Mathematics

This introductory course covers mathematical topics closely related to computer science. Topics include: logic, sets, functions, relations, countability, combinatorics, proof techniques, mathematical induction, recursion, recurrence relations, graph theory, and number theory. The course emphasizes the context and applications of these concepts within computer science. Prerequisites: No prior programming experience is assumed. 

This course provides a basic introduction to mathematical theory and methods in biology, with enough scope to enable the student to handle biologically phrased problems. Topics covered include population models with discrete or continuous time, pharmacokinetics and -dynamics, qualitative analysis of systems of differential equations, modelling of the spread of infectious diseases, bifurcations, limit cycles, and excitable media with applications to, e.g., predator-prey models, spatial methods with application to diffusion, and nerve conduction.

The course covers properties of the real numbers R: completeness axiom, Cauchy sequences, cardinality of rational, and irrational numbers; Topology in Rn: open and closed sets, p-norms, convergence, compactness, the Bolzano-Weierstrass theorem, and connected sets; Continuous functions in Rn: intermediate value theorem, min-max theorem, uniform continuity, continuity of inverse functions, implicit function theorem; Convergence of sequences and series of functions: pointwise, absolute, and uniform convergence, term wise differentiation and integration, power series;  and examples of applications to selected topics relevant to mathematical research at the center for mathematical sciences. Admission to the course requires at least 30 credits in mathematics including knowledge corresponding to MATA31 Analysis in One Variable, 15 credits, MATA32 Algebra and Vector Geometry, 7.5 credits and NUMA01 Computational Programming with Python, 7.5 credits.

In this course, students learn about vectors spaces, subspaces, bases, inner products, linear transformations, rank/nullity, matrices of linear maps, change of basis, eigenvalues/eigenvectors, Jordan normal form, diagonalization, and special classes of linear transformations and their matrices.

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.