## Yorck Sommerhäuser

### Seminars

- Linear topologies on algebraic objects (Lineare Topologien auf algebraischen Objekten)
- Winter semester 1996/97
- Time: Friday, 2:00 pm
- Room: 251
- Planning session:
- Contents: The seminar deals with topologies on algebraic objects -groups, rings, fields and vector spaces. It will also treat applications, for example to infinite Galois theory, to the theory of distributions and in particular to the duality theory of Hopf algebras. No prior knowledge of these fields is required, but some knowledge of point set topology is needed (topological spaces, continuity, compactness). The seminar is intended for students that have completed their Vordiplom (graduate students).
- Seminar program (german, dvi)

- Completion of quantum groups (Vervollständigung von Quantengruppen)
- Summer semester 1997
- Time: Friday, 2:00 pm
- Room: 251
- Planning session: Friday, May 2, 1997, 2:00 pm, Room 251
- Contents: The seminar starts with an introduction to the theory of deformed enveloping algebras of semisimple Lie algebras. Afterwards, we consider completions of these algebras and construct the universal R-matrix, which is contained in this completion. The seminar is a continuation of the seminar held last semester, nevertheless it is accessible for newcomers since the theory of deformed enveloping algebras is developed from the very beginning. Prerequisites comprise the courses `Linear algebra I,II', including the tensor product. Parts of the seminar require some point set topology. The seminar is intended for graduate students that are interested in an introduction to the fast developing field of quantum groups.
- Seminar program (german, dvi)

- Jones theory (Jones-Theorie)
- Winter semester 1997/98
- Time: Friday, 2:00 pm
- Room: 251
- Planning session: Friday, November 8, 1997, 2:00 pm, Room 251
- Contents:
The seminar deals with Jones' approach to the so-called Jones polynomial, an invariant of knots discovered by him, which in contrast to the older Alexander polynomial can distinguish between a knot and its mirror image. Jones' approach to this invariant will lead us to extensions of algebras, Dynkin diagrams and Hecke algebras. The seminar is based on the book:
F. Goodman/P. de la Harpe/V. Jones: Coxeter graphs and towers of algebras, MSRI Publications 14, Springer, Berlin 1989

Apart from a good knowledge about linear algebra, no prerequisites are required.

- Seminar program (german, dvi)

- Fundamental groups and Tannakian categories (Fundamentalgruppen und Tannaka-Kategorien)
- Winter semester 1997/98
- Time: Tuesday, 2:00 pm
- Room: 138
- Contents: The finite-dimensional complex representations of a group form a category. Every representation has an underlying vector space, and one can form the tensor product of two representations - the category is Tannakian. Conversely, the question arises whether one can reconstruct the group from its representations, i.e., whether for a given category one can find a group so that the given category consists exactly of the representations of the group. For this, it is important to construct first a functor to the category of vector spaces - a fiber functor. In the seminar, we discuss these and related questions.

- Jones theory (Jones-Theorie)
- Summer semester 1998
- Time: Thursday, 3:15 pm
- Room: 251
- Planning session: Thursday, May 7, 1998, 9:15 am, Room 133
- Contents: The seminar deals with Jones' approach to the so-called Jones polynomial, an invariant of knots discovered by him, which in contrast to the older Alexander polynomial can distinguish between a knot and its mirror image.
The seminar is a continuation of the seminar held last semester, however it is also accessible for newcomers since the topics are largely independent from the material developed in the previous seminar. The topics discussed include Hecke algebras, Temperley-Lieb algebras and the traces admitted by these algebras, which are used in the construction of the Jones polynomial. The seminar is based on the book:
F. Goodman/P. de la Harpe/V. Jones: Coxeter graphs and towers of algebras, MSRI Publications 14, Springer, Berlin 1989

The seminar requires a basic knowledge about matrix rings.

