External Collaboration. With Ethan Berkove (Lafayette College, Pennsylvania), I have written a preprint,The mod 2 cohomology rings of SL2 of the imaginary quadratic integers,http: hal.archives-ouvertes.fr hal-00769261conditionally accepted for publication. A joint paper extending these investigations to congruence subgroups is in advanced progress together with Ethan Berkove and Grant Lakeland (East Illinois University).__________________________________________________�With Matthias Wendt (Universitaet Duisburg-Essen), I apply the techniques of my paperAccessing the cohomology of discrete groups above their virtual cohomological dimension, Journal of Algebra, Volume 404, 15 February 2014, Pages 152�175,to large classes of arithmetic SL_n -groups. Two joint papers have been written up so far: A refinement of a conjecture of Quillen, joint with Matthias Wendt,accepted for publication at Comptes Rendus Math�matique de l�Acad�mie des Sciences,https: hal.archives-ouvertes.fr hal-01083049,and On Farrell-Tate cohomology of SL_2 over S-integers, joint with Matthias Wendt, recently submitted,https: hal.archives-ouvertes.fr hal-01081081____________________________________With Rob de Jeu (VU University Amsterdam), I search for elements in H_3(GL_2 (Q(sqrt{ #8722;m}))) related to the algebraic K-theory of imaginary quadratic rings. Rob de Jeu has found a method to relate these elements to geometric images coming from the Bianchi groups. I have determined these images using my answer to a question of Serre, which had been open for 40 years. See our recent joint preprint, http: hal.archives-ouvertes.fr hal-00975454For an extension of this research project, I have been granted the MFO-hosted research project Constructing explicit elements in algebraic K-theory, joint with Rob de Jeu, Herbert Gangl (University of Durham) and Dan Yasaki (University of North Carolina at Greensboro) at Mathematisches Forschungsinstitut Oberwolfach (MFO), April 17 to 30, 2016.
___________________________________________With� Mehmet Haluk Seng�n (University of Sheffield), I search for modular forms whichcannot be obtained as lifts with the Langlands base change procedure, see our paperOn Level One Cuspidal Bianchi Modular Forms, joint with Mehmet Haluk Seng�nLMS Journal of Computation and Mathematics, volume 16 (2013), pp. 187-199.Currently, we are evaluating the results from numerical experiments for which I have obtained an ICHEC grant of 900,000 processor hours (electricity and material replacement costs of 30,000 Euro). We are preparing these results for publication in the LMFDB database.______________________________________________With Fabio Perroni (University of Trieste), I study Ruan�s conjecture on crepant resolutions for stringy orbifold cohomology. I have recently submitted a paper about this, Chen Ruan orbifold cohomology of the Bianchi orbifolds,http: hal.archives-ouvertes.fr hal-00627034 _________________________________________________ With Nicolas Bergeron (Universit� Paris 6) and Aurel Page (University of Warwick) (link to his previous homepage), I study the asymptotics of the Abelianization of the Bianchi groups, based on my paperHigher torsion in the Abelianization of the full Bianchi groups, LMS J. of Computation and Mathematics.__________________________________________________With Ruben Sanchez-Garcia (University of Southampton), Jean-Francois Lafont (Ohio State University) and Ivonne Ortiz (Miami University, Oxford, Ohio), I compute the Bredon homology and equivariant K-homology of all Coxeter tetrahedral groups, as well as more Coxeter hyperbolic groups, using an adaptation to Bredon homology of my Torsion Subcomplex Reduction technique.