Functional Analysis (STÆ507M)

Banach spaces and Hilbert spaces and their main properties. Duals of Banach spaces. Convolutions. The Fourier transform on LBanach spaces and Hilbert spaces and their main properties. Duals of Banach spaces. Convolutions. The Fourier transform on L 1(R). The Plancherel theorem. Equicontinuity, the Arzelà-Ascoli theorem. The Stone-Weierstrass approximation theorem. Linear operators on Hilbert spaces, in particular compact operators. The spectral theorem. The Hahn-Banach theorem. The Baire category theorem. The uniform boundedness theorem, the open mapping theorem and the closed graph theorem.

Partial Differential Equations (STÆ505M)

The object of the course is to give a firm and rigorous foundation for more advanced studies in partial differential equations. Contents: first order equations; the Cauchy-Kowalevski theorem; techniques of analysis (Lebesgue-integral, convolutions, Fourier-transform); distributions; fundamental solutions; the Laplace operator; the heat operator.  The course is mainly intended for postgraduate students with a good background in analysis.

Distributions (STÆ523M)

Fundamentals of distribution theory with applications to partial differential equations

Subject matter: Test funcitons, distributions, differnetiation of distributions, convergence of sequences of distributions, Taylor expansions in several variables, localization, distributions with compact support, multiplication by functions, transpostition: pullback and push-forward of distributions, convolution of distributions, fundamental solutions, Fourier transformation, Fourier series, and fundamental solutions and Fourier transforms.

Numerical Linear Algebra (STÆ511M)

Iterative methods for linear systems of equations.  Decompositions of matrices: QR, Cholesky, Jordan, Schur, spectral and singular value decomposition (SVD) and their applications.  Discrete Fourier transform (DFT) and the fast Fourier transform (FFT).  Discrete cosine transform (DCT) in two-dimensions and its application for the compression of images (JPEG) and audio (MP3, AAC).  Sparse matrices and their representation.

Special emphasis will be on the application and implementation of the methods studied.

Practical Applied Mathematics (STÆ514M)

The main aim of this course is to introduce some techniques of applied mathematics and show how they can be applied to
practical problems. The course is offered for Master and PhD students in  engineering, science and mathematics. (It is also open for 3rd year  students of mathematics.)

Subject matter: Examples of mathematical models in engineering and physics and their theoretical and numerical solutions, calculus in Banach spaces  and Newton's method, Hilbert spaces, basic approximate methods of analysis, distributions and and some aspects of Fourier analysis.

Numerical Methods for Partial Differential Equations (STÆ537M)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Mathematical Colloquium (STÆ001M)

Current research in mathematics.

Final project (STÆ441L)

Geometry and differential equations (STÆ534M)

This course is an introduction to Lie group methods in differential equations. Lie groups, Lie algebras, symmetry groups of differential equations, symmetry groups and conservation laws (calculus of variations, variational symmetries, conservation laws, Noether’s theorem).

Thesis skills: project management, writing skills and presentation (VON001F)

Introduction to the scientific method. Ethics of science and within the university community.
The role of the student, advisors and external examiner. Effective and honest communications.
Conducting a literature review, using bibliographic databases and reference handling. Thesis structure, formulating research questions, writing and argumentation. How scientific writing differs from general purpose writing. Writing a MS study plan and proposal. Practical skills for presenting tables and figures, layout, fonts and colors. Presentation skills. Project management for a thesis, how to divide a large project into smaller tasks, setting a work plan and following a timeline. Life after graduate school and being employable.

Differential Geometry (STÆ519M)

Differentiable manifolds, tangent space, cotangent space, differentiation over manifolds, vector fields, differential forms and exterior derivative, partition of unity, integration over manifolds, Stoke’s theorem, elements of Riemannian geometry.

