Graduate level courses, mathematics, 2009-2010

Fall 2009

Våren 2010

Commutative Algebra

SU, Rikard Bögvad
Room 306, House 6 (SU), Tuesdays at 13-15. Course start: September 8

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During the fall two courses in commutative algebra and algebraic geometry respectively will be run in parallel. The two subjects are intimately related and the courses will be coordinated so that a participant in both of them will benefit particularly.

Commutative algebra is concerned with the theory of commutative rings with (at least originally) a view towards applications in algebraic geometry. The course will present the basic results of the subject. The coordination mentioned above will not mean that attendance at the course in algebraic geometry will be a requirement but some understanding of the geometric picture will be an aid to understand the concepts presented.

Literature:

1. Steps in commutative algebra by Ronald Sharp, London Mathematical Society Student Texts

Algebraic Geometry

SU, Torsten Ekedahl
Room 306, House 6 (SU), Fridays at 13-15. Course start: September 11

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During the fall two courses in commutative algebra and algebraic geometry respectively will be run in parallel. The two subjects are intimately related and the courses will be coordinated so that a participant in both of them will benefit particularly.

Algebraic geometry which generally can be said to deal with the geometry of zero sets of polynomials can be today be based either on the theory of complex analytic functions and differential geometry or on commutative algebra. The course will exclusively deal with the second type of foundation. The prospective attendant who does not know commutative algebra beforehand is strongly recommended to also attend the course in commutative algebra.

Literature:

1. Algebraic Geometry by Robin Hartshorne, Springer Verlag

Geometric Multilinear Analysis

SU, Andreas Axelsson
Room 306, House 6 (SU), Wednesdays at 10-12. Course start: September 9

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Obstacle problems in mathematical physics and industry

KTH, Henrik Shahgholian
Room 3733 (KTH), Tuesdays 13-15. Course start: September 15

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To goal of this course is to learn about certain problems in mathematical physics related to industrial problems. The prime goal, besides learning about how mathematics and especially partial differential equations can be used to formulate problems in physics, mechanics, finance, biology, and industry, is to introduce students to real-world problems and problems in the frontier of active research. The course can be seen as an introduction to the topic “Free Boundary Problems” and there are possibilities of further study and “Examensarbete” in mathematics.

Literature:

1. Obstacle problems in mathematical physics by Jose-Francisco Rodrigues (1986), North-Holland, Mathematics studies 134. (The book is out of print, but we can provide with some copies from the library.)
2. Additional handouts and research papers

Computational number theory (independent study)

KTH, Pär Kurlberg
Weekly meetings in room 3721 (KTH), Tuesdays 15.15-17.15

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Literature:

1. Prime numbers : a computational perspective by Richard Crandall and Carl Pomerance, Springer Verlag.
2. A course in computational algebraic number theory by Henri Cohen, Springer Verlag

Inverse problems

KTH, Anders Szepessy
Room 3721 (KTH), Thursdays 10-12. Course start: September 10

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The goal of this course is to understand basic mathematical and numerical methods to solve inverse problems related to partial differential equations.

Some topics: ill-posed problems and their numerical solution by regularization methods, regularization of linear problems, Tikhonov regularization, regularization of non linear problems.

Some applications: differentation as an invers problem, X-ray tomography, data-assimilation for weather and climate prediction, inverse scattering, optimal design, image processing, parameter identification.

Literature:

1. Regulation of Invers Problems by Heinz W. Engl, Martin Hanke and Andreas Neubauer, Kluwer Academic Publishers. (Main course book.)
2. Computational Methods for Inverse Problems by Curtis R. Vogel, SIAM.
3. Inverse Problem Theory by Albert Tarantola, SIAM

Elliptic Curves

KTH, Carel Faber

Dynamics of Strings and Membranes

KTH, Jens Hoppe
Room 3733 (KTH), Tuesdays and Thursdays 15-17. Course start: January 12. Course end: March 26

The course will be an introduction, including M(atrix)-theory, to the theory of relativistic extended objects (classically: time-like manifolds that are stationary points of the volume-functional), and will be given (with different levels of examination) both as a higher level course (avancerad kurs) for undergraduate students, and as a graduate course for PhD students.

Literature:

1. A First Course in String Theory by B. Zwiebach
2. Original research articles

Coxeter groups

KTH, Axel Hultman
Room 3721 or 3733 (KTH), Thursdays 10–12. Course start: January 26 in room 3721.

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This graduate course concerns basic Coxeter group theory and should be suitable for graduate students with interests in combinatorics, algebra or Lie theory.

The theory of Coxeter groups provides a common framework for important classes of groups such as symmetric groups, Weyl groups, reflections groups (finite, affine, hyperbolic) and symmetry groups of regular polytopes. These objects are central in Lie theory, algebraic geometry and combinatorial geometry among other subjects. We shall study Coxeter groups using mainly combinatorial techniques, although connections with other areas will be prevalent. This is a thriving field of research, and we shall touch upon very recent results on several occasions.

Literature:

1. A. Björner and F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics 231, Springer, New York, 2005.
2. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge studies in advanced mathematics 29, Cambridge University Press, 1990.

Topics in advanced analysis

SU, Jan-Erik Björk
Wednesdays at 10-12. Course start: January 20