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, 2010, 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)
Start/End Dates: 23-MAY-14 / 24-MAY-14
Project: Dimension computation of Bianchi modular forms ( ngmat003b)
Start/End Dates: 01-JUL-12 / 30-JUN-14
Peer Reviewed Journals
|Alexander D. Rahm (2014) 'Complexifiable characteristic classes'. Journal of Homotopy and Related Structures, . [Details]|
|Alexander D. Rahm (2014) 'Accessing the cohomology of discrete groups above their virtual cohomological dimension'. Journal Of Algebra, Accepted for publication . [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 (2013) 'THE HOMOLOGICAL TORSION OF PSL2 OF THE IMAGINARY QUADRATIC INTEGERS'. Transactions Of The American Mathematical Society, 365 :1603-1635. [DOI] [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]|
|A.~D.~Rahm and M.~H.~Sengun (2013) 'On Level One Cuspidal Bianchi Modular Forms'. Lms Journal Of Computation And Mathematics, . [DOI] [ARAN Link] [Details]|
|Rahm, AD (2013) 'Higher torsion in the Abelianization of the full Bianchi groups'. Lms Journal Of Computation And Mathematics, 16 :344-365. [DOI] [Details]|
|Rahm, AD,Sengun, MH (2013) 'On level one cuspidal Bianchi modular forms'. Lms Journal Of Computation And Mathematics, 16 :187-199. [DOI] [Details]|
|Rahm, AD (2012) 'On a question of Serre'. Comptes Rendus Mathematique, 350 :741-744. [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]|
|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]|
|Alexander D. Rahm (2007) Characteristic classes of complexified bundles. Lille 2007: Conference Paper [Details]|
|Rob de Jeu and Alexander D. Rahm (2014) The image of the Borel-Serre bordification in algebraic K-theory. Research Paper [Details]|
|Ethan Berkove and Alexander D. Rahm (2014) The mod 2 cohomology rings of SL$_2$ of the imaginary quadratic integers. Research Paper [Details]|
|Alexander D. Rahm (2013) Chen/Ruan orbifold cohomology of the Bianchi groups. Research Paper [Details]|
|Alexander D. Rahm (2013) Seminar of the Institute of Physics and Mathematics at Universidad Austral de Chile (UACh), Valdivia: Group Actions and Orbit Spaces. Research Seminar [Details]|
|Alexander D. Rahm (2013) Séminaire Algèbre et topologie, IRMA Strasbourg: Calculs efficaces au dessus de la dimension cohomologique virtuelle. Research Seminar [Details]|
|Alexander D. Rahm (2012) Twenty more research seminar talks, please look up the list linked from here. Research Seminar [Details]|
|Alexander D. Rahm (2010) (Co)homologies et K-théorie de groupes de Bianchi par des modèles géométriques calculatoires. Thesis [ARAN Link] [Details]|
Honours and Awards
| Year: 2010.
Title: Certificat de Compétences en Calcul Intensif (C3I)
|Association: Irish Mathematical Society (IMS), Function/Role: Member|
|Association: European Mathematical Society (EMS), Function/Role: Member|
|Association: Société Mathématique de France (SMF), Function/Role: Member|
|Association: Deutsche Mathematiker-Vereinigung (DMV), Function/Role: Member|
|Alexander D. Rahm (2012) Closing Talk: A new technique to extract invariants of matrix groups. [Keynote Address], 6th de Brun Workshop Linear Algebra and Matrix Theory: connections, applications and computations, NUI Galway , 03-DEC-12 - 07-DEC-12.|
|Alexander D. Rahm (2012) Accessing the Farrell-Tate cohomology of discrete groups. [Invited Oral Presentation], Torsion in the homology of arithmetic groups: geometry, arithmetic, and computation, Banff International Research Station for Mathematical Innovation and Discovery, Canada , 01-JUL-12 - 06-JUL-12.|
|Alexander D. Rahm (2011) Implications of hyperbolic geometry to operator K-theory of artihmetic groups. [Invited Oral Presentation], London Mathematical Society - EPSRC Durham Symposium: Geometry and Arithmetic of Lattices, Durham, England , 04-JUL-11 - 14-JUL-11.|
|Alexander D. Rahm (2009) The integral homology of PSL_2 of imag. quad. int. of class number two. [Invited Oral Presentation], 17th öMG Congress and Annual DMV Meeting, Graz, Austria, TU Graz, Autria , 20-SEP-09 - 25-JAN-09.|
|Alexander D. Rahm (2007) Characteristic classes of complexified bundles. [Invited Oral Presentation], Summer School in Algebraic Topology: Sheaf theoretic methods in the theory of characteristic classes, Université Lille 1 , 11-JUN-07 - 15-JUN-07.|
|Alexander D. Rahm (2006) Characteristic classes of real generator bundles. [Invited Oral Presentation], Young Topologists and new topology at the Banach Mathematical Center and Conference Center, Bedlewo, Poland , 28-AUG-06 - 02-SEP-06.|
I am teaching the following modules:
This semester, starting January 2014: Calculus II
September to December 2013: Quantitative Techniques for Business
Mathematical and Logical Aspects of Computing.
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)