Mathematical Physics (EÐL612M)

Continuum mechanics: Stress and strain, equations of motion. Seismic waves. Maxwell's equations and electromagnetic waves. Plane waves, reflection and refraction. Distributions and Fourier transforms. Fundamental solutions of linear partial differential equation. Waves in homogeneous media. Huygens' principle and Ásgeirsson's mean value theorem. Dispersion, phase and group velocities, Kramers-Kronig equations. The method of stationary phase. Surface waves on liquids.

Complex Analysis II (STÆ606M)

The course is a continuation of Complex Analysis I (STÆ301G) and covers various topics in complex analysis of one variable and its connections to other branches of mathematics.

Topics: Approximation of analytic functions by polynomials and rational functions, Runge's theorem. Infinite series and infinite products of meromorphic functions. Existence theorems of Mittag-Leffler and Weierstrass. Elliptic functions. The Gamma function. Simply and multiply connected domains, and connections to topology. Normal families and Montel's theorem. Conformal mappings, the Riemann sphere, fractional linear transformations, Schwarz's lemma, analytic automorphisms and hyperbolic geometry in the unit disk. Riemann's mapping theorem; boundary continuity. Schwarz's reflection principle, the modular function, the Great Picard theorem. Analytic continuation and compact Riemann surfaces of analytic functions. Subharmonic and harmonic functions, the Dirichlet problem and its solution using Perron's method. Hilbert spaces of analytic functions and the Bergman kernel.

Mathematical Colloquium (STÆ002M)

Current research in mathematics.

Final project (STÆ441L)

Functional Analysis (STÆ507M)

Banach spaces and Hilbert spaces and their main properties. Duals of Banach spaces. Convolutions. The Fourier transform on LBanach spaces and Hilbert spaces and their main properties. Duals of Banach spaces. Convolutions. The Fourier transform on L 1(R). The Plancherel theorem. Equicontinuity, the Arzelà-Ascoli theorem. The Stone-Weierstrass approximation theorem. Linear operators on Hilbert spaces, in particular compact operators. The spectral theorem. The Hahn-Banach theorem. The Baire category theorem. The uniform boundedness theorem, the open mapping theorem and the closed graph theorem.

Partial Differential Equations (STÆ505M)

The object of the course is to give a firm and rigorous foundation for more advanced studies in partial differential equations. Contents: first order equations; the Cauchy-Kowalevski theorem; techniques of analysis (Lebesgue-integral, convolutions, Fourier-transform); distributions; fundamental solutions; the Laplace operator; the heat operator.  The course is mainly intended for postgraduate students with a good background in analysis.

Practical Applied Mathematics (STÆ514M)

The main aim of this course is to introduce some techniques of applied mathematics and show how they can be applied to
practical problems. The course is offered for Master and PhD students in  engineering, science and mathematics. (It is also open for 3rd year  students of mathematics.)

Subject matter: Examples of mathematical models in engineering and physics and their theoretical and numerical solutions, calculus in Banach spaces  and Newton's method, Hilbert spaces, basic approximate methods of analysis, distributions and and some aspects of Fourier analysis.

Numerical Methods for Partial Differential Equations (STÆ537M)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Numerical Methods for Partial Differential Equations (STÆ537M)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Mathematical Colloquium (STÆ001M)

Current research in mathematics.

Final project (STÆ441L)

Geometry and differential equations (STÆ534M)

This course is an introduction to Lie group methods in differential equations. Lie groups, Lie algebras, symmetry groups of differential equations, symmetry groups and conservation laws (calculus of variations, variational symmetries, conservation laws, Noether’s theorem).

Thesis skills: project management, writing skills and presentation (VON001F)

Introduction to the scientific method. Ethics of science and within the university community.
The role of the student, advisors and external examiner. Effective and honest communications.
Conducting a literature review, using bibliographic databases and reference handling. Thesis structure, formulating research questions, writing and argumentation. How scientific writing differs from general purpose writing. Writing a MS study plan and proposal. Practical skills for presenting tables and figures, layout, fonts and colors. Presentation skills. Project management for a thesis, how to divide a large project into smaller tasks, setting a work plan and following a timeline. Life after graduate school and being employable.

