Related papers: Singularities and Characteristic Classes for Diffe…
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 first described using words, and how several…
Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schuermann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern…
In this paper, we prove the Eichler cohomology theorem of weakly parabolic generalized modular forms of real weights on subgroups of finite index in the full modular group. We explicitly establish the isomorphism for large weights by…
We obtain restrictions on the rational homotopy types of mapping spaces and of classifying spaces of homotopy automorphisms by means of the theory of positive weight decompositions. The theory applies, in particular, to connected components…
In analogy to what happens in finite dimensions we state the Normal Form Theorem for k-singularities, introduced in the previous paper of the series. By means of that we study the local behaviour near a singularity i.e. we deduce local…
We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the…
We adapt algorithms for resolving the singularities of complex algebraic varieties to prove that the natural map of homology theories from complex bordism to the bordism theory of complex derived orbifolds splits. In equivariant stable…
These are extended notes of a course given at Tulane University for the 2015 Clifford Lectures. Their aim is to present structure results for group schemes of finite type over a field, with applications to Picard varieties and automorphism…
Special generic maps are generalizations of Morse functions with exactly two singular points on spheres and canonical projections of unit spheres. They restrict the manifolds of the domains strongly in considerable cases and are important…
For any (n+1)-dimensional weight vector {\chi} of positive integers, the weighted projective space P(\chi) is a projective toric variety, and has orbifold singularities in every case other than CP^n. We study the algebraic topology of…
We consider the problem of computing the Euler characteristic of an abstract simplicial complex given by its vertices and facets. We show that this problem is #P-complete and present two new practical algorithms for computing Euler…
To an arbitrary variety over a field of characteristic zero, we associate a complex of Chow motives, which is, up to homotopy, unique and bounded. We deduce that any variety has a natural Euler characteristic in the Grothendieck group of…
We construct characteristic classes for singular algebraic varieties in motivic Borel-Moore homology, extending the motivic Euler class of the tangent bundle defined for smooth varieties. The two classes we define refine, in the setting of…
We study the topology of complements of caustics of function singularities of low codimensions, in particular 1) complete the enumeration of connected components of the complements of caustics of {\em simple} (in the sense of V.Arnold)…
We study the notion of singular tropical hypersurfaces of any dimension. We characterize the singular points in terms of tropical Euler derivatives and we give an algorithm to compute all singular points. We also describe non-transversal…
Waring problem for homogeneus forms asks for additive decomposition of a form $f$ into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this paper I answer this question when the degree of…
The singularity confinement test is very useful for isolating integrable cases of discrete-time dynamical systems, but it does not provide a sufficient criterion for integrability. Quite recently a new property of the bilinear equations…
We construct a singular homology theory on the category of schemes of finite type over a Dedekind domain and verify several basic properties. For arithmetic schemes we construct a reciprocity isomorphism between the integral singular…
We use a version of localization in equivariant cohomology for the norm-square of the moment map, described by Paradan, to give several weighted decompositions for simple polytopes. As an application, we study Euler-Maclaurin formulas.
We study the topology of some simple infinite dimensional singularities arising from spaces of \emph{algebraic formal loops}. We prove that in some simple cases the natural analogue of nearby cycles cohomology for a function on the loop…