Tropical geometry is a systematic development of the fundamental concept, "degenerations". In this talk, I will make this statement precise in the algebraic view point, namely ''Initial ideals".

We apply the tropical approach to the classical objects in algebraic geometry, "Severi varieties". Severi varieties are spaces whose points correspond to the plane curves with a given number of nodal singular points. In this tropical approach, we can obtain purely combinatorial results on Severi varieties which involve subdivisions of polygons.

I shall finish proof of the Hirzebruch signature theorem today, using index theorem. I shall give a few elementary applications, including divisibility results for Pontryagin numbers and a proof of the fact that CP^2 does not admit any spin structure. On Friday we shall see a more dramatic application in the talk by Mincong.

Why the cylinder is a straight line (thoughts on a modern interpretation of affine Kac-Moody Lie algebras)

One interesting application of Hirzebruch signature theorem is the construction of exotic differential structure on $S^7$. In this talk I will first show the construction of the exotic $S^7$, which is the sphere bundle of the 4 dimensional vector bundle over $S^4$ by using of the first Pontrjagin class and Euler class. Then for proving it has an exotic differential structure, we use Hirzebruch signature theorem to construct a invariant and compute this invariant for standard $S^7$ and our sphere bundle to see they are different.