相关论文: Resolution of Singularities -- Seattle Lecture
Using a version of Hironaka's resolution of singularities for real-analytic functions, any elliptic multiplier $\mathrm{Op}(p)$ of order $d>0$, real-analytic near $p^{-1}(0)$, has a fundamental solution $\mu_0$. We give an integral…
We give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie…
This is the manuscript for Proceedings of International Conference and Workshop on Valuation Theory held at University of Saskachewan, Canada in 1999. I have succeeded in showing that any two-dimensional hypersurface singularities of germs…
This is a note on my mini-course in the International Workshop on Real and Complex Singularities held at ICMC-USP (Sao Carlos, Brazil) in July 2012. Here we introduce a new branch of the Thom polynomial theory for singularities of…
These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic…
This article is an introduction to causal properties of General Relativity. Topics include the Raychaudhuri equation, singularity theorems of Penrose and Hawking, the black hole area theorem, topological censorship, and the Gao-Wald…
We proposed, in our previous paper, to characterize the Hirota-Miwa equation by means of the theory of triangulated category. We extend our argument in this paper to support the idea. In particular we show in detail how the singularity…
We study the local properties of positive solutions of the equation $-\Delta u+ m\abs{\nabla u}^q-e^{u}=0$ in a punctured domain $\Omega\setminus\{0\}$ of $R^N$, $N\geq 2$, where $m$ is a positive parameter and $q>1$. We study particularly…
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 classes of singularities, this leads to an embedded…
We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity…
In this paper we characterize the regularity structure, as well as show the global-in-time existence and uniqueness, of (energy) conservative solutions to the Hunter-Saxton equation by using the method of characteristics. The major…
An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified…
In this article we consider the connected component of the identity of $G$-character varieties of compact Riemann surfaces of genus $g > 0$, for connected complex reductive groups $G$ of type $A$ (e.g., $SL_n$ and $GL_n$). We show that…
Hironaka's concept of characteristic polyhedron of a singularity has been one of the most powerful and fruitful ideas of the last decades in singularity theory. In fact, since then combinatorics have become a major tool in many important…
In this article we apply the results in the article "On Isolated Real Singularities I" to the study of real $ADE$-singularities. We show that said results enables us to find the homology groups of the Milnor fibres of real…
Building upon works of Hironaka, Bierstone-Milman, Villamayor and Wlodarczyk, we give an a priori estimate for the complexity of the simplified Hironaka algorithm. As a consequence of this result, we show that there exists canonical…
We show that relativistic strings of open and closed types in Minkowski space-time of dimension 3 and 4 have topologically stable singular points. This paper describes the structure of singularities, derives their normal forms, and…
In this short note, a brief overview with a critical appraisal of the acclaimed singularity theorems, the most genuine post-Einsteinian result of General Relativity, is presented.
On the occasion of Sir Roger Penrose's 2020 Nobel Prize in Physics, we review the singularity theorems of General Relativity, as well as their recent extension to Lorentzian metrics of low regularity. The latter is motivated by the quest to…
This is an overview of math.AG/0310186, math.AG/0309290, math.AG/0501247, math.AG/0401002 and math.AG/0504584 written for the Proceedings of the AMS Meeting on Algebraic Geometry, Seattle, 2005.