- Seminar program (german, dvi)

- Hopf algebras (Hopf-Algebren)
- Summer semester 1998
- Time: Wednesday, 9:15 am
- Room: 252
- Planning session: Wednesday, February 25, 1998, 11:15 am, Room E 39
- Contents: The seminar deals with the foundations of the theory of Hopf algebras as well as with newer developments which touch upon open problems. Topics discussed include the Nichols-Zoeller theorem, trace formulas, and the structure theory of commutative and cocommutative Hopf algebras. The seminar requires a good knowledge about linear algebra, including the tensor product. Knowledge about abstract algebra is helpful, but not necessary.
- Seminar program (german, dvi)

- Selected topics in Galois theory
(Ausgewählte Kapitel aus der Galoistheorie)
- Summer semester 1999
- Time: Tuesday, 11:15 am
- Room: 252
- Planning session: Tuesday, May 4, 1999, 11:15 am, Room 252
- Contents: We treat methods and examples from Galois theory, which shall extend the knowledge from the standard algebra courses, for example the calculation of concrete Galois groups of polynomials. Equations of third and fourth degree are covered in detail. We introduce fundamental notions such as norm, trace, and discriminant and explain them by examples. The seminar is directed in particular to future teachers that want, together with their seminar certificate, aquire knowledge relevant for the state examination. We offer the possibility to write a thesis connected to some of the talks.
- Seminar program (german, dvi)

- Vertex algebras and conformal field theory (Vertexalgebren und konforme Feldtheorie)
- Winter semester 1999/2000
- Time: Tue 11:15 am
- Room: 134
- Planning session: Friday, July 30, 1:30 pm, Room 251
- Course number: 16214
- Contents:
Vertex algebras are algebraic structures that are designed to set the formalism
of conformal field theory in physics onto a rigorous mathematical foundation. During the investigation of these structures, there appeared astonishing connections to other areas of mathematics, for example to the theory of finite simple groups or the theory of Hopf algebras. The great interest that arose for this comparatively new fields has manifested itself in the award of the fields medal to R. Borcherds for his contributions to the theory of vertex algebras in 1998.
The seminar is addressed to students of mathematics and physics after their Vordiplom. The seminar is based on the book:

V. Kac: Vertex Algebras for Beginners

University Lecture Notes Series, Vol. 10

American Mathematical Society

Providence, USA - Seminar program (german, dvi)

- Quantum groups and conformal field theory (Quantengruppen und konforme Feldtheorie)
- Winter semester 2000/2001
- Time: Tue 2:00 pm
- Room: 251
- Planning session: Thursday, October 18, 3:15 pm, Room 251
- Contents: The aim of the seminar is to explain the connections between the theory of quantum groups, modular categories, and conformal field theory. We want to show how deformed enveloping algebras of semisimple Lie algebras as well as the Wess-Zumino-Witten model from conformal field theory lead to the notion of a modular category and a modular functor. The general principles will be explained by simple examples.
The seminar is based on the books of B. Bakalov und A. Kirillov on the one hand and the book of V. G. Turaev on the other hand. A good command of mathematics is required for participation.

- Seminar program

- Cristallographic groups (Kristallographische Gruppen)
- Summer semester 2001
- Time: Wednesday, 11:15 am
- Room: 251
- Planning session: Tuesday, February 6, 2001, 1:15 pm, Room 251
- Contents: The goal of the undergraduate seminar is to classify the plane cristallographic groups, i.e., the symmetry groups of ornaments in the plane. Although there are plenty of ornaments of different colours and shapes, it turns out that, for the symmetries underlying these ornaments, there are only 17 distinct types. It seems that this fact was already known to the arabic craftsmen during the middle ages, since in their buildings of that time all these types can be found. The rigourous mathematical treatment of this classification, however, was first carried out at the end of the nineteenth century. In our times, these symmetries have attracted greater attention through the work of M. C. Escher.
The undergraduate seminar is directed at students of mathematics before the Vordiplom from their second semester on upwards. It provides a good opportunity the apply the notions learned in linear algebra to an attractive problem and thereby to enhance their understanding. In the planning of the talks, care has been taken to partially repeat the tools needed from linear algebra, so that the seminar is also helpful to rework the linear algebra course.

