Advanced Real Analysis I HT22

Course Content

The course covers integration and measure theory and functional analysis, integration of measurable functions (Lebesgueintegraler), convergence theorems, product measure, Fubini's theorem, Banach spaces including LP-room and fundamental theorems on linear operators and functionals. Areas of application are in Fourier analysis, ergodic theory, probability theory, Sobolev spaces and partial differential equations.

Please note that self-enrollment on the course page is not the same as course registration in Ladok.

Mathematical Communication HT22

The course covers:

* TeX and LaTeX. Presentations in Beamer and PowerPoint.
* Orientation and published in mathematical journals. Writing scientific papers in mathematics.
* Mathematical popular science. Writing a popular science article in mathematics.
* To hold lectures and presentations in mathematics.

Please note that self-enrollment on the course page is not the same as course registration in Ladok.

Logic II, HT22

Logic II is a second tier logic course, on the advanced level, which gives an introduction to modern mathematical logic. It includes Gödel incompleteness theorems, computability theory, model theory, nonstandard analysis, axiomatic set theory, ordinal and cardinal numbers, equivalents of the axiom of choice.

Course Literature

R. Cori, D. Lascar; Recursion Theory, Godel's Theorems, Set Theory, Model Theory. Oxford university press

Please note that self-enrollment on the course page is not the same as course registration in Ladok.

Linear Algebra and Learning from Data HT22

Kursen innehåller: Grundläggande beräkningseffektiva algoritmer för stora matriser; Principalkomponentanalys; Glesa och underbestämda system och deras relation till komprimering av data; Konstruktion av neurala nätverk och modeller för djup inlärning; Anpassning av hyperparametrar; Valda ämnen om särskilda matristyper.

Kurslitteratur: G. Strang, Linear algebra and learning from data

Please note that self-enrollment on the course page is not the same as course registration in Ladok.

Optimization HT22

The main focus of the course is convex analysis and a rather modern treatment of optimization problems.  It covers basic convex analysis and Lagrange duality theory with their applications in linear and nonlinear programming problems with and without constraints and a touch to modern convex optimization theory.  It also provides links to other specific optimization problems such as matrix game, integer programming and dynamic programming. 

The contents of the course may be applied in modelling and computation nearly everywhere when mathematical models or computations can not be made exact.  In particular it provides a solid theoretic background and skill for understanding  nature and mathematical structure of different problems so that practical problems can be tackled successfully.  An apparent example of such is understanding (big) data to make optimization algorithms work for example in Machine learning.    

Course Literature

Bazaraa, Sherali & Shetty: Nonlinear programming, Theory & Algorithms. John Wiley and Sons Ltd

Please note that self-enrollment on the course page is not the same as course registration in Ladok.

Spectral Geometry of Graphs HT22

Please note that self-enrollment on the course page is not the same as course registration in Ladok.

Topology HT22

The course treats the foundations of general set topology (topological spaces, continuity, compactness, connectedness, quotient topologies), the fundamental group, classification of closed surfaces.
Course Literature
John M. Lee: Introduction to Topological Manifolds, 2nd edition. Springer

Please note that self-enrollment on the course page is not the same as course registration in Ladok.

Algebraic Geometry and Commutative Algebra HT22

Course Content

The course covers rings, ideals, prime ideals, nilpotents, zero-divisors, modules, Noetherian rings, Hilbert's basis theorem, finite extensions and Noetherian normalization, varieties, Nullstellensatz, prime ideal spectra, localization, primary decomposition. Algebraic geometry is the study of solutions to systems of polynomial equations. Commutative algebra is the underlying machinery. The course will give an introduction to these areas.

Please note that self-enrollment on the course page is not the same as course registration in Ladok.

Galois theory HT22

The course treats Galois theory for finite field extensions.

Please note that self-enrollment on the course page is not the same as course registration in Ladok.

Reductive Algebraic Groups HT22

Please note that self-enrollment on the course page is not the same as course registration in Ladok.