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| 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: 2003-2004) | | 2003 - 2004 | DFG Research Fellow, University of Leeds, England, UK | | Since 2005 | Professor (W2), University of Bonn |
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My research area is the representation theory of finite-dimensional algebras and quivers. I focus particularly on the numerous deep connections to the representation theory of Kac-Moody Lie algebras. Various crucial geometric constructions (Nakajima quiver varieties, Kashiwara-Saito's geometric crystal graphs, semicanonical bases for enveloping algebras, generic bases for cluster algebras) can only be realized for symmetric Kac-Moody 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, non-symmetric cases. This should also trigger a new research field inside the classical representation theory of finite-dimensional algebras, namely the study of generalized modulated graphs. I'm also interested in classical homological conjectures for finite-dimensional 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 finite-dimensional algebras whose module category contains all module categories of all finite-dimensional 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 Auslander-Reiten Theory.
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DFG Collaborative Research Center Transregio SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties”
Principal Investigator
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Research Area DE
My research group works on the representation theory of finite-dimensional 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 Kac-Moody 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 Kac-Moody 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 Auslander-Reiten theory, initiated by Iyama and others involved in the categorification project of Fomin-Zelevinsky cluster algebras. |
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[ 1] Christof Geiss, Bernard Leclerc, Jan Schröer
Quivers with relations for symmetrizable Cartan matrices I: Foundations
Invent. Math. , 209: (1): 61--158
2017
DOI: 10.1007/s00222-016-0705-1[ 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: : 375--413
2016
DOI: 10.1090/ert/487[ 4] Christof Geiß, Daniel Labardini-Fragoso, Jan Schröer
The representation type of Jacobian algebras
Adv. Math. , 290: : 364--452
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): 337--397
2013
DOI: 10.1007/s00029-012-0099-x[ 6] Christof Geiss, Bernard Leclerc, Jan Schröer
Generic bases for cluster algebras and the Chamber ansatz
J. Amer. Math. Soc. , 25: (1): 21--76
2012
DOI: 10.1090/S0894-0347-2011-00715-7[ 7] Christof Geiß, Bernard Leclerc, Jan Schröer
Kac-Moody groups and cluster algebras
Adv. Math. , 228: (1): 329--433
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): 589--632
2006
DOI: 10.1007/s00222-006-0507-y[ 9] Christof Geiss, Bernard Leclerc, Jan Schröer
Semicanonical bases and preprojective algebras
Ann. Sci. École Norm. Sup. (4) , 38: (2): 193--253
2005
DOI: 10.1016/j.ansens.2004.12.001[ 10] William Crawley-Boevey, Jan Schröer
Irreducible components of varieties of modules
J. Reine Angew. Math. , 553: : 201--220
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): 651--674
2000
DOI: 10.1112/S0024611500012600 |
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| 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 Mittag-Leffler Institute, Stockholm, Sweden |
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| 2008 | University of Dortmund (W3) | | 2009 | University of Bielefeld (W3) |
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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
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- Master theses: 29, currently 7
- Diplom theses: 16
- PhD theses: 7, currently 2
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