- Seminar program (german, dvi)

- Morita theory and deformation quantisation
(Morita-Theorie und Deformationsquantisierung)

- Winter semester 2001/2002
- Time: Friday, 11:15 am
- Room: E 27
- Planning session: Thursday, October 18, 2001, 11:15 pm, Room E 39
- Contents: In the first part of the seminar, we explain the ordinary Morita theory for rings, following lecture notes of Hyman Bass. After a brief introduction to the theory of operator algebras, we then treat the analog of this theory for C
^{*}-algebras, as developed by Marc Rieffel. In the third part of the seminar we will make the connection with deformation quantisation. In deformation quantisation, one introduces on an a priori commutative algebra a new product, a so-called star product, which is no longer commutative. Often there are several possibilities for the introduction of such a star product. In some cases, it is possible to describe the circumstances under which the algebras endowed with the various star products are Morita-equivalent. This problem will be approached with the aid of recent research articles. - Seminar program (german, dvi)

- Azumaya Algebras and Brauer Groups
(Azumaya-Algebren und Brauergruppen)

- Summer semester 2002
- Time: Friday, 11:15 am
- Room: 251
- Planning session: Friday, April 19, 2002, 11:15 am, Room 251
- Contents: While over algebraically closed fields every semisimple algebra is a product of matrix algebras over the base field, over not algebraically closed fields matrix rings over division rings also occur, in the case of the real numbers for example the quaternions. The question which semisimple algebras and which division rings are possible over a given base field leads to interesting connections to other properties of the field, for example to its Galois cohomology groups. In the seminar, we first develop this theory over fields and afterwards consider its generalization for rings. This theory, which is comparatively old for the rings considered in number theory, has recently attacted the attention of the string theorists through its application to rings of differentiable functions.
- Seminar program (german, dvi)
- Seminar workout

- K-Theory (K-Theorie)

- Winter semester 2002/2003
- Time: Friday, 2:15 pm
- Room: E 45
- Planning session: Friday, October 18, 2002, 2:15 pm, Room E 45
- Contents:
In this seminar we give an introduction to K-theory. K-theory is a part of ring theory that assigns to every ring certain abelian groups, the so-called K-groups, that describe certain properties of the ring. The origin of this theory lies in topology, where the base ring is the ring of continuous functions on a topological space. The K-groups then become invariants of the topological space.
The focus of the seminar will be on algebraic K-theory, which, however, will be compared with topological K-theory and the K-theory of operator algebras. A central topic of the seminar is the connection between the second K-group and the Brauer group, which is given by the Merkuriev-Suslin theorem. From this theorem, we will proceed on the one hand to the discussion of the new progress in motivic cohomology that has recently attracted attention, on the other hand to the consideration of the role that K-theory plays in string theory.

The seminar is addressed at graduate students who want to learn about a subject that is relevant for algebra, topology, and differential geometry. Besides a certain mathematical maturity, there are no special prerequisites necessary.

- Seminar program (german, dvi)

- Galois theory of inseparable extensions (Galoistheorie inseparabler Erweiterungen)
- Summer semester 2003
- Time: Friday, 2:15 pm
- Room: 132
- Planning session: Friday, February 4, 2003, 1:45 pm, Room 138
- Contents:
It is well-know that a field extension is Galois if it is finite, normal, and separable. In this case, the Galois correspondence yields a one-to-one correspondence between the subgroups of the Galois group and the intermediate fields of the field extension.
In the case of purely inseparable field extensions, one can sometimes also establish a Galois correspondence by working instead of the automorphism group with different objects. In the case of extensions of exponent 1, the automorphism groups are replaced by certain Lie algebras of derivations; in the case of higher exponents one has to work instead with so-called higher derivations. Derivations and higher derivations can be understood as elements of a Hopf algebra, so that all these Galois correspondences can be subsumed under the Hopf-Galois theory.

The seminar is addressed at graduate students who have attended the course "Algebra I" and now want to extend the knowledge aquired there. As prerequisites, linear algebra and the foundations of Galois theory are sufficient.

- Seminar program (german, dvi)

- Seminar on algebra
- Autumn quarter 2006
- Time: Monday, 1:00 pm-2:00 pm (every second week)
- Room: Braunstein 324

- Seminar on algebra
- Winter quarter 2007
- Time: Friday, 12:00 m-1:00 pm (every second week)
- Room: Rieveschl 422-F

- Seminar on algebra
- Spring quarter 2007
- Time: Tuesday, 10:00 am-11:00 am (every second week)
- Room: Rieveschl 422-F

- USA Algebra Seminar
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- Last modification: January 29, 2012