Leibniz Universität Hannover

Kolloquium der Fakultät für Mathematik und Physik
Dienstag, 23.6.2009, ab 17:15 Uhr im Kleinen Physik-Hörsaal F 342

The periodicity conjecture via 2-Calabi-Yau categories

Prof. Dr. Bernhard Keller (Paris)

The periodicity conjecture was formulated in mathematical physics at the beginning of the 1990s, in the work of Zamolodchikov, Kuniba-Nakanishi and Ravanini-Valleriani-Tateo. It asserts that a certain discrete dynamical system associated with a pair of Dynkin diagrams is periodic and that its period divides the double of the sum of the Coxeter numbers of the two diagrams. The conjecture was proved by Frenkel-Szenes and Gliozzi-Tateo for the pairs (A_n, A_1), by Fomin-Zelevinsky in the case where one of the diagrams is A_1 and by Volkov and independently Szenes when both diagrams are of type A. We will sketch some of the ideas of the proof in the general case. It is based on Fomin-Zelevinsky's work on cluster algebras and on the theory relating cluster algebras to triangulated 2-Calabi-Yau categories. An important role is played by Coxeter transformations.