Resolution Of Singularities Research Articles

Mean curvature flow from conical singularities

AbstractWe prove Ilmanen’s resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface $M^{n}\subset \mathbb{R}^{n+1}$ M n ⊂ R n + 1 , $2\leq n\leq 6$ 2 ≤ n ≤ 6 , with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens.

Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement

Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement

On Bruhat-Tits theory over a higher dimensional base

Let $k$ be a perfect field. Assume that the characteristic of $k$ satisfies certain tameness assumptions \eqref{tameness}. Let $\mathcal O_{_n} := k\llbracket z_{_1}, \ldots, z_{_n}\rrbracket$ and set $K_{_n} := \text{Fract}~\cO_{_n}$. Let $G$ be an almost-simple, simply-connected affine Chevalley group scheme with a maximal torus $T$ and a Borel subgroup $B$. Given a $n$-tuple ${\bf f} = (f_{_1}, \ldots, f_{_n})$ of concave functions on the root system of $G$ as in Bruhat-Tits \cite{bruhattits1}, \cite{bruhattits}, we define {\it {\tt n}-bounded subgroups ${\tt P}_{_{\bf f}}\subset G(K_{_n})$} as a direct generalization of Bruhat-Tits groups for the case $n=1$. We show that these groups are {\it schematic}, i.e. they are valued points of smooth {\em quasi-affine} (resp. {\em affine}) group schemes with connected fibres and {\it adapted to the divisor with normal crossing $z_1 \cdots z_n =0$} in the sense that the restriction to the generic point of the divisor $z_i=0$ is given by $f_i$ (resp. sums of concave functions given by points of the apartment). This provides a higher-dimensional analogue of the Bruhat-Tits group schemes with natural specialization properties. In \S\ref{mixedstuff}, under suitable assumptions on $k$ \S \ref{charassum}, we extend all these results for a $n+1$-tuple ${\bf f} = (f_{_0}, \ldots, f_{_n})$ of concave functions on the root system of $G$ replacing $\mathcal O_{_n}$ by ${\cO} \llbracket x_{_1},\cdots,x_{_n} \rrbracket$ where $\cO$ is a complete discrete valuation ring with a perfect residue field $k$ of characteristic $p$. In the last part of the paper, we give applications in char zero to constructing certain natural group schemes on wonderful embeddings of groups and also certain families of {\tt 2-parahoric} group schemes on minimal resolutions of surface singularities that arose in \cite{balaproc}.

Cosmological singularity and power-law solutions in modified gravity

A bouncing Universe avoids the big-bang singularity. Using the time-like and null Raychaudhuri equations, we explore whether the bounce near the big-bang, within a broad spectrum of modified theories of gravity, allows for cosmologically relevant power-law solutions under reasonable physical conditions. Our study shows that certain modified theories of gravity, such as Stelle gravity, do not demonstrate singularity resolution under any reasonable conditions, while others including f(R) gravity and Brans–Dicke theory can demonstrate singularity resolution under suitable conditions. For these theories, we show that the accelerating solution is slightly favoured over ekypyrosis.

A resolution of singularities of Drinfeld compactification with an Iwahori structure

A resolution of singularities of Drinfeld compactification with an Iwahori structure

Classical and quantum analysis of gravitational singularity from Raychaudhuri equation

The present work deals with the Raychaudhuri equation (RE) to examine the space-time singularity both from classical and quantum point of view. The RE has been looked upon in terms of a classical linear Harmonic oscillator and Focusing theorem has been studied. Also, the Raychaudhuri scalar has been shown to affect the convergence and singularity formation in Black-holes and cosmology. Finally, resolution of initial big-bang singularity has been attempted quantum mechanically in two ways namely by canonical quantization and by formulation of Bohmian trajectories.

Black holes and wormholes beyond classical general relativity

In the paper only Static Spherically Symmetric space–times in four dimension are considered within modified gravity models. The non-singular static metrics, including black holes not admitting a de Sitter core in the center and traversable wormholes, are reconsidered within a class of higher-order F(R), satisfying the constraints F(0)=dFdR(0)=0. Furthermore, making use of the so called effective field theory formulation of gravity, the quantum corrections to Einstein–Hilbert action due to higher-derivative terms related to curvature invariants are investigated. In particular, in the case of Einstein–Hilbert action plus cubic curvature Goroff–Sagnotti contribution, the second order correction in the Goroff–Sagnotti coupling constant is computed. In general, it is shown that the effective metrics, namely Schwarzschild expression plus small quantum corrections are related to black holes, and not to traversable wormholes. In this framework, within the approximation considered, the resolution of singularity for r=0 is not accomplished. The related properties of these solutions are investigated.

