Abstract

Theory of quantum multiplication and Gromov-Witten invariants of a smoothprojective algebraic variety X deals with enumerative questions concerningcurves in X. A long time ago physicists E. Witten, R. Dijkgraaf, E. Verlinde, andH. Verlinde discovered a remarkable system of non-linear differential equations on the generating series for Gromov-Witten invariants. In particular, onecan calculate the number of rational curves of degree N in projective planepassing through generic 3N-1 points via a certain solution of the Painleve Viequation.

The topic of my lectures is an application of the theory of quantum multiplica-tion to birational geometry (joint work with L. Katzarkov, T. Pantev, and TYu), based on the recent blowup formula by H. Iritani. In particular, we solveda long-standing conjecture on the non-rationality of a generic cubic hypersur.face in 5-dimensional projective space..

In my first lecture, I'll give a brief introduction to the theory of quantum cohomology and introduce a notion of atoms which play the central role in rationality questions. In the second lecture, I'll propose a series of conjectures re.lating atoms and derived category of coherent sheaves.

About the Speaker

Professor Kontsevich is one of the leading mathematicians and mathematicaphysicists of the 20th and 21st centuries with outstanding contributions to algebraic geometry, algebraic topology, and geometric aspects of mathematicalphysics, such as knot theory, quantization and mirror symmetry.He received the Henri Poincaré Prize in 1997, the Fields Medal in 1998, theCrafoord Prize in 2008, the Shaw Prize and Fundamental Physics Prize in2012, and the Breakthrough Prize in Mathematics in 2014.