相关论文: Streamlining resolution of singularities with weig…
Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let…
We show how to use the arguments of [CM2] to get a stronger effective version of uniqueness of blowups that has a number of consequences.
Despite much research, hard weighted problems still resist super-polynomial improvements over their textbook solution. On the other hand, the unweighted versions of these problems have recently witnessed the sought-after speedups.…
We consider the Calogero--Moser derivative nonlinear Schr\"odinger equation (CM-DNLS), which is an $L^{2}$-critical nonlinear Schr\"odinger equation with explicit solitons, self-duality, and pseudo-conformal symmetry. More importantly, this…
In this paper we report on numerical studies of formation of singularities for the semilinear wave equations with a focusing power nonlinearity $u_{tt} - \Delta u = u^{p}$ in three space dimensions. We show that for generic large initial…
This note revisits the ideas in an earlier (2007) paper on orbifolds and branched manifolds, showing how the constructions can be simplified by using a version of the Kuranishi atlases recently developed by McDuff--Wehrheim. We first show…
An exact solution of Einstein's equations which represents a pair of accelerating and rotating black holes (a generalised form of the spinning C-metric) is presented. The starting point is a form of the Plebanski-Demianski metric which, in…
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…
We construct Pleba\'nski-Demia\'nski stationary and axisymmetric solutions with two expanding and double principal null directions in the framework of Metric-Affine gauge theory of gravity. Starting from the new improved form of the metric…
We present analytic solutions to a class of cosmological models described by a canonical scalar field minimally coupled to gravity and experiencing self interactions through a hyperbolic potential. Using models and methods inspired by…
We construct complete Kahler metrics of Saper type on the nonsingular set of a subvariety X of a compact Kahler manifold using (a) a method for replacing a sequence of blow-ups along smooth centers, used to resolve the singularities of X,…
Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow,…
In the paper "Uniformity of Mordell-Lang" by Vesselin Dimitrov, Philipp Habegger and Ziyang Gao (arXiv:2001.10276), they use Silverman-Tate's Height Inequality and they give a proof of the same which makes use of Cartier divisors and hence…
We refine the iterated blow-up techniques. This technique, combined with a rigidity result and a specific choice of the kernel projection in the Poincar\'e inequality, might be employed to completely linearize blow-ups along at least one…
We study mean field equations with singular sources on a compact Riemann surface with boundary $(\Sigma,g)$, subject to homogeneous Neumann boundary conditions: \[ -\Delta_g v = \rho\left( \frac{V e^{v}}{\int_\Sigma V e^{v}\, d v_g} -…
With the help of the notion of weighted sharing of sets, this paper dealt with the question posed by \emph{Yi} \cite{Yi-SC-1994} regarding the uniqueness of meromorphic functions concerning three set sharing. A result has been proved which…
We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the solution. In this paper, we are interested in…
We consider the blow-up problem for discretized scale-invariant nonlinear dissipative wave equations. It is known that the critical exponents for undiscretized equations (continuous equations) are given by Fujita and Strauss exponents…
We study traveling fronts in a system of one dimensional reaction-diffusion-advection equations motivated by problems in reactive flows. In the limit as a parameter tends to infinity, we construct the approximate front profile and determine…
In this paper we develop symbolic computation algorithms to investigate finiteness of central configurations for the planar $n$-body problem. Our approach is based on Albouy-Kaloshin's work on finiteness of central configurations for the…