’s Minimax Theorem. The course explores cooperation in games and investigates the interesting Nash bargaining solution which arises from reasonable bargaining axioms. Students also explore the concept of a congestion game, often applied to situations involving traffic flow, where they see the counterintuitive Braess paradox emerge and prove Nash’s theorem in another context. 

This course looks at definition of a curve, arc length, curvature, and torsion of a curve, Frenet-Serret equations. It also looks at definition of a surface patch, first and second fundamental forms, isometries, conformal maps, area, Gaussian curvature, mean curvature, principal curvatures, Gauss map, geodesics, and Theorema Egregium.

Quantitative finance remains one of the fastest growing areas in modern finance. Alternative names are financial engineering, mathematical finance, or financial mathematics. This is an application-based course on the mathematical and computational aspects of derivative pricing. It lies at the heart of mathematics, computing, finance, and economics. Both theory and numerical techniques are presented, with computer simulations performed on MS Excel. If you are interested in technical finance and have wondered what Brownian Motion is, or how Monte Carlo methods are used to price options; then this module is precisely what you are looking for – covering Itô Calculus, Black-Scholes world and Monte Carlo simulations. This is not a theorem-proof based course, but all results are derived.

This course provide students with basic training on modern financial mathematics methods, which covers an overview of data analysis, principles of actuarial modelling and financial transactions, the understanding of real and nominal interest rates, the time value of money methods, bond pricing methods, assets replication methods, the equation of value methods, and project appraisals methods. This course focuses on applying the above methods to the mathematical modelling of financial markets.

This course examines the theory of systems of ordinary differential equations. The emphasis will not be on finding explicit solutions, but instead on the qualitative features of these systems, such as stability, instability and oscillatory behavior. The applications are from biology, physics, chemistry, and engineering, including population dynamics, epidemics, chemical reactions, and simple mechanical systems.