FÖRDJUPNINGS OCH DOKTORANDKURSER 07/08
VT 2008
FÖRDJUPNINGSKURSER
Torbjörn Kolsrud Fourieranalys (SF 2705)
Literature: E.M. Stein and R. Shakarchi, Fourier Analysis, An Introduction. Princeton University Press 2003. Notes on the Fourier transform on finite groups. Examination: Homework plus oral exam on theory. Lectures: Fridays 1012, in 3733.

Ari Laptev Funktionalanalys (SF 2707)
The main goal is to give an introduction to the basics of functional analysis and operator theory, and to some of their (very numerous) applications. First lecture will be on January 18 between 1517 in the seminar room 3733, Institutionen för matematik. We continue our lectures every second Monday starting from January 28 between 1015 with one hour break for lunch in the seminar room 3721 Institutionen för matematik.
Literature: Avner Friedman, Foundations of Modern Analysis, Dover Publications, Inc., 1982.

Axel Hultman Kombinatorik (SF 2708)
Abstract: We will study basic techniques in enumerative combinatorics. Examples include the ``twelvefold way'' (i.e. counting functions subject to various restrictions), sieve methods such as various versions of the inclusionexclusion principle, the involution principle and determinantal lattice path counting. We also take a peek into the rich theory of partially ordered sets (posets). Our initial motivation is the general Möbius inversion theorem for posets which is a common generalisation of the Möbius inversion theorem from number theory and the principle of inclusionexclusion, but posets turn out to have many more applications.
Essentially, we will cover the first three chapters in Stanley's book.
Literature: Richard P. Stanley, Enumerative Combinatorics, Volume I, 2nd edition, Cambridge University Press, 1998.

Sandra Di Rocco Valda ämnen i matematik: Toric geometry (SF 2716).
In recent years toric geometry has become increasingly visible in mathematics. Its importance comes from the bridge that the toric action provides between a discrete setting (polytopes) and an algebraic setting (complex algebraic varieties). The aim of this course is to give an introduction to toric geometry with a view towards applications. This course is for both undergraduates and PhD students. The PhD students will present a "research" topic in toric geometry at the end of the course, as part of the course requirement.
Facts: Teacher: Sandra Di Rocco, contact. Secretary: RoseMarie Jansson, rom 3527, tel: 7907201, contact. Book: Notes from the upcoming book on "toric geometry" by CoxLittleSchenck, to be handed out in class. Prerequisite: Basic algebra (like SF2703) and discrete mathematics (like SF1630). Exam: takehome assignments + presentation. Additional references: Introduction to toric varieties (W. Fulton), Combinatorial convexity and Algebraic geometry (G. Ewald).
The course will be given during period 3 and 4. There will be one twohour lecture every week, preliminarily on Monday 1012. Start: Jan. 20.

DOKTORANDKURSER
Jens Hoppe Liealgebror Depending on the participants' interests, there are various possible routes to take, resp. weights to assign to The geometrical and analytical origins of the theory of
continuous transformation groups (Lie's ingenious response to Jacobi)
 The structure (+representation) theory of finite dimensional (complex,
semi)simple Liealgebras
 Applications and extensions (Liesuperalgebras? infinitedimensional
Liealgebras?)
The course will start on Tuesday, January 15, 15.1517.00 (and continue, from January 22nd on, on Tuesdays and Thursdays, 15.1517.00), Seminarierum 3733, Institutionen f. Matematik, KTH, Lindstedtsvägen 25, plan 7.
 Anders Szepessy Partiella differentialekvationer (period 3) Graduate course, 7.5 ECTS credits, starting 2008 Friday January 18th 13.1515 room 4523, KTH, CSC Lindstedsvägen 3. The weekly time for lectures will be decided jointly at the first meeting. The course will include basic representation formulas for linear and some nonlinear PDE, basic theory for linear PDE and some methods for nonlinear PDE. This means chapter 1 to 4 and parts of chapter 5 to 11 in Evans's book "Partial Differential Equations". Evans's book is modern and broad with a careful choice of methods, that guide the reader to the state of the art in mathematical research on PDE. The purpose of the course is to achieve this goal.
Prerequisites: undergraduate ordinary and partial differential equations, some functional analysis. Literature: Evans, Lawrence C. Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. ISBN: 0821807722.
 Henrik Shahgholian Homogenization, oscillation and randomness in PDE and FBP
Start: Monday February 4, 13.1515.00, at room 3733, building of the Dept. of Mathematics. This course is partly selfstudying. The lectures will take place Mondays: February 4,11,18,25, March 3,17,24, April 7, Student presentations: May 5,12, 19, 26. There will be a break of 3 weeks and during this period the students will pick up a certain material for presentation (1h). See Schedule. Language: English. Goal: To learn about certain problems in classical homogenization and oscillation (hopefully something about random media). The core application will be towards free boundary problems.

