Assume that, in the near future, someone can prove resolution of singularities in arbitrary
characteristic and dimension. Then one may want to know why the case of positive …

The Łojasiewicz inequalities for real analytic functions on Euclidean space were first proved
by Stanisław Łojasiewicz (1959, 1965) using methods of semianalytic and subanalytic sets …

Our goal in this work is to develop aspects of Bialynicki-Birula and Morse-Bott theory that
can be extended from the classical setting of smooth manifolds to that of complex analytic …

We study a natural generalization of transversally intersecting smooth hypersurfaces in a
complex manifold: hypersurfaces whose components intersect in a transversal way but may …

This manuscript originates from a series of lectures the author1 gave at the Clay Summer
School on Resolution of Singularities at Obergurgl in the Tyrolean Alps in June 2012. A …

The objective of this article is to give an effective algebraic characterization of normal
crossing hypersurfaces in complex manifolds. It is shown that a divisor (= hypersurface) has …

We propose two local invariants for the inductive proof of the embedded resolution of purely
inseparable surface singularities of order equal to the characteristic. The invariants are built …

We give an introduction to the McKay correspondence and its connection to quotients of Cn
by finite reflection groups. This yields a natural construction of noncommutative resolutions …

We discuss to what extent the local techniques of resolution of singularities over fields of
characteristic zero can be applied to improve singularities in general. For certain interesting …

In this text I present some problems which led to the introduction of special kinds of graphs
as tools for studying singular points of algebraic surfaces. I explain how such graphs were …