

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 C (until 10/2012) We established the polynomiality of generalized Tsystems. These are recursively defined systems of equations arising in statistical mechanics but also in the representation theory of quantum affine algebras. We proved that Tsystems also play an important role in the categorification of cluster algebras arising in Lie theory and (using representation theory of preprojective algebras) we showed that there exist polynomial solutions to any Tsystem.  Former Research Area E Starting in 2005, in an extensive and ongoing project with Bernard Leclerc (Caen) and Christof Geiss (UNAM, Mexico City) we established or strengthened connections between the representation theory of KacMoody Lie algebras, the representation theory of quivers and preprojective algebras, the geometry of Lusztig's nilpotent varieties, Lusztig's semicanonical basis of enveloping algebras, Fomin and Zelevinsky's theory of cluster algebras, module varieties of clustertilted algebras, and CalabiYau categories of dimension 2. As one application we managed to construct semicanonical bases for a large class of cluster algebras, and we proved that this class of cluster algebras arises naturally in the theory of KacMoody groups. This was achieved by a categorification of cluster algebras using CalabiYau categories associated to Weyl group elements of KacMoody Lie algebras. We proved that our semicanonical bases of cluster algebras can be seen as generic bases, where “generic” refers to certain module varieties of clustertilted algebras. 


[ 1] 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[ 2] Christof Geiss, Bernard Leclerc, Jan Schröer
Quivers with relations for symmetrizable Cartan matrices I: Foundations arXiv preprint arXiv:1410.1403 2014[3] 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 [ 4] C. Geiß, B. Leclerc, J. Schröer
Cluster structures on quantum coordinate rings Selecta Math. (N.S.) , 19: (2): 337397 2013 DOI: 10.1007/s000290120099x[ 5] 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[ 6] 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[ 7] Christof Geiß, Bernard Leclerc, Jan Schröer
Rigid modules over preprojective algebras Invent. Math. , 165: (3): 589632 2006 DOI: 10.1007/s002220060507y[ 8] 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[ 9] William CrawleyBoevey, Jan Schröer
Irreducible components of varieties of modules J. Reine Angew. Math. , 553: : 201220 2002 DOI: 10.1515/crll.2002.100[ 10] 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) 


Philipp Lampe (2010): “Quantum cluster algebras and the dual canonical basis”,
now Postdoc, University of Bielefeld
Jan Geuenich (January/February 2017): “Quiver Mutations and Potentials”,
afterwards Postdoc, University of Bielefeld


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


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