... 1.The homological mirror symmetry (HMS) conjecture. ... Here the following results were obtained by the author.: 1) a proof of Kontsevich-Soibelman conjecture which allows one to rigorously define Fukaya A -category; 2) a proof of HMS conjecture for curves of higher genus treated symplectically; 3) a proof of HMS conjecture for punctured spheres treated symplectically (joint with A. Abouzaid, D. Auroux, L. Katzarkov and D. Orlov, in preparation). ... 2.Exceptional col lections. ...