University of Toronto's Symplectic Geometry Seminar

Jan. 13 2003, 2pm

Grisha Mikhalkin

University of Utah

Toric surfaces, Gromov-Witten invariants and tropical algebraic geometry

Abstract: The talk presents a new formula for the Gromov-Witten invariants of arbitrary genus in the projective plane as well as for the related enumerative invariants in other toric surfaces. The answer is given in terms of certain lattice paths in the relevant Newton polygon. The length of the paths turns out to be responsible for the genus of the holomorphic curves in the count. The formula is obtained by working in terms of the so-called tropical algebraic geometry. This version of algebraic geometry is simpler than its classical counterpart in many aspects. In particular, complex algebraic varieties themselves become piecewise-linear objects in the real space. The transition from the classical geometry is provided by consideration of the "large complex limit" or "worst possible degeneration of complex structure" (also known as "dequantization" or "patchworking" in some other areas of Mathematics).