Spherical objects in dimensions two and three

This paper classifies spherical objects in various geometric settings in dimensions two and three, including both minimal and partial crepant resolutions of Kleinian singularities, as well as arbitrary flopping 3 -fold contractions with only Gorenstein terminal singularities. The main result is much more general: in each such setting, we prove that all objects in the associated null category \mathcal{C} with no negative Ext groups are the image, under the action of an appropriate braid or pure braid group, of some object in the heart of a bounded t-structure. The corollary is that all objects x which admit no negative Exts, and for which \mathrm{Hom}_{\mathcal{C}}(x,x)=\mathbb{C} , are the images of the simples. A variation on this argument goes further and classifies all bounded t-structures on \mathcal{C} . There are multiple geometric, topological and algebraic consequences, primarily to autoequivalences and stability conditions. Our main new technique also extends into representation theory, and we establish that in the derived category of a finite-dimensional algebra which is silting discrete, every object with no negative Ext groups lies in the heart of a bounded t-structure. As a consequence, every semibrick complex can be completed to a simple-minded collection.

Categorical resolutions of filtered schemes

We give an alternative proof of the theorem by Kuznetsov and Lunts, stating that any separated scheme of finite type over a field of characteristic zero admits a categorical resolution of singularities. Their construction makes use of the fact that every variety (over a field of characteristic zero) can be resolved by a finite sequence of blow-ups along smooth centres. We merely require the existence of (projective) resolutions. To accomplish this we put the \mathcal{A} -spaces of Kuznetsov and Lunts in a different light, viewing them instead as schemes endowed with finite filtrations. The categorical resolution is then constructed by gluing together differential graded categories obtained from a hypercube of finite length filtered schemes.

Black hole bulk-cone singularities

Lorentzian correlators of local operators exhibit surprising singularities in theories with gravity duals. These are associated with null geodesics in an emergent bulk geometry. We analyze singularities of the thermal response function dual to propagation of waves on the AdS Schwarzschild black hole background. We derive the analytic form of the leading singularity dual to a bulk geodesic that winds around the black hole. Remarkably, it exhibits a boundary group velocity larger than the speed of light, whose dual is the angular velocity of null geodesics at the photon sphere. The strength of the singularity is controlled by the classical Lyapunov exponent associated with the instability of nearly bound photon orbits. In this sense, the bulk-cone singularity can be identified as the universal feature that encodes the ubiquitous black hole photon sphere in a dual holographic CFT. To perform the computation analytically, we express the two-point correlator as an infinite sum over Regge poles, and then evaluate this sum using WKB methods. We also compute the smeared correlator numerically, which in particular allows us to check and support our analytic predictions. We comment on the resolution of black hole bulk-cone singularities by stringy and gravitational effects into black hole bulk-cone “bumps”. We conclude that these bumps are robust, and could serve as a target for simulations of black hole-like geometries in table-top experiments.

Black hole singularity resolution in Wheeler–DeWitt quantum gravity

Ever since Hawking highlighted the puzzle of the existence of a singularity in the classical spacetime of general relativity, implying breakdown of all laws of physics living in the classical spacetime, singularity resolution became a problem of significantly high importance. This undesirable feature, indicating loss of predictability, has intrigued physicists over several decades. It has been hoped that the classical singularity would be resolved in a quantum theory of gravity. However, no appropriate wave function in the vicinity of the black hole singularity has been obtained so far to reach a definite conclusion.In this paper, we focus upon the interior of the Schwarzschild black hole, plagued by a non-removable classical singularity. We consider the interior spacetime of the black hole to be represented by the Kantowski–Sachs metric. Since in a quantum mechanical scenario, existence of spontaneous fluctuations of matter fields should not be ignored, we include a Klein–Gordon field in the system. Quantizing this simple gravity-matter model in the canonical scheme, we obtain the Wheeler–DeWitt equation and find an exact solution in the minisuperspace variables.We find that there exist three classes of solutions belonging to three different subregions of the eigenvalue space. Two of these classes of solutions admit the DeWitt criterion, a necessary condition for singularity resolution, implied by vanishing of the wave function at the singularity. These solutions are well-behaved and finite in the vicinity of the singularity and they indicate the existence of regular black holes in quantum gravity. In these classes of solutions, we further find that the expectation value of the Kretschmann operator is well-behaved and regular near the singularity, confirming a definite resolution to the puzzle of classical black hole singularity. On the other hand, there exists a small subregion in the eigenvalue space where the solution does not satisfy both conditions, the DeWitt criterion and finiteness of the Kretschmann expectation value at the singularity. This last class of solutions does not represent regular quantum black holes.

