Related papers: Simultaneous Resolution of Singularities
Let $M$ be a complex- or real-analytic manifold, $\theta$ be a singular distribution and $\mathcal{I}$ a coherent ideal sheaf defined on $M$. We prove the existence of a local resolution of singularities of $\mathcal{I}$ that preserves the…
The Raychaudhuri equation has seen extensive use in general relativity, most notably in the development of various singularity theorems. In this rather technical article we shall generalize the Raychaudhuri equation in several ways. First…
We prove Grothendieck's Conjecture on Resolution of Singulari-ties for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X = SpecA, A a reduced complete Noetherian local ring of…
In the classification of real singularities by Arnold et al. (1985), normal forms, as representatives of equivalence classes under right equivalence, are not always uniquely determined. We describe the complete structure of the equivalence…
We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.
We construct singular solutions of a complex elliptic equation of second order, having an isolated singularity of any order. In particular, we extend results obtained for the real partial differential equation in divergence form by…
An algorithmic proof of the General Neron Desingularization theorem is given for $2$-dimensional local rings and morphisms with small singular locus.
We develop a local theory for the construction of singular spacetimes in all spacetime dimensions which become asymptotically self-similar as the singularity is approached. The techniques developed also allow us to construct and classify…
We answer positively a question of B. Teissier on existence of resolution of singularities inside an equivariant map of toric varieties.
We study singular solutions to the fractional Laplace equation and, more generally, to nonlocal linear equations with measurable kernels. We establish B\^ocher type results that characterize the behavior of singular solutions near the…
We provide a new algebraic solution procedure for the global positioning problem in $n$ dimensions using $m$ satellites. We also give a geometric characterization of the situations in which the problem does not have a unique solution. This…
We show that the universal abelian cover of the complement to a germ of a reducible divisor on a complex space $Y$ with isolated singularity is $(dimY-2)$-connected provided that the divisor has normal crossings outside of the singularity…
A general method for the construction of solutions of the d'Alambertian and double d'Alambertian (harmonic and bi-harmonic) equations with local dependence of arbitrary functions upon two independent arguments is proposed. The connection…
We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the…
This paper is concerned with the transient dynamics described by the solutions of the reaction-diffusion equations in which the reaction term consists of a combination of a superlinear power-law absorption and a time-independent point…
For the completion B of a local geometric normal domain, V. Srinivas asked which subgroups of Cl B arise as the image of the map from Cl A to Cl B on class groups as A varies among normal geometric domains with B isomorphic to the…
The goal of this paper is to establish Beilinson-Bernstein type localization theorems for quantizations of some conical symplectic resolutions. We prove the full localization theorems for finite and affine type A Nakajima quiver varieties.…
We prove nonuniqueness results for complete metrics with constant positive fractional curvature conformal to the round metric on $S^n \setminus S^k$, using bifurcation techniques. These are singular (positive) solutions to a non-local…
We consider cosmological solutions of string and M-theory compactified to four dimensions by giving a general prescription to construct four-dimensional modular cosmologies with two commuting Killing vectors from vacuum solutions. By…
A common strategy to investigate the fate of gravitational singularities in asymptotically AdS spacetimes is to translate the question from the gravitational side to a dual field theory using the gauge/gravity correspondence and to do a…