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A central part of the course is existence- and uniqueness theorems about measures defined on sigma-algebras, integration theory, conditional expectation and weak convergence in metric spaces.

The course treats: Convergence of sequences of analytic functions. Locally uniform convergence, normal families, Montel’s theorem. In-depth study of conformal mappings. Riemann sphere, stereographic projection, Möbius…

The course treats the geometry of curves and surfaces, especially in three dimensions. In particular, we study the concepts of curvature and torsion. The course covers: The geometry of curves in Euclidean space, their curvature and…

The course treats: Finitely generated groups: free groups, Nielsen-Schreier Theorem, group presentations, finitely presented groups. Free products: free products with amalgamation, HNN extensions. Solvable groups: polycyclic groups,…

The course treats: topological spaces and metric spaces with examples, product topologies, continuity of functions, connectedness, completeness, and compactness, including the Arzela-Ascoli Theorem, examples of applications and topological…

The course treats: Fourier series, Fourier transform and finite Fourier transform, L^2 convergence of Fourier series, pointwise convergence, Cesàro means and Fejer’s theorem, Weyl’s criterion, the Fourier inversion theorem, Parseval’s and…

This course treats the general notion of a measure defined on a sigma-algebra, construction of measures with help of outer measures, in particular the Lebesgue measure in Rd. These concepts are then used to define the integral of a…

The course treats: Linear systems of ordinary differential equations in the complex plane. Power series solutions. Singular points. Frobenius method. Some classical differential equations and special functions. Boundary value…

The course covers: Differentiable manifolds, their tangent spaces and tangent bundles. Riemannian metrics and their unique Levi-Civita connection. Geodesics and the important Riemann curvature tensor and its influence on the local…

The course treats: • homotopy theory, the fundamental group, covering spaces • the Brouwer fixed point theorem, the Borsuk-Ulam theorem • deformation retracts, fundamental groups and homology of surfaces • surgery of manifolds, also called…

The course treats: Group representations: linear representations, equivalent representations, regular representation, induced and restricted representations, irreducible representations, Maschke’s Theorem, dual representations and…

The course treats: Groups: Permutation groups. Burnside's lemma with application to Pólya arithmetic. Sylow's theorems. Symmetric and alternating groups. The structure of finitely generated Abelian groups. Rings: Noetherian and…

The course treats fundamental properties of Banach and Hilbert spaces and the bounded linear operators defined on them: Banach spaces, the Hahn-Banach Theorem, weak convergence and  weak precompactness of the unit ball. Hilbert…

The course treats: The method of characteristics and nonlinear equations of the first order. Laplace’s equation. The heat equation. The wave equation. The Cauchy–Kowalevski theorem. Sobolev spaces. Existence, uniqueness and…

The course treats basic properties of signed and complex measures: Hahn and Jordan decomposition, absolute continuity and the Radon-Nikodym theorem, singularity, Lebesgue decomposition of measures, differentiability of finite Borel…

The course treats applications of the Hahn-Banach theorem, weak convergence and compactness, the Riesz representation theorem, the use of orthonormal bases, boundedness, compactness and spectra of integral operators, the…

This course provides opportunities for participants to enhance their knowledge and skills in sustainability science through a deeper comprehension of the strengths and weaknesses of participatory methods. The course is based on…

The themes are covered significantly through a variety of scientific readings and reflection that exemplify different approaches and perspectives for analysing and interpreting present and future water resource availability, the political…