Impact of Cluster Algebras

At present, more than 50 researchers and around 20 Ph.D. students and (recent) postdocs distributed all over the world are working on topics closely related to cluster algebras.

It is possibly due to the fact that cluster like structures were for quite a while implicit in many areas of mathematics that this field has developed with such an amazing speed in the last years. By the end of October 2007, MathSciNet lists 100 articles published since 2002 related to cluster algebras whereas the arXiv server lists 134 preprints concerning this topic posted in the same period.

Year	Number of preprints
2001	4
2002	5
2003	8
2004	17
2005	33
2006	39
2007	29 (October)

Many of the articles were published in journals with the highest reputation, as for example:

• S. Fomin, A. Zelevinsky: $Y$-systems and generalized associahedra. Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
• S. Fomin, A. Zelevinsky: Cluster algebras. II. Finite type classification. Invent. Math. 154 (2003), no. 1, 63-121.
• C. Geiss, B. Leclerc, J. Schröer: Rigid modules over preprojective algebras. Invent. Math. 165 (2006), no. 3, 589-632.
• V. Fock, A. Goncharov: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. No. 103 (2006), 1-211.
• P. Caldero, B. Keller: From triangulated categories to cluster algebras. To appear in Invent. Math.

It is worth mentioning that the paper of Fock and Goncharov takes a whole volume of Publ. Math. IHES, which only publishes 4 volumes each year.

The paper by Buan, Marsh, Reiten, Reineke and Todorov: Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572-618 is featured in Essential Science Indicators
http://www.esi-topics.com/fbp/2007/october07-Buan_etal.html
as ``October 2007 Fast Breaking Paper''.

It is also worth mentioning that Fields medalist Maxim Kontsevich incorporates the concepts of cluster algebras in his recent work with Y. Soibelman on Donaldson-Thomas invariants.

To date, at more than 20 conferences cluster algebras played an important role or were indeed the topic. Notably this was the case at the last two ICRA conferences (Mexico 2003 and Poland 2007), which are traditionally conferences dedicated to the representation theory of finite dimensional algebras. Parallel to this development at many places all over the world working seminars on different aspects of cluster algebras were held. See S. Fomin's cluster portal mentioned in the Overview 1 for details on these activities.

According to in-cites, A. Zelevinsky featured in 2006 the highest percent increase in total citations in his field, see:
http://www.in-cites.com/scientists/AndreiZelevinksy.html

Osama Iyama (Nagoya) received the 2007 ICRA award ``for his original and influential work on developing a higher theory of almost split sequences and Auslander correspondence, and his subsequent work on Calabi-Yau categories, which have strong connections with the Cluster algebras of Fomin-Zelevinsky.''

This short overview shows that the field has developed with amazing speed and attracted the attention of a great variety of prestigious researchers all over the world.

We are planning a truly comprehensive event bringing together the ideas and results from different fields connected with cluster algebras. The preceding events were on a much smaller scale, and only treated separate aspects. Geographically, this will be the first event of this kind targeting mathematicians from Mexico and other Latin American countries. Moreover, it should also be stressed that the field is still sufficiently new so that students can catch up with recent developments in a reasonable time.