

1997  Dr. math., University of Bielefeld (advisor: C.M. Ringel)  1997  1998  Research Fellow, University of Bielefeld  1998  1999  DAAD Postdoctoral Fellow, National Autonomous University of Mexico, Mexico City, Mexico  1999  2000  Research Fellow, University of Bielefeld  2000  2005  Lecturer/Reader, University of Leeds, England, UK (Temporary leave: 20032004)  2003  2004  DFG Research Fellow, University of Leeds, England, UK  Since 2005  Professor (W2), University of Bonn 


My research area is the representation theory of finitedimensional algebras and quivers. I focus particularly on the numerous deep connections to the representation theory of KacMoody Lie algebras. Various crucial geometric constructions (Nakajima quiver varieties, KashiwaraSaito's geometric crystal graphs, semicanonical bases for enveloping algebras, generic bases for cluster algebras) can only be realized for symmetric KacMoody Lie algebras. In an extensive project with Geiss and Leclerc, we are currently developing a general framework for all of the above (using quivers with loops and relations) which covers all symmetrizable, nonsymmetric cases. This should also trigger a new research field inside the classical representation theory of finitedimensional algebras, namely the study of generalized modulated graphs. I'm also interested in classical homological conjectures for finitedimensional algebras.
The project described above will keep us busy for several years. A related topic of future investigation is the representation theory of wild quivers or more generally of wild algebras. Roughly speaking these are finitedimensional algebras whose module category contains all module categories of all finitedimensional algebras via suitable embedding functors. This fractal behaviour of module categories is quite common and should also occur in many other areas of mathematics. As a research group we would like to “start again from zero” and develop a vision for the future of this research area. The methods will include Schofield induction, Kerner bijections and AuslanderReiten Theory.


DFG Collaborative Research Center Transregio SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties”
Principal Investigator


Research Area DE My research group works on the representation theory of finitedimensional algebras and quivers. In an extensive joint work with Geiss and Leclerc, we introduced a a new class of quivers with relations (see [1]) which yields a strong link to the representation theory of symmetrizable KacMoody Lie algebras. For example, in a current preprint arXiv:1702.07570 [2] we use these quivers with relations to obtain a geometric realization of the crystal graph of any symmetrizable KacMoody algebra. (Such results were before only available for the symmetric case.) We also laid the groundwork for a new direction in the representation theory of modulated graphs, working not only with bimodules over division rings but over more general rings. Jasso (long term postdoc in Bonn) developed together with Külshammer a theory of Higher Nakayama algebras, contributing to the development of Higher AuslanderReiten theory, initiated by Iyama and others involved in the categorification project of FominZelevinsky cluster algebras. 


[ 1] Christof Geiss, Bernard Leclerc, Jan Schröer
Quivers with relations for symmetrizable Cartan matrices I: Foundations Invent. Math. , 209: (1): 61158 2017 DOI: 10.1007/s0022201607051[ 2] Christof Gei, Bernard Leclerc, Jan Schröer
Quivers with relations for symmetrizable Cartan matrices IV: Crystal graphs and semicanonical functions eprint, arXiv:1702.07570 2017[3] Christof Geiß, Bernard Leclerc, Jan Schröer
Quivers with relations for symmetrizable Cartan matrices III: Convolution algebras Represent. Theory , 20: : 375413 2016 DOI: 10.1090/ert/487 [ 4] Christof Geiß, Daniel LabardiniFragoso, Jan Schröer
The representation type of Jacobian algebras Adv. Math. , 290: : 364452 2016 DOI: 10.1016/j.aim.2015.09.038[ 5] C. Geiß, B. Leclerc, J. Schröer
Cluster structures on quantum coordinate rings Selecta Math. (N.S.) , 19: (2): 337397 2013 DOI: 10.1007/s000290120099x[ 6] Christof Geiss, Bernard Leclerc, Jan Schröer
Generic bases for cluster algebras and the Chamber ansatz J. Amer. Math. Soc. , 25: (1): 2176 2012 DOI: 10.1090/S089403472011007157[ 7] Christof Geiß, Bernard Leclerc, Jan Schröer
KacMoody groups and cluster algebras Adv. Math. , 228: (1): 329433 2011 DOI: 10.1016/j.aim.2011.05.011[ 8] Christof Geiß, Bernard Leclerc, Jan Schröer
Rigid modules over preprojective algebras Invent. Math. , 165: (3): 589632 2006 DOI: 10.1007/s002220060507y[ 9] Christof Geiss, Bernard Leclerc, Jan Schröer
Semicanonical bases and preprojective algebras Ann. Sci. École Norm. Sup. (4) , 38: (2): 193253 2005 DOI: 10.1016/j.ansens.2004.12.001[ 10] William CrawleyBoevey, Jan Schröer
Irreducible components of varieties of modules J. Reine Angew. Math. , 553: : 201220 2002 DOI: 10.1515/crll.2002.100[ 11] Jan Schröer
On the infinite radical of a module category Proc. London Math. Soc. (3) , 81: (3): 651674 2000 DOI: 10.1112/S0024611500012600





2000  Plenary lecture at the ICRA, Beijing, China  2002  Plenary lecture at the ICRA, Toronto, ON, Canada  2004  Plenary lecture at the ICRA, Pátzcuaro, Mexico  2005  Morning Speaker at the British Mathematical Colloquium, Liverpool, England, UK  2011  Lecture at the Abel Symposium, Balestrand, Norway  2013  Mathematisches Kolloquium, Bern, Switzerland  2014  Lecture series at the ICRA, Sanya, China  2015  Lecture at the MittagLeffler Institute, Stockholm, Sweden 


2008  University of Dortmund (W3)  2009  University of Bielefeld (W3) 


Jan Geuenich (January/February 2017): “Quiver Mutations and Potentials”,
afterwards Postdoc, University of Bielefeld
Sondre Kvamme (October 2017): “Comonads and Gorenstein Homological Algebra”,
now Postdoc, Département de Mathématiques d’Orsay


 Master theses: 29, currently 7
 Diplom theses: 16
 PhD theses: 7, currently 2


Download Profile 