Dr Alexander Rahm Doctor of Mathematics
| Binational doctoral degree (Cotutelle de thèse), with the advisors
Prof. Dr. Philippe Elbaz-Vincent , at the Institut Fourier, Université Joseph Fourier Grenoble, France
and Prof. Dr. Thomas Schick, Mathematisches Institut, Universität Göttingen, Germany.
Supported by a research grant of the Ministère de l'Enseignement Supérieur et de la Recherche, and partially supported by DFH-UFA grant CT-26-07-I and by a DAAD (German Academic Exchange Service) grant.
The intensive computations award C3I has been attributed to this thesis.
The thesis was defended October 15th, Grenoble. It can be downloaded from the server TEL of the CNRS.
The iterated advisors of my advisors can be found in my mathematics genealogy graph.I am grateful for a Research stay at the I3M Montpellier during my doctoral studies.
Post-doctoral scientific research:Former IRCSET Research fellow at the National University of Ireland at Galway, Department of Mathematics. Mentor: Prof. Dr. Graham J. Ellis.
Former Research fellow of the Weizmann Institute of Science, Department of Mathematics. Mentor: Prof. Dr. Stephen S. Gelbart.
- Classifying Space for proper actions.
- Discrete subgroups of Lie groups.
- Discontinuous groups of transformations.
- Isometries of Hyperbolic Space.
- And many others.
You can watch a video of a talk I gave at a Banff workshop
on the Torsion Sub-complex Reduction technique that I have introduced.
Project: Conference "Groups in Galway" ( Groups in Galway)
Description: The annual conference "Groups in Galway" takes place on a weekend in May at NUI Galway.
Start/End Dates: 16-MAY-14 / 17-MAY-14
Project: Dimension computation of Bianchi modular forms ( ngmat003b)
Role: Principal Investigator
Description: Joint project with Mehmet Haluk Sengun (University of Warwick) at the Irish Centre for High-End Computing (ICHEC)
Start/End Dates: 01-JUL-12 / 30-JUN-14
Peer Reviewed Journals
|Alexander D. Rahm (2013) 'Complexifiable characteristic classes'. accepted for publication in Journal of Homotopy and Related Structures, . [Details]|
|A.~D.~Rahm and M.~H.~Sengun (2013) 'On Level One Cuspidal Bianchi Modular Forms'. LMS J. Comp. Math, . [ARAN Link] [Details]|
|Rahm, Alexander D. and Fuchs, Mathias (2011) 'The integral homology of $\rm PSL_2$ of imaginary quadratic integers with nontrivial class group'. J. Pure Appl. Algebra, 215 (6):1443-1472. [DOI] [ARAN Link] [Details]|
|Rahm, Alexander D. (2011) 'Homology and $K$-theory of the Bianchi groups'. C. R. Math. Acad. Sci. Paris, 349 (11-12):615-619. [DOI] [ARAN Link] [Details]|
|A.~D.~Rahm (2013) 'Higher torsion in the Abelianization of the full Bianchi groups'. LMS J. Comp. Math, . [Details]|
|A.~D.~Rahm (2013) 'The subgroup measuring the defect of the Abelianization of SL_2(Z[i])'. Journal of Homotopy and Related Structures, . [ARAN Link] [Details]|
|Rahm, Alexander D. (2013) 'The homological torsion of $\rmPSL_2$ of the imaginary quadratic integers'. Trans. Amer. Math. Soc, 365 (3):1603-1635. [DOI] [ARAN Link] [Details]|
|Rahm, AD (2012) 'On a question of Serre'. Comptes Rendus Mathematique, 350 :741-744. [DOI] [ARAN Link] [Details]|
|Association: Société Mathématique de France (SMF), Function/Role: Member|
|Association: Deutsche Mathematiker-Vereinigung (DMV), Function/Role: Member|
| This semester, starting September 2, 2013: Quantitative Techniques for Business.
Summer semester 2013: Lecture series "Discrete groups and torsion subcomplexes" at the Graduiertenkolleg Cohomological Methods in Geometry, University of Freiburg, Germany.
2012/2013: Quantitative Techniques for Business.
2011/2012: Graduate course in Arithmetic.
Diplom thesis of Hendrik Demmer (defended 2011, Universität Göttingen),
In his Diplom thesis, Hendrik Demmer studied the modular group: the quotient of the group SL2 (Z) by its centre. He constructed a cellular model for the action of the modular group, as a bridge between the classical geometrical model - the modular tree of Serre - and Kulkarni's arithmetic model. The latter admits the advantage that for every subgroup of finite index in the modular group, a Farey symbol can be computed in an efficient way, containing the essential information about the group structure. Demmer's model is a graph, constituting the 1-skeleton of a two-dimensional cell complex dual to the modular tree. Its set of edges is the set of elements of the modular group itself; and its 0-skeleton is the projective line over the rational numbers, to which Hendrik Demmer lends additional arithmetic structure as the set of "Stern-Brocot fractions". This enables Demmer to compute the Farey symbol of a subgroup from a fundamental domain for it in the modular tree. To arrive there, he shows that he can choose fundamental domains in the modular tree particularly appropriately. Demmer's fundamental domains incorporate the bifurcation points into their interior and hence avoid that a bifurcation in the quotient graph becomes visible only after carrying out the identifications. Also, bifurcations of the modular tree are either incorporated entirely into Demmer's fundamental domain, or as a single edge in the case that the orbit of the latter contains the whole bifurcation. By the strictness concerning the edges, it is achieved that the number of edges of such a fundamental domain equals the index of the concerned subgroup. The information attached to the corners in terms of Stern-Brocot fractions now allows to compute the identifications among the endpoints of the fundamental domain as matrices, via an algorithm conceived by Hendrik Demmer. Furthermore, Hendrik Demmer has elaborated algorithms for the conversion between matrices of the modular group and words in the free product (isomorphic to the latter group) of the two finite groups of orders 2 and 3.
|2013/2014 Core module for Bachelor students of Business Information Systems: Quantitative Techniques for Business|
| Graham Ellis (National University of Ireland at Galway)
Bui Anh Tuan (National University of Ireland at Galway)
| Ethan Berkove (Lafayette College, Pennsylvania)
Mehmet Haluk Sengun (University of Warwick)
Rob de Jeu (VU University Amsterdam)
Matthias Wendt (Universitaet Freiburg)
Noam Solomon (Tel Aviv University)
Barak Weiss (Ben Gurion University of the Negev)