Differential Geometry (STÆ519M)

Differentiable manifolds, tangent space, cotangent space, differentiation over manifolds, vector fields, differential forms and exterior derivative, partition of unity, integration over manifolds, Stoke’s theorem, elements of Riemannian geometry.

General Relativity (EÐL610M)

This course provides a basic introduction to Einstein's relativity theory: Special relativity, four-vectors and tensors. General relativity, spacetime curvature, the equivalence principle, Einstein's equations, experimental tests within the solar system, gravitational waves, black holes, cosmology.

Teachers: Benjamin Knorr and Ziqi Yan, postdocs at Nordita

Mathematical Physics (EÐL612M)

Continuum mechanics: Stress and strain, equations of motion. Seismic waves. Maxwell's equations and electromagnetic waves. Plane waves, reflection and refraction. Distributions and Fourier transforms. Fundamental solutions of linear partial differential equation. Waves in homogeneous media. Huygens' principle and Ásgeirsson's mean value theorem. Dispersion, phase and group velocities, Kramers-Kronig equations. The method of stationary phase. Surface waves on liquids.

Mathematical Colloquium (STÆ002M)

Current research in mathematics.

Final project (STÆ441L)

Stochastic Processes (STÆ415M)

Introduction to stochastic processes with main emphasis on Markov chains.

Subject matter: Hitting time, classification of states, irreducibility, period, recurrence (positive and null), transience, regeneration, coupling, stationarity, time-reversibility, coupling from the past, branching processes, queues, martingales, Brownian motion.

Functional Analysis (STÆ507M)

Banach spaces and Hilbert spaces and their main properties. Duals of Banach spaces. Convolutions. The Fourier transform on LBanach spaces and Hilbert spaces and their main properties. Duals of Banach spaces. Convolutions. The Fourier transform on L 1(R). The Plancherel theorem. Equicontinuity, the Arzelà-Ascoli theorem. The Stone-Weierstrass approximation theorem. Linear operators on Hilbert spaces, in particular compact operators. The spectral theorem. The Hahn-Banach theorem. The Baire category theorem. The uniform boundedness theorem, the open mapping theorem and the closed graph theorem.

Mathematical Colloquium (STÆ001M)

Current research in mathematics.

Final project (STÆ441L)

Geometry and differential equations (STÆ534M)

This course is an introduction to Lie group methods in differential equations. Lie groups, Lie algebras, symmetry groups of differential equations, symmetry groups and conservation laws (calculus of variations, variational symmetries, conservation laws, Noether’s theorem).

Thesis skills: project management, writing skills and presentation (VON001F)

Introduction to the scientific method. Ethics of science and within the university community.
The role of the student, advisors and external examiner. Effective and honest communications.
Conducting a literature review, using bibliographic databases and reference handling. Thesis structure, formulating research questions, writing and argumentation. How scientific writing differs from general purpose writing. Writing a MS study plan and proposal. Practical skills for presenting tables and figures, layout, fonts and colors. Presentation skills. Project management for a thesis, how to divide a large project into smaller tasks, setting a work plan and following a timeline. Life after graduate school and being employable.

Introduction to Measure-Theoretic Probability (STÆ418M)

Probability based on measure-theory.

Subject matter: Probability, extension theorems, independence, expectation. The Borel-Cantelli theorem and the Kolmogorov 0-1 law. Inequalities and the weak and strong laws of large numbers. Convergence pointwise, in probability, with probability one, in distribtution, and in total variation. Coupling methods. The central limit theorem. Conditional probability and expectation.

Differential Geometry (STÆ519M)

Differentiable manifolds, tangent space, cotangent space, differentiation over manifolds, vector fields, differential forms and exterior derivative, partition of unity, integration over manifolds, Stoke’s theorem, elements of Riemannian geometry.

Mathematical Colloquium (STÆ002M)

Current research in mathematics.

Final project (STÆ441L)