Profile
Profile

Prof. Dr. Jan Schröer

E-mail: schroer(at)math.uni-bonn.de
Phone: +49 228 73 7786
Homepage: http://www.math.uni-bonn.de/~schroer/
Room: 3.006
Location: Mathematics Center
Institute: Mathematical Institute
Research Areas: Research Area DE (Leader)
Research Area C (until 10/2012)
Former Research Area E
Birthdate: 04.May 1971
Mathscinet-Number: 633566

Academic Career

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

Research Profile

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.

Research Projects and Activities

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

Contribution to Research Areas

Research Area C (until 10/2012)
We established the polynomiality of generalized T-systems. These are recursively defined systems of equations arising in statistical mechanics but also in the representation theory of quantum affine algebras. We proved that T-systems 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 T-system.
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 Kac-Moody 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 cluster-tilted algebras, and Calabi-Yau 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 Kac-Moody groups. This was achieved by a categorification of cluster algebras using Calabi-Yau categories associated to Weyl group elements of Kac-Moody 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 cluster-tilted algebras.
Research Area DE

Selected Publications

[1] Christof Geiß, Bernard Leclerc, Jan Schröer
Quivers with relations for symmetrizable Cartan matrices III: Convolution algebras
Represent. Theory , 20: : 375--413
2016
ISSN: 1088-4165
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 Labardini-Fragoso, Jan Schröer
The representation type of Jacobian algebras
Adv. Math.
, 290: : 364--452
2016
ISSN: 0001-8708
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): 337--397
2013
ISSN: 1022-1824
DOI: 10.1007/s00029-012-0099-x
[5] Christof Geiss, Bernard Leclerc, Jan Schröer
Generic bases for cluster algebras and the Chamber ansatz
J. Amer. Math. Soc. , 25: (1): 21--76
2012
ISSN: 0894-0347
DOI: 10.1090/S0894-0347-2011-00715-7
[6] Christof Geiß, Bernard Leclerc, Jan Schröer
Kac-Moody groups and cluster algebras
Adv. Math. , 228: (1): 329--433
2011
ISSN: 0001-8708
DOI: 10.1016/j.aim.2011.05.011
[7] Christof Geiß, Bernard Leclerc, Jan Schröer
Rigid modules over preprojective algebras
Invent. Math. , 165: (3): 589--632
2006
ISSN: 0020-9910
DOI: 10.1007/s00222-006-0507-y
[8] Christof Geiss, Bernard Leclerc, Jan Schröer
Semicanonical bases and preprojective algebras
Ann. Sci. École Norm. Sup. (4) , 38: (2): 193--253
2005
ISSN: 0012-9593
DOI: 10.1016/j.ansens.2004.12.001
[9] William Crawley-Boevey, Jan Schröer
Irreducible components of varieties of modules
J. Reine Angew. Math. , 553: : 201--220
2002
ISSN: 0075-4102
DOI: 10.1515/crll.2002.100
[10] Jan Schröer
On the infinite radical of a module category
Proc. London Math. Soc. (3) , 81: (3): 651--674
2000
ISSN: 0024-6115
DOI: 10.1112/S0024611500012600

Publication List

Selected Invited Lectures

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

Offers

2008

University of Dortmund (W3)

2009

University of Bielefeld (W3)

Selected PhD students

Phillip Lampe (2010): “Quantum cluster algebras and the dual canonical basis”,
now Postdoc, University of Bielefeld

Supervised Theses

  • Diplom theses: 14, currently 1
  • PhD theses: 4, currently 2
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