Non-commutative resolutions of linearly reductive quotient singularities

ABSTRACT We prove the existence of non-commutative crepant resolutions (in the sense of Van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension 2, we relate these to resolutions of singularities provided by G-Hilbert schemes and F-blowups. As an application, we establish and recover results concerning resolutions for toric singularities, as well as canonical, log terminal and F-regular singularities in dimension 2.

Black hole interior quantization: a minimal uncertainty approach

In a previous work we studied the interior of the Schwarzschild black hole implementing an effective minimal length, by applying a modification to the Poisson brackets of the theory. In this work we perform a proper quantization of such a system. Specifically, we quantize the interior of the Schwarzschild black hole in two ways: once by using the standard quantum theory, and once by following a minimal uncertainty approach. Then, we compare the obtained results from the two approaches. We show that, as expected, the wave function in the standard approach diverges in the region where classical singularity is located and the expectation value of the Kretschmann scalar also blows up on this state in that region. On the other hand, by following a minimal uncertainty quantization approach, we obtain 5 new and important results as follows. (1) All the interior states remain well-defined and square-integrable. (2) The expectation value of the Kretschmann scalar on the states remains finite over the whole interior region, particularly where used to be the classical singularity, therefore signaling the resolution of the black hole singularity. (3) A new quantum number is found which plays a crucial role in determining the convergence of the norm of states, as well as the convergence and finiteness of the expectation value of the Kretschmann scalar. (4) A minimum for the radius of the (2-spheres in the) black holes is found (5) By demanding square-integrability of states in the whole interior region, an exact relation between the Barbero-Immirzi parameter and the minimal uncertainty scale is found.

Semiclassical resolution of the black hole singularity inspired in the minimal uncertainty approach

We propose a new lapse function that simplifies the Hamiltonian constraint, describing the interior of the black hole in terms of the Ashtekar-Barbero variables, into a more straightforward form. The new Hamiltonian leads to different equations of motion than those found in the literature, but through a suitable transformation between temporal parameters, it is found that such a choice leads us to the classical solutions of the Schwarzschild metric, still preserving the physical singularity. In order to resolve this singularity, and inspired by the minimal uncertainty approach, we modify the classical algebra between the dynamic variables of the model, imposing an effective dynamics within the black hole. As a consequence, one of the dynamic variables, denoted by pb, acquires a minimum value at the singularity t=0, and on the other hand, the variable related to the radius of the 2-sphere, pc, leads to the resolution of the classical singularity of the black hole by replacing it with a bounce that connects the interior of the black hole with the interior of the white hole. This bounce occurs in the Planck-scale region, where a new event horizon manifests. Upon crossing this horizon, the nature of the interval changes from spatial to temporal outside the white hole.

Reflexive Modules on Normal Gorenstein Stein Surfaces, Their Deformations and Moduli

In this paper we generalize Artin-Verdier, Esnault and Wunram construction of McKay correspondence to arbitrary Gorenstein surface singularities. The key idea is the definition and a systematic use of a degeneracy module, which is an enhancement of the first Chern class construction via a degeneracy locus. We study also deformation and moduli questions. Among our main result we quote: a full classification of special reflexive MCM modules on normal Gorenstein surface singularities in terms of divisorial valuations centered at the singularity, a first Chern class determination at an adequate resolution of singularities, construction of moduli spaces of special reflexive modules, a complete classification of Gorenstein normal surface singularities in representation types, and a study on the deformation theory of MCM modules and its interaction with their pullbacks at resolutions. For the proof of these theorems it is crucial to establish several isomorphisms between different deformation functors, that we expect that will be useful in further work as well.

