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. |