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 field of interest is the representation theory of algebras. I am especially interested in quivers, cluster algebras, and quantum groups. My Ph.D. studies have focused on the connection between dual canonical bases and quantum cluster algebras. For an overview please look at two posters I have created for the annual poster exhibition of the Bonn International Graduate School in Mathematics (BIGS). More recently, I became interested in ring theoretic properties of cluster algebras. For an overview please loook at a slide I made for a five-minutes talk.

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**, to appear in Journal of Algebra. 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. [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 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.