Mathematics

This course is divided into two parts: Linear algebra and optimization of several variable functions. Part one offers a study of the properties of matrices and determinants and systems of linear equations. Part two discusses the concept of continuity and calculus of several variables (partial derivatives), among other topics. Students solve linear systems of equations, describe the properties of functions, and approximate a function of several variables. They address economic problems by means of abstract models, interpret and classify different solutions, and apply appropriate conclusions to social contexts.

This is the first semester in a mainstream calculus sequence. It covers limits of one variable functions; continuity and differentiability; implicit differentiation; differentiation of trigonometric, exponential, and logarithmic functions; higher derivatives; applications of derivatives: related rates, linear approximations, the Mean Value Theorem, l’Hopital’s Rule, maxima and minima, curve sketching and optimization problems; definite and indefinite integrals, Riemann Sums, the Fundamental Theorem of Calculus.

This course provides a comprehensive introduction to Mathematical Logic, serving as a toolkit for clear thinking and a foundation for advanced studies in philosophy and sciences. We will begin with the basics of argumentation and traditional syllogistic logic, then move to the core of modern symbolic logic: Propositional Logic and First-Order Predicate Logic. Students will learn how to analyze the syntax and semantics of these systems and construct formal proofs. The course also covers essential mathematical tools such as basic Set Theory and mathematical induction. Furthermore, we will explore the frontiers of logic, including an introduction to Type Theory, the connection between logic and computation, and the relation of logic with computation and modern Artificial Intelligence.

This course develops both theoretical understanding and, above all, practical skills in ordinary differential equations (ODEs). It begins with a review of the concept of integration as the area under a curve and introduces numerical methods for computing integrals using a computer. Fundamental results on the existence and uniqueness of solutions to ODEs are presented, along with several explicit solution methods. The course also includes elements of qualitative analysis of differential equations, such as barrier methods, phase portraits, and stability analysis. Hands-on computer sessions focus on the implementation of numerical methods in Python for solving ODEs. At the end of the semester, a project allows students to apply these tools to concrete problems arising in various scientific fields, including physics, biology, chemistry, and meteorology.

The course extends the methods of calculus of one variable to calculus for functions of many variables, that is, calculus on higher dimensional spaces. This involves concepts such as multiple integrals and partial derivatives, which enable us to make sense of the idea of length of a curve, area of a surface, and maxima and minima of functions of many variables. The final part of the course presents the great integral theorems: Green’s Theorem, Stokes’ theorem and the Divergence Theorem which form a cornerstone of mathematics. 

The course covers Number theory including the fundamental theorem of arithmetic, and modular arithmetic; Group definition, basic examples of groups, subgroups, normal subgroups, factor groups, isomorphisms and homomorphisms, Lagrange's theorem, permutation groups, symmetric and alternating groups, finitely generated Abelian groups; Ring definition, basic examples of rings, isomorphisms and homomorphisms, ideals, factor rings, polynomial rings, factorization of polynomials as products of irreducible polynomials; and Field characteristic, simple field extensions, and finite fields. For admission to the course, English 6 is required and at least 60 credits in Natural Science or Engineering studies, of which at least 30 credits in mathematics, including the courses MATA32 Algebra and Vector Geometry, 7.5 credits and MATB32 Linear algebra, 7.5 credits or equivalent courses.

This is a course in abstract algebra, although connections with other fields are stressed as often as possible. It covers some of the jewels in the crown of undergraduate mathematics, drawing together groups, rings, and fields to solve problems that resisted the efforts of mathematicians for many centuries. The powerful central ideas of this course are now crucial to many modern problems in algebra, differential equations, geometry, number theory, and topology. 

This course is structured into four hands-on modules. Three modules are dedicated to using and developing different types of models typically used by Earth Scientists. Many models rely on solving Partial Differential Equations (PDE). They either represent an efficient simplification thereof, e.g., in the form of box modelling (module 2 or directly write and solve (un)coupled sets of PDE’s (module 3). A new branch of modelling constitutes data-driven modelling, which exploits artificial intelligence to build models based on observations (module 2). These first three modules integrate the learning of more advanced programming with acquiring the basics of different methods used for modelling. The last module focuses on the process of modelling itself, such that you can translate Earth Science problems into effective modelling strategies. Together these modules complete the programming, modelling and data skill learning lines. Participants develop and write their own box model with a complete numerical model, which quantifies to what depth the water level would fall if the Mediterranean Sea was disconnected from the Atlantic Ocean. The course teaches the development and uses data-driven models. Learn the basics of different machine learning models, including how neural networks, random forests, and long short-term memory algorithms work. Train models using hydrological data and optimize their performance on different data sets. Finally, you build a machine learning model from scratch to simulate a new hydrological data set. Participants also develop or program various codes from scratch, starting from physical conservation laws to writing codes to simulate 1D and 2D flow; heat, chemical and pressure diffusion; and coupled momentum and continuity equations with variable viscosity. Developments are based on a staggered grid, finite difference formulation, which facilitates understanding of numerical mathematics for Earth Science students. Through gradually increasing code complexity students are introduced to spatial and temporal discretization, initial and boundary conditions, and basic numerical solvers. Finally, learnings is applied to perform a numerical modelling research project. The thermomechanical code used for this module is an extension of the building blocks you have programmed yourself in the PDE-module. You can choose to apply this code to simulate subduction over millions of years to understand earthquake sizes, flow of a mountain glacier, or tsunami propagation over a lake. Students are tasked to formulate a research question in one of these Earth Science domains based on literature and develop and execute a modelling strategy to answer it. Execute your research project step-by-step accumulating into a poster presentation. Assumed previous knowledge includes GEO1-1135, GEO2-1230 and GEO2-1301. 

This course introduces the science of strategic decision-making. Students learn to identify the key elements of any strategic situation: who the players are, what actions they can take, what information they have, and how they value different outcomes. Building on these foundations, students explore core concepts such as Nash equilibrium, the power of commitment, and the principles of effective mechanism design, all with minimal mathematical prerequisites but strong emphasis on conceptual insight.

The course moves from basic models to more sophisticated scenarios, each grounded in real-world applications. It examines how telecoms spectrum auctions raised billions, why Brexit negotiations unfolded as they did, how matching algorithms assign doctors to hospitals and support kidney exchanges, and how AI agents learn to interact strategically. Course outcomes include thinking more clearly and systematically about strategic situations in any context: equipping students with frameworks that are valuable across economics, business, politics, and technology.

Machine learning tools are widely used across various aspects of contemporary society. A solid understanding of the foundations that support these tools is essential for developing new methods and assessing existing ones. This course introduces the mathematical and statistical theory behind modern machine learning methods. The course explores the why and how of machine learning methods from a theoretical perspective considering generalization, regularization, and optimization. Key techniques including kernel-based methods, tree-based methods, and neural networks are discussed, along with recent developments.