KTH/SU Mathematics Colloquium

05-11-16

Mattias Jonsson, KTH

Singularities in geometry, complex dynamics and analysis

A curve in the plane is said to have normal crossings if it looks like $y=0$ or $xy=0$ in suitable local coordinates $(x,y)$ at every point. An old theorem, going back at least to Max Noether, asserts that every curve in the plane---no matter how singular---can be turned into one having normal crossings by a sequence of point blowups, that is, a composition of coordinate changes of the form $(x,y)\to(x,xy)$. This result, known as embedded resolution of curve singularities, was later vastly generalized in the celebrated work by Hironaka in the 1960's. I will discuss two recent (two-dimensional) extensions of Noether's theorem obtianed jointly with C. Favre (CNRS, Paris VII). The first one concerns resolutions of singularities of positive closed currents (these can be viewed as "limits" of curves). The second is of dynamic nature and applies to fixed point germs $f:(\mathbf{C}^2,0)\to(\mathbf{C}^2,0)$.