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 10-12, in 3733.

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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 15-17 in the seminar room 3733, 
Institutionen för matematik. We continue our lectures every second Monday 
starting from January 28 between 10-15 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.

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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 inclusion-exclusion
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 inclusion-exclusion, 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.

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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: Rose-Marie Jansson, rom 3527, tel: 7907201, contact.
Book: Notes from the upcoming book on "toric geometry" by
Cox-Little-Schenck, to be handed out in class.
Prerequisite: Basic algebra (like SF2703) and discrete mathematics
(like SF1630).
Exam: take-home 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
two-hour lecture every week, preliminarily on Monday 10-12. Start: Jan. 20.

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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 Lie-algebras
Applications and extensions (Lie-superalgebras? infinite-dimensional
Lie-algebras?)
The course will start on Tuesday, January 15, 15.15-17.00 (and continue, from 
January 22nd on, on Tuesdays and Thursdays, 15.15-17.00), Seminarierum 3733, 
Institutionen f. Matematik, KTH, Lindstedtsvägen 25, plan 7.

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Anders Szepessy
Partiella differentialekvationer (period 3)
 
Graduate course, 7.5 ECTS credits, starting 2008 Friday January 18th 13.15-15 
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: 0-8218-0772-2.

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Henrik Shahgholian
Homogenization, oscillation and randomness in PDE and FBP

Start:  Monday February 4, 13.15-15.00, at room 3733, building of the Dept. of 
Mathematics.
This course is partly self-studying. 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.

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HT 2007

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, Radon-Nykodim theorem, Lebesgue decomposition,
absolutely continuous functions, functions of bounded variation, Fubini's
theorem. L_p spaces. Hölder and Minkowski inequalities. Metric spaces.
Arzela-Ascoli theorem.

Prerequisites:
Analys grundkurs 5B1303.

Literature:
A. Friedman, Foundations of Modern Analysis, Dover 1982
                  

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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 7-13.
                  

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Jan-Olov 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 time-frequence 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)
                  

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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.
                  

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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.
                  

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Torbjörn Kolsrud

Stokastisk analys

Med anledning av höstens verksamhet vid Mittag-Lefflerinstitutet kommer
jag att föreläsa ca en gång i vecka under höstterminen. Ingen
särskild kursbok, ingenting om finans. Material i urval från böcker
av Dellacherie-Meyer, Emery, Ikeda-Watanabe, Malliavin (Montreal Lecture
notes), Protter, Revuz-Yor. Jag vill också koppla samman med klassisk
analys, harmoniska funktioner, konformalitet, maximalolikheten mm, samt ODE.
Start i september.

Tentamen i form av föredrag.
                  

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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://www-math.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 13.15-15.00.
The first lecture will be on Thursday 30/8 13.15-15.00 in room 3733.
                  

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