HT 2007
FÖRDJUPNINGSKURSER
Henrik Shahgholian
Integration Theory (SF 2709)
Goal: To learn about the notions of measure and Lebesgue integral. Emphasis will be on the measure and Lebesgue integral on the real line.
Topics: Riemann integral, Lebesgue measure, measurable functions. Egoroff's theorem, signed measures, Jordan decomposition, absolutely continuous and singular measures, RadonNykodim theorem, Lebesgue decomposition, absolutely continuous functions, functions of bounded variation, Fubini's theorem. L_p spaces. Hölder and Minkowski inequalities. Metric spaces. ArzelaAscoli theorem.
Prerequisites: Analys grundkurs 5B1303.
Literature: A. Friedman, Foundations of Modern Analysis, Dover 1982

Roy Skjelnes
Algebra II (SF 2706)
Abstract: The course covers basic concepts about polynomial rings and modules. This includes the symmetric and exterior algebras, Jordan canonical form, and algebraic field extensions.
We will use the book "Abstract Algebra" authored by Dummit and Foote, and the course material is roughly covered by chapters 713.

JanOlov Strömberg
Wavelets, 6 hp, SF2702 (5B1308)
Abstract: The course covers the basic concept of orthonormal wavelet functions and wavelet filterbanks; local trigonometrical function bases and also timefrequence analysis with continuous wavelet transforms. We will give applications of these concepts in signal and image processing. During the course the students will gain a lot of experience of Matlab programming.
Course literature: Bergh/Ekstedt/Lindberg: Wavelets (for sale at Kårbokhandeln)

DOKTORANDKURSER
Michael Benedicks
Geometry of fractal sets and measures (Fraktal geometri och måtteori)
Abstract: The aim of the course is to cover aspects of the geometry of fractal sets and measures in Euclidean spaces, in particular results by Besicovitch and Marstrand. The main reference for the course is the book
P. Mattila: "The geometry of sets and measures in Euclidean spaces", Cambridge University Press, 1995,
but the two more expository books by Falconer (1985, 1990) are also valuable. The encyclopedic book by Federer from 1969 is still the main reference in the field.
Application of the theory to topics such as dynamical systems and random processes (Stochastic Loewner evolutions) will also be given.

Wojciech Chachólski
Topological vector bundles and characteristic classes
Abstract: The aim of the course is to present topological methods to study vector bundles. We will be using mainly homological techniques. Thus one purpose of the course is to talk about homology and cohomology of spaces. We will then illustrate the use of these methods to study geometric questions related to vector bundles.
I will use several sources for the course, among them Characteristic classes by Milnor and Stasheff, and Algebraic topology by Hatcher.

Torbjörn Kolsrud
Stokastisk analys
Med anledning av höstens verksamhet vid MittagLefflerinstitutet kommer jag att föreläsa ca en gång i vecka under höstterminen. Ingen särskild kursbok, ingenting om finans. Material i urval från böcker av DellacherieMeyer, Emery, IkedaWatanabe, Malliavin (Montreal Lecture notes), Protter, RevuzYor. Jag vill också koppla samman med klassisk analys, harmoniska funktioner, konformalitet, maximalolikheten mm, samt ODE. Start i september.
Tentamen i form av föredrag.

Svante Linusson
Hyperplane arrangements (Hyperplansarrangemang)
The first half of the course will go through the fundamental combinatorics of arrangements of affine hyperplanes. For this part we will use the the lecture notes of Richard P. Stanley "Introduction to hyperplane arrangements" available on http://wwwmath.mit.edu/~rstan/arr.html
For the second half we will go through research papers in the area for more detailed combinatorics, or topological and algebraic aspects of hyperplane arrangements or subspace arrangements. This part may depend on the interest of the students.
The students of the course are expecxted to take active part in the presentation and discussion of the material. This will also be a large part of the examination.
The lectures are planned to be on Thursdays 13.1515.00. The first lecture will be on Thursday 30/8 13.1515.00 in room 3733.