Compact convergence, deformation of the $L^{2}$-$\overline{\partial}$-complex and canonical $K$-homology classes

Let (X,\gamma) be a compact, irreducible Hermitian complex space of complex dimension m and with \dim(\mathrm{sing}(X))=0 . Let (F,\tau)\rightarrow X be a Hermitian holomorphic vector bundle over X , and let us denote by \overline{\eth}_{F,m,\mathrm{abs}} the rolled-up operator of the maximal L^{2} - \overline{\partial} -complex of F -valued (m,\bullet) -forms. Let \pi:M\rightarrow X be a resolution of singularities, g a metric on M , E:=\pi^{*}F and \rho:=\pi^{*}\tau . In this paper, under quite general assumptions on \tau , we prove the following equality of analytic K -homology classes [\overline{\eth}_{F,m,\mathrm{abs}}]=\pi_{*}[\overline{\eth}_{E,m}] , with \overline{\eth}_{E,m} the rolled-up operator of the L^{2} - \overline{\partial} -complex of E -valued (m,\bullet) -forms on M . Our proof is based on functional analytic techniques developed in Kuwae and Shioya (2003) and provides an explicit homotopy between the even unbounded Fredholm modules induced by \overline{\eth}_{F,m,\mathrm{abs}} and \overline{\eth}_{E,m} .

Singular contact varieties

In this note we propose the generalization of the notion of a holomorphic contact structure on a manifold (smooth variety) to varieties with rational singularities and prove basic properties of such objects. Natural examples of singular contact varieties come from the theory of nilpotent orbits: every projectivization of the closure of a nilpotent orbit in a semisimple Lie algebra satisfies our definition after normalization. We show the correspondence between symplectic varieties with the structure of a C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^*$$\\end{document}-bundle and the contact ones along with the existence of stratification à la Kaledin. In the projective case we demonstrate the equivalence between crepant and contact resolutions of singularities, show the uniruledness and give a full classification of projective contact varieties in dimension 3.

Quotient singularities in the Grothendieck ring of varieties

Let G G be a finite group, X X be a smooth complex projective variety with a faithful G G -action, and Y Y be a resolution of singularities of X / G X/G . Larsen and Lunts asked whether [ X / G ] − [ Y ] [X/G]-[Y] is divisible by [ A 1 ] [\mathbb {A}^1] in the Grothendieck ring of varieties. We show that the answer is negative if B G BG is not stably rational and affirmative if G G is abelian. The case when X = Z n X=Z^n for some smooth projective variety Z Z and G = S n G=S_n acts by permutation of the factors is of particular interest. We make progress on it by showing that [ Z n / S n ] − [ Z ⟨ n ⟩ / S n ] [Z^n/S_n]-[Z\langle n\rangle / S_n] is divisible by [ A 1 ] [\mathbb {A}^1] , where Z ⟨ n ⟩ Z\langle n\rangle is Ulyanov’s polydiagonal compactification of the n n th configuration space of Z Z .

Variational Bayesian Neural Networks via Resolution of Singularities

In this work, we advocate for the importance of singular learning theory (SLT) as it pertains to the theory and practice of variational inference in Bayesian neural networks (BNNs). To begin, we lay to rest some of the confusion surrounding discrepancies between downstream predictive performance measured via the test log predictive density and the variational objective. Next, we use the SLT-corrected asymptotic form for singular posterior distributions to inform the design of the variational family itself. Specifically, we build upon the idealized variational family introduced in Bhattacharya, Pati, and Plummer which is theoretically appealing but practically intractable. Our proposal takes shape as a normalizing flow where the base distribution is a carefully-initialized generalized gamma. We conduct experiments comparing this to the canonical Gaussian base distribution and show improvements in terms of variational free energy and variational generalization error. Supplemental appendices and code for the article are available online.

New methods for quasi-interpolation approximations: Resolution of odd-degree singularities

In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange functions when the orders of the singularities of the radial function’s Fourier transforms at zero do not match the parity of the dimension of the space, and therefore new expansions and coefficients are needed to overcome this problem. We develop explicit constructions of infinite Fourier expansions that provide these coefficients and make an extensive comparison of the approximation qualities and – with a particular focus – polynomial reproduction and uniform approximation order of the various formulae. One of the interesting observations concerns the link between algebraic conditions of expansion coefficients and analytic properties of localness and convergence.