BIREP – Representations of finite dimensional algebras at Bielefeld

Faculty of Mathematics

Bielefeld University

Postfach 100 131

D-33501 Bielefeld

Office: V5-229

Phone: +49 521 106 5036

Office hours: Tuesdays, 14:00 - 16:00

Email: lampe@math.uni-bielefeld.de

I am a postdoc in the Bielefeld Representation Theory Research group (BIREP) under Professor Dr. Henning Krause at the Faculty of Mathematics at Bielefeld University.

I graduated from the University of Bonn, where I used to be a student of Professor Dr. Jan Schröer in the Algebra and Representation Theory Research group.

My research lies at the interface between algebra and combinatorics. In particular, I am interested in Fomin-Zelevinsky's theory of cluster algebras, in Lie theory, and in the representation theory of quivers.

Here you can find a short curriculum vitae, and informations about my travels including selected talks.

**Maximum antichains in posets of quiver representations**. This joint work with Florian Gellert concerns posets of quiver representations. Sperner's theorem asserts that the poset of subrepresentations of a given set-valued representation of A_{1} is Sperner. We prove Sperner theorems for subrepresentations posets attached to set-valued representations of certain star-shaped quivers and to certain linear representations of A_{2}. Moreover, we construct maximum antichains in monomorphism posets of indecomposable representations for certain orientations of A_{n}. [arXiv]
**On the approximate periodicity of sequences attached to noncrystallographic root systems**, to appear in Experimental Mathematics. We study Fomin-Zelevinsky's mutation rule in the context of noncrystallographic root systems. Whereas the cluster variables in a finite-type cluster algebra of rank 2 (attached to a crystallographic root system) form a periodic sequence, we observe an approximate periodicity for noncrystallographic root systems of rank 2. Moreover, we describe matrix mutation classes for type H3 and H4. [arXiv]
**Diophantine equations via cluster transformations**, Journal of Algebra **462** (2016), 320-337. Motivated by Fomin and Zelevinsky's theory of cluster algebras we introduce a variant of the Markov equation; we show that all natural solutions of the equation arise from an initial solution by cluster transformations. [Journal], [arXiv]
**The divisor class group of a cluster algebra**, Oberwolfach Reports **8** (2014), 484-485. The divisor class group is a useful tool to decide whether a given algebra is a unique factorization domain. We use the divisor class group to study the ring theoretic nature of cluster algebras. In particular, we give a sufficient and computer-checkable criterion to decide whether an acyclic cluster algebra is a UFD. [Journal]
**Quantisation Spaces of Cluster Algebras**, January 2014. This joint work with Florian Gellert concerns the question: When does a cluster algebra have a quantization and how unique is it? Florian maintains a complementary webpage. [arXiv]
**Acyclic cluster algebras from a ring theoretic point of view**, October 2012. Many authors have studied Fomin-Zelevinsky's cluster algebras combinatorially (what do Laurent coefficients count?), representation-theoretically (what triangulated categories are they a shadow of?) and linear algebraically (what are good bases?). We want to know: What is a cluster algebra as an algebra? We focus on two questions: When is a cluster algebra a unique factorization domain? What are irreducible elements? [arXiv]
**Quantum cluster algebras and dual canonical bases**, Oberwolfach Reports **8** (2011), no. 10, 564-565. A short survey about the connections between Lusztig's canonical basis and Fomin-Zelevinsky's cluster algebra. [Journal]
**Quantum cluster algebras of type A and the dual canonical basis**, Proceedings of the London Mathematical Society **108** (2014), no. 1, 1-43. The article concerns the subalgebra U_{v}(w) of the quantized universal enveloping algebra of the complex Lie algebra sl_{n+1} associated with a particular Weyl group element of length 2n. We verify that U_{v}(w) can be endowed with the structure of a quantum cluster algebra of type A_{n}. [Journal], [arXiv].
**A quantum cluster algebra of Kronecker type and the dual canonical basis**, International Mathematics Research Notices **2011** (2011), no. 13, 2970-3005. The article studies the quantized cluster algebra structure induced by the dual canonical basis associated with a terminal module over the path algebra of the Kronecker quiver. [Journal], [arXiv]

Im Wintersemester 2016/17 leite ich eine Übung zur Vorlesung Lineare Algebra I bei Henning Krause. [eKVV]

Im Sommersemester 2016 biete ich ein Proseminar zu Algebraischen Methoden an. [eKVV]

Im Wintersemester 2015/16 leite ich den Übungsbetrieb zur Vorlesung Darstellungstheorie von Algebren III bei Henning Krause. [eKVV]

Im Sommersemester 2015 leite ich den Übungsbetrieb zur Vorlesung Darstellungstheorie von Algebren II bei Dieter Vossieck [eKVV], [Evaluation]; ferner organisiere ich mit Mario Kieburg ein Seminar Mathematische Physik. [eKVV]

Im Wintersemester 2014/15 leite ich den Übungsbetrieb zur Vorlesung Darstellungstheorie von Algebren I bei Henning Krause. [eKVV], [Evaluation]

Im Sommersemester 2014 biete ich zusammen mit Henning Krause ein Bachelorseminar zur Algebra [eKVV] an; ferner leite ich eine Übung zur Vorlesung Lineare Algebra II bei Stefan Bauer. [eKVV], [Evaluation]

Im Wintersemester 2013/2014 lese ich die Vorlesung Cluster-Algebren. [eKVV], [Evaluation]

Im Sommersemester 2013 biete ich ein Bachelor-Seminar zur Kombinatorik und Gruppentheorie an. [eKVV]

Im Wintersemester 2012/2013 lese ich die Vorlesung Vertiefung Gruppentheorie. [eKVV], [Evaluation]

Im Sommersemester 2012 biete ich zusammen mit Henning Krause und Nils Mahrt ein Proseminar: Einführung in Cluster-Algebren [eKVV] an; ferner leite ich eine Übung zur Vorlesung Lineare Algebra II bei Henning Krause. [eKVV], [Evaluation]

Im Wintersemester 2011/2012 leite ich eine Übung zur Vorlesung Lineare Algebra I bei Henning Krause. [eKVV]

Im Sommersemester 2011 leite ich den Übungsbetrieb zur Vorlesung Einführung in die Darstellungstheorie bei Henning Krause. [eKVV]

I am a coorganizer of the Maurice Auslander Memorial Workshop, which was held in Bielefeld on November 13-15, 2014.

Mathe+ ist die Mathe-AG für Schülerinnen und Schüler in der Uni Bielefeld.

The BIREP group hosted the Workshop and International Conference on Representations of Algebras (ICRA 2012) in August 2012.

I was a coorganizer of the 16th NWDR Workshop which was held in Bielefeld on July 06, 2012.

Together with colleagues from Bielefeld I have organized a Summer School on Polynomial Representations of the General Linear Group and a Summer School on Koszul duality which were held in Bad Driburg in August 2011 and August 2015.

A selection of links that you might find interesting.

- [FDlist] Miscellaneous items related to the representation theory of finite-dimensional algebras.
- [Cluster Algebras Portal] A website maintained by Sergey Fomin with a large collection of links on cluster algebras.
- [Quiver Mutation in Java] An applet created by Bernhard Keller; it implements Fomin-Zelevinsky's quiver mutation.
- [My diploma thesis] from November 2007 and [my Ph.D. thesis] from February 2011.
- [Genealogy] Me on the Mathematics Genealogy Project.
- [arXiv] Me on arXiv.
- [Detexify] A device for finding LaTex symbols.
- [Mathematik-Olympiade] A competition for high school students I am active in.