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Kursen behandlar: Lineära system av ordinära differentialekvationer i det komplexa planet. Potensserielösningar. Singulära punkter. Frobenius metod. Några klassiska differentialekvationer och speciella funktioner. Randvärdesproblem. Egenfunktionsutvecklingar. Spektralsatsen. Sturm-Liouville-teori. 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 problems. Eigenfunction expansions. Spectral theorem. Sturm-Liouville theory.

Kursen behandlar Differentierbara mångfalder, deras tangentrum och tangentknippen. Riemannska metriker och deras unika Levi-Civita-förbindelse. Geodeter och den viktiga Riemannska krökningsten sorn samt dess betydelse för den lokala geometrin. 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 geometry.

Kursen behandlar: • homotopiteori, fundamentalgruppen, övertäckningsrum • Brouwer's fixpunktssats, Borsuk-Ulams sats • deformationsretraktion, fundamentalgrupper och ytors homologi • mångfalders kirurgi, också kallad "klippa och klistra" • konstruktion och klassificering av kompakta ytor.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 "cutting and pasting" • the construction and classification of compact surfaces.

Kursen behandlar: Grupprepresentationer: linjära representationer, ekvivalenta representationer, regulära representationer, inducerade och begränsade representationer, irreducibla representationer, Maschkes sats, dualrepresentationer och tensorprodukt av representationer. Karaktärteori: karaktärer av representationer, inducerade karaktärer, irreducerbara karaktärer, ortogonalitetsrelationer,The course treats: Group representations: linear representations, equivalent representations, regular representation, induced and restricted representations, irreducible representations, Maschke’s Theorem, dual representations and tensor product of representations. Character theory: characters of representations, induced characters, irreducible characters, orthogonality relations, integralit

Kursen behandlar: Grupper: Konjugatklasser. Burnsides lemma med tillämpning på Polyaräkning.Sylows satser. Strukturen hos ändligt genererade abelska grupper. Ringar: Noetherska och Artinska ringar och moduler. Artin-Wedderburns sats. Ändligt genererade moduler över en huvudidealring med tillämpning på Jordans normalform. Lineär algebra: Multilineära avbildningar. Tensorproduct 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 Artinian rings and modules. Artin-Wedderburn's theorem. Finitely generated modules over a principal ideal domain with application to the Jordan's normal form of matri

Kursen behandlar grundläggande egenskaper hos Banach- och Hilbertrum och begränsade lineära operatorer definierade på sådana rum: Banachrum, Hahn-Banachs sats, svag konvergens och svag prekompakthet av enhetsklotet. Hilbertrum. Exempel inklusive L2-rum. Ortogonalitet, ortogonalt komplement, slutna underrum, projektionssatsen. Riesz representationsteorem. Ortonormala mängder, Pythagoras satThe 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 spaces. Examples including L2 spaces. Orthogonality, orthogonal complement, closed subspaces, projection theorem. Riesz Representation Theorem. Orthonormal set

Kursen behandlar: Karakteristikmetoden och icke-linjära ekvationer av första ordningen. Laplaces ekvation. Värmeledningsekvationen. Vågekvationen. Cauchy-Kowalevskis sats. Sobolevrum. Existens, entydighet och regularitet för svaga lösningar till linjära elliptiska,paraboliska och hyperboliska ekvationer av andra ordningen. Maximalprinciper för elliptiska och paraboliska ekvationer. 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 regularity for weak solutions to linear second order elliptic, parabolic and hyperbolic equations. Maximum principles for elliptic and parabolic equations.

Kursen behandlar grundläggande egenskaper hos mått med tecken och komplexamått: Hahn- och Jordanuppdelning, absolut kontinuitet och Radon-Nikodyms teorem, singularitet, Lebesgue-uppdelning av mått, deriverbarhet av ändliga Borel-mått på R^d, deriverbarhet av absolutkontinuerliga funktioner, Hardy-Littlewoods maximalfunktion och uppskattningar för densamma, Hardy-Littlewoods maximalfunkThe 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 measures on Rd, differentiability of absolutely continuous functions, the Hardy-Littlewood maximal function and the weak type estimate for it, the Hardy-Lit

Kursen behandlar tillämpningar av: Hahn-Banachs sats, svag konvergens och kompakthet, Riesz representationssats, Ortonormala baser, Integraloperatorers begränsning, kompakthet och spektra, Spektralsatsen för kompakta, självadjungerade operatorer. 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 spectral theorem for compact, self-adjoint operators.

