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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 maximalfunktionen, BMO-rummet (rummet av funktioner av begränsad medeloscillation) och Carlesons inbäddningssats, Muckenhoupt-vikter, grundläggande operatorer inom harmonisk analys såsom singulära integraloperatorer (särskilt Hilbert-transformen) och kvadratfunktionen. Kursen ger en introduktion till moderna diskretiseringstekniker för harmonisk analys, till exempel wavelets (i syThe course treats the maximal function, the space BMO (the space of functions of bounded mean oscillation) and the Carleson embedding theorem, Muckenhoupt weights, fundamental operators of harmonic analysis such as singular integral operators (in particular the Hilbert transform) and the square function. The course gives an introduction to modern discretisation techniques of harmonic analysis

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.

Kursen syftar till att ge fördjupade kunskaper i fysiken kring acceleratorer och erfarenhet om beräkningar och modellering av acceleratorsystem. Dessutom syftar kursen till en grundläggande förståelse av frielektronlasern och dess acceleratorsystem.  The aim of the course is to give deepened knowledge in the physics of accelerators and experience in both calculations and modeling of accelerator systems. In addition the course aims at providing a fundamental understanding of the Free Electron Laser and its accelerator systems.

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 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 beskriver olika typer av acceleratorer (cyklotroner, mikrotroner, linac, linjär acceleratorer, lagringsringar, synkrotroner) i olika applikationer (synkrotronljus, högenergifysik, colliders, medicin, tumörterapi, industri). Vidare behandlas elektronstråleoptik och partikelstråleoptik och dynamik (magneter, dipoler, quadrupoler, fokusering, matrisformalism, betatronsvängningar, betatrontonerThis course describes different types of accelerators (cyclotrons, microtrons, linacs, linear accelerators, storage rings, synchrotrons) used in various applications (synchrotron radiation, high energy physics, colliders, medicine, tumor therapy, industry). The description of electron beam optics, particle beam optics and dynamics (magnets, dipoles, quadrupoles, focusing, matrix formulation, betat

Kursen behandlar synkrotronljuskällors egenskaper och deras användning inom modern forskning. Uppbyggnaden av strålrör med de speciella optiska komponenter som behövs för fokusering, avbildning eller diffraktion är en central del i kursen. En översikt av experimentella tekniker och metoder för spektroskopi, strukturbestämning, avbildning, mikroskopi, tomografi och så vidare ingår också. KurseThis course is about the properties and use of synchrotron radiation in modern science. The design and construction of the optical elements required for focusing, imaging and diffraction constitute a central part of this course. An overview of experimental techniques and methods used in spectroscopy, structure determination, imaging, microscopy, and tomography is also part of the curriculum. Furth

A particular focus is put on energy transitions, including both societal and environmental dimensions. In this context we will discuss for example energy justice dimensions and spatial aspects (land demand) in the context of the transition. Further the course will examine the present debate regarding energy demand, energy efficiency and energy end-use technology status and potential, as well as iA particular focus is put on energy transitions, including both societal and environmental dimensions. In this context we will discuss for example energy justice dimensions and spatial aspects (land demand) in the context of the transition. Further the course will examine the present debate regarding energy demand, energy efficiency and energy end-use technology status and potential, as well as i

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

By highlighting four key emerging and persistent global health challenges: The unavoidable health impacts of climate change (heat-stress, extreme cold, flooding, rising incidence of vector borne-diseases) The prevailing sanitation crises (causing preventable deaths and disease outbreaks, contributing to the poverty trap, affecting economic output, polluting the environment) The global dietBy highlighting four key emerging and persistent global health challenges: The unavoidable health impacts of climate change (heat-stress, extreme cold, flooding, rising incidence of vector borne-diseases) The prevailing sanitation crises (causing preventable deaths and disease outbreaks, contributing to the poverty trap, affecting economic output, polluting the environment) The global diet

Kursen behandlar distributionsteorins grunder, testfunktioner, distributionsbegreppet, distributioner med kompakt stöd, operationer på distributioner, faltning, homogena distributioner och Fouriertransformen.The course treats the foundations of distribution theory test functions, the concept of a distribution, distributions with compact support, operations on distributions, convolution, homogeneous distributions and the Fourier transform.

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 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 ger en introduktion till användningen av synkrotronljus och till användningen av högbriljant vakuumultraviolett- och röntgenstrålning. Kursen ger en beskrivning av uppbyggnaden av strålrör och en orientering om hur olika experiment utnyttjar en facilitet. En kort genomgång av synkrotronljusets betydelse för forskning inom fysik, kemi och biologi ges. Några exempel på metodik som fotonelektrThis course gives an introduction to the use of synchrotron radiation with different wavelengths. The design of beam lines and an introduction to different experiments at a synchrotron radiation facility is part of the course. The importance of synchrotron radiation for research in the fields of physics, chemistry and biology is discussed and examples are given of various experimental techniques (

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

As a result, the notion of inner transformation (or inner transition) has emerged as a new area of exploration. The course explores this new area and creates space and opportunities for learning and knowledge development on this topic. Inner transformation, as used here, describes changes in the sphere of human interiority related to people’s (expanded) consciousness, mindsets, values, and beAs a result, the notion of inner transformation (or inner transition) has emerged as a new area of exploration. The course explores this new area and creates space and opportunities for learning and knowledge development on this topic. Inner transformation, as used here, describes changes in the sphere of human interiority related to people’s (expanded) consciousness, mindsets, values, and be

The course deals with how theories about the resilience concept have developed within sustainability science. The course starts with a historical overview of the resilience concept within development- and sustainability science. The course then goes on to critically examine resilience as a concept and as an approach. Resilience has come to mean many things. There is for example, a resistance amonThe course deals with how theories about the resilience concept have developed within sustainability science. The course starts with a historical overview of the resilience concept within development- and sustainability science. The course then goes on to critically examine resilience as a concept and as an approach. Resilience has come to mean many things. There is for example, a resistance amon