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 $f:(\mathbf{C}^2,0)\to(\mathbf{C}^2,0)$.