The thesis should be a synthesis and further development of the knowledge and competences gained during the three terms of course work in LUMES on top of undergraduate studies. The thesis work should develop the student’s potential for professional sustainability work and/or for further research. The process puts great demands on the capacity to wThe thesis should be a synthesis and further development of the knowledge and competences gained during the three terms of course work in LUMES on top of undergraduate studies. The thesis work should develop the student’s potential for professional sustainability work and/or for further research. The process puts great demands on the capacity to w

I kursen diskuteras olika politiska ansatser och förhållningssätt till hållbarhetsfrågor i relation till den faktiska förmågan att hantera hållbarhetsutmaningar. Kursen tar upp centrala politiska faktorer samt hur vi kan förstå och skapa alternativa vägar till förändring. I kursen behandlas framför allt: Begreppet politik och dess historik och beståndsdelar. Aktuella politiska former och utmaniThe course addresses political approaches to and perspectives on sustainability issues and place these in relation to the actual ability to address sustainability challenges. It tackles both central political factors as well the creation of alternative pathways towards change. The main focuses of the course are: the concept of politics, its history and components; current political forms and ch

Kursen erbjuder tvärvetenskapliga perspektiv från miljögeografi för att studera sociala och miljömässiga processer i specifika geografiska sammanhang. Perspektiv från en rad olika forskningsfält, såsom markförändringsvetenskap, landskapsgeografi och politisk ekologi, behandlas. Kursen syftar till att fördjupa studentens förståelse för och förmåga att utforska de sociala, ekonomiska och miljömThe course provides interdisciplinary perspectives from the field of environmental geography aimed at studying the dynamics of social and environmental change in particular places. It covers perspectives from a variety of research fields, such as land-change science, landscape geography, and political ecology. It aims to advance students’ understanding of the spatial dimensions of social, economic

Kursen ger en översikt över ekonomiska teorier och ekonomisk tillväxt och analyserar deras miljömässiga och sociala konsekvenser, så som degradering av miljön eller inverkan på social välfärd och jämlikhet. Studenter utvecklar förmågan att kritiskt utvärdera implikationer från olika ekonomiska paradigm för transformativ förändring samt att kunna avväga deras respektive för- och nackdelar. Under kuThe course briefly reviews economic theories and economic growth and analyses their environmental and social consequences, for instance, environmental degradation, social welfare and equity. Students will develop competencies in critically assessing the implications of different economic paradigms for transformative change and weighing the trade-offs for each. During the course, students will: C

Kursen ger deltagarna möjlighet att förbättra sina kunskaper och färdigheter inom hållbarhetsvetenskap genom en djupare förståelse av styrkorna och svagheterna med deltagande metoder. Kursen bygger på transformativt studentlärande där studenterna utvecklar kunskaper och färdigheter i problemlösningsprocesser för hållbarhet genom iterativa aktiviteter som idéskapande, reflektion, teoribildning, intThis 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 transformative student learning whereby students develop knowledge and skills in sustainability problem solving processes through iterative activities such as idea generati

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 ecology of water resource challenges, water-related vulnerability and adaptation measures, and uncertainties in climate modelling coupled to freshwater resourcesThe 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 ecology of water resource challenges, water-related vulnerability and adaptation measures, and uncertainties in climate modelling coupled to freshwater resources

The main aim of the course is to provide conceptual frameworks for studying and evaluating the impact that social movements for sustainability can have on the public agenda and decision-making. The course is a seminar course, although it starts with a few lectures on the history and different theoretical perspectives on social movements, introducing the students to the relevant literature. TThe main aim of the course is to provide conceptual frameworks for studying and evaluating the impact that social movements for sustainability can have on the public agenda and decision-making. The course is a seminar course, although it starts with a few lectures on the history and different theoretical perspectives on social movements, introducing the students to the relevant literature. T

In the course we will discuss the broad field of gender studies and the central issues, key concepts and analytical tools in the current debates regarding inequalities, diversities and social change relevant for sustainability. The course draws on global, regional and local empirical examples to illustrate issues of how gender and sustainability interact. We will analyse similarities, differencesIn the course we will discuss the broad field of gender studies and the central issues, key concepts and analytical tools in the current debates regarding inequalities, diversities and social change relevant for sustainability. The course draws on global, regional and local empirical examples to illustrate issues of how gender and sustainability interact. We will analyse similarities, differences

The course also provides an understanding of and an ability to analyse and understand political questions and decision making in relation to climate mitigation and climate adaptation. Based on this, the course aims to provide a deepened understanding of the most important scientific questions and how these are reflected and dealt with in the international climate policy debate. The course also incThe course also provides an understanding of and an ability to analyse and understand political questions and decision making in relation to climate mitigation and climate adaptation. Based on this, the course aims to provide a deepened understanding of the most important scientific questions and how these are reflected and dealt with in the international climate policy debate. The course also inc