# University of Toronto's Symplectic Geometry Seminar

Jan. 13 2003, 2pm

SS5017A

## 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).