Smooth geometric shapes
(manifolds) are extensively studied and used in mathematic,
physics, computer science, etc. Often a geometric shape (variety), for example,
the one given by an equation f(x,y,z)
in the three-dimensional space, is singular: it may contain self-intersections,
cusps, pinch points, etc. Resolution of singularities is a classical branch of
algebraic geometry which studies how a variety can be modified to a smooth
manifold, and such modifications are a very useful tool for working with
general varieties.
First resolution of
singularities over fields containing the rational numbers (the characteristic
zero case) was obtained by Hironaka in 1964 and
awarded him a Fields medal. Until very recently, Hironaka's
method was polished and improved but mathematicians have known essentially a
unique resolution algorithm.
The first half of this
project addresses the characteristic zero case and its goal is to obtain new
methods, which are faster, use no memory, also apply to maps between manifolds,
take into account logarithmic structures, etc. This
provides new results even over the field of complex numbers and might have
applications also outside of mathematics.
The second half of the
project plans to study the arithmetic cases of positive and mixed
characteristic. After extensive work of many mathematicians only the cases of
dimensions 1,2 and 3 were solved so far. I hope that the new tools designed in
the first half of the project will be useful here too. These goals of this
project are very important for various fields of mathematics, such as
arithmetical geometry.