Martin Hertweck's homepage

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Last update: March 19, 2009

PHOTO I am Privatdozent (unsalaried lecturer) within the Department of Mathematics at the University of Stuttgart, located on Campus Vaihingen.

Postal address:

Universität Stuttgart
Fachbereich Mathematik
Institut für Geometrie und Topologie
Lehrstuhl für Differentialgeometrie
70569 Stuttgart
Germany
Picture of Institute

E-mail: hertweck@mathematik.uni-stuttgart.de

Publications   Manuscripts

black board RESEARCH INTERESTS
The theory of finite groups and their representation theory, in particular the structure of (integral) group rings, and interactions.
A group theorist might find some of my papers a bit strange, but group ring people do "applied group theory". (As an anonymous referee told me once. For example, finite conjugate groups, hypercentrality, Hamiltonian groups and even Blackburn groups occur naturally in the study of group rings, and researchers in the field have to come to grips with these concepts.)
Together with Dr. Soriano, I'm currently working on the modular isomorphism problem. What is it about? Click here. I'm also working on the Zassenhaus conjecture concerning rational conjugacy of torsion units in integral group rings (this is "applied representation theory"). Units in integral group rings? Click here.



PUBLICATIONS
* Publication list (all papers for download)
* My Erdös Number
* Some recent manuscripts (with abstracts in HTML)
* Slides of some talks I have given.

Please follow IMU's recommendation and make available electronically as much of your own work as feasible. Perhaps a good place to post preprints is the arXiv. I try my best!



Lecture notes for a course on group theory in Stuttgart. In summer term 2008, I gave a course on finite group theory. I started from scratch, ending with a proof of Burnside's paqb theorem (after Goldschmidt and Matsuyama). This should give a rough idea of what's in my lecture notes (written in german) I am posting here online.
Version DVI (for printing) and version PDF (for viewing).



In winter term 2001/02 I've given a course Algebra I at the University of Stuttgart.

The excercises, together with solutions (in german!): DVI or PS.



TWO The modular isomorphism problem asks for whether two finite p-groups G and H are isomorphic provided their modular group algebras (over the field with p elements) are isomorphic:
$ \mathbb{F}_{p}G\cong\mathbb{F}_{p}H\overset{\text{?}}{\Longrightarrow} G\cong H$
I started collaboration with Marcos Soriano (University of Hannover) on this fascinating problem which is still so poorly understood. We have written a note on the groups of order 26 which also contains a brief introduction into the subject: On the modular isomorphism problem: groups of order 26 (version dvi and version pdf).

In a preprint from 2003, Inger Christin Borge and Olav Arnfinn Laudal claimed to have answered the modular isomorphism problem in the affirmative. But see their own remark (seen on Benson's Preprint Archive).
Marco and I have written a note, Central p-group extensions: Laudal obstruction spaces revisited (version DVI and version PDF), where we at least give an accessible and unified presention of Borge's and Laudal's approach, and also briefly explain at what point the proof breaks down.

THE MODULAR ISOMORPHISM PROBLEM IS STILL OPEN.

A joke, only for the few who can understand it: I omitted the word "WIDE".



TWO

Units in integral group rings? The group of units in ZS3, the integral group ring of the symmetric group on three letters, has some well known descriptions. I like in particular the geometric approach given by Marciniak and Sehgal [Units in group rings and geometry, Methods in ring theory (Levico Terme, 1997)] which yields nice generators and relations for the group of units. I have written it up for my own, Units in ZS3 (after Marciniak and Sehgal) (version DVI and PDF), since I wanted to see some pictures included. The original (often the better one) is available from Sehgal's homepage.

Problem/Challenge (Zassenhaus Conjecture for some "small groups"): Suppose that G is a finite group having a normal abelian 2-group A with quotient G/A isomorphic to the symmetric group S3. Is it true that units of 2-power order in ZG are rationally conjugate to elements of ±G?


Integral group ring community. Research groups in different places of the world join efforts in the area of (integral) group rings, discussing topics such as algebraic structure, representation theory, and units, of group rings. Here is an incomplete list of those people. (You are missing? Something wrong? Then please mail to me.)



Conferences:
*   FDLIST Representation theory of finite dimensional algebras: Informations on conferences, maintained by the BIREP group at University of Bielefeld.


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Preprint archives: 
* Dave Benson's Preprint Archive (Groups, Representations and Cohomology)
* ArXiv, e-print archive owned, operated and funded by Cornell University
People, Search in membership lists:
* Combined membership list (CML) of AMS, SIAM, MAA, AMATYC, AWM. 
* Search a member of the DMV? click here
* Allen Bell's list of ring theorists
Mathematical reviews on the Web:
* MathSciNet of the American Mathematical Society (AMS).
* Zentralblatt MATH - database of the European Mathematical Society (EMS).

More
* GAP, a system for computational discrete algebra, with particular emphasis on Computational Group Theory.
* The ATLAS of Finite Group Representations prepared by Robert Wilson et al.
* Group Pub Forum for the discussion of any aspect of Group Theory
* Homepage of the Deutsche Mathematiker-Vereinigung
* Homepage of the European Mathematical Society
* Homepage of the American Mathematical Society
* German-English Online Dictionary, a service of TU Chemnitz


*

If I were a Springer-Verlag Graduate Text in Mathematics, I would be Saunders Mac Lane's Categories for the Working Mathematician.

I provide an array of general ideas useful in a wide variety of fields. Starting from foundations, I illuminate the concepts of category, functor, natural transformation, and duality. I then turn to adjoint functors, which provide a description of universal constructions, an analysis of the representation of functors by sets of morphisms, and a means of manipulating direct and inverse limits.

Which Springer GTM would you be? The Springer GTM Test

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Eulenspiegel, a german satiric magazine. Unbestechlich aber käuflich!


Finally, want to see some math pictures? Then click here.