Strong Conjecture Research Articles

Strong cosmic censorship in near-extremal Kerr-Sen-de Sitter spacetime

In this work, we first calculate equations of motion for particles in the Kerr-Sen-de Sitter black hole spacetime. Then, in the eikonal regime, we analytically obtain the quasi-normal resonant modes of massless neutral scalar field perturbation and find its imaginary part to be characterized by the surface gravity of a near-extremal Kerr-Sen-de Sitter black hole with the Cauchy horizon approaching the event horizon. We further show that the Penrose strong cosmic censorship conjecture is thus respected in this spacetime with dilaton scalar field and axion pseudoscalar field.

Some Results on Number Theory and Differential Equations

In this work, using the basic tools of functional analysis, we obtain a technique that allows us to obtain important results, related to quadratic equations in two variables that represent a natural number and differential equations. We show the possible ways to write an even number that ends in six, as the sum of two odd numbers and we establish conditions for said odd numbers to be prime, also making use of a suitable linear functional <img src=image/13423281_01.gif> we obtain representations of natural numbers of the form <img src=image/13423281_02.gif> in order to obtain positive integer solutions of the equation quadratic <img src=image/13423281_03.gif> where <img src=image/13423281_04.gif> is a natural number given that it ends with one. And finally, we show with three examples the use of the proposed technique to solve some ordinary and partial linear differential equations. We believe that the third corollary of our first result of this investigation can help to demonstrate the strong Goldbach conjecture.

Blowup of the local energy of linear waves at the Reissner–Nordström–AdS Cauchy horizon

We show that generic linear perturbations ψ solving on Reissner–Nordström–AdS black holes have infinite local energy at the Cauchy horizon. Combined with the result of Kehle (2020 Commun. Math. Phys. 376 145–200) that such perturbations ψ remain uniformly bounded and extend continuously across the Cauchy horizon, this settles the linear analog of the strong cosmic censorship conjecture for Reissner–Nordström–AdS: the C 0-formulation is false but the H 1-formulation is true.

The Quantum Supremacy Tsirelson Inequality

A leading proposal for verifying near-term quantum supremacy experiments on noisy random quantum circuits is linear cross-entropy benchmarking. For a quantum circuit C on n qubits and a sample z∈{0,1}n, the benchmark involves computing |⟨z|C|0n⟩|2, i.e. the probability of measuring z from the output distribution of C on the all zeros input. Under a strong conjecture about the classical hardness of estimating output probabilities of quantum circuits, no polynomial-time classical algorithm given C can output a string z such that |⟨z|C|0n⟩|2 is substantially larger than 12n (Aaronson and Gunn, 2019). On the other hand, for a random quantum circuit C, sampling z from the output distribution of C achieves |⟨z|C|0n⟩|2≈22n on average (Arute et al., 2019).In analogy with the Tsirelson inequality from quantum nonlocal correlations, we ask: can a polynomial-time quantum algorithm do substantially better than 22n? We study this question in the query (or black box) model, where the quantum algorithm is given oracle access to C. We show that, for any ε≥1poly(n), outputting a sample z such that |⟨z|C|0n⟩|2≥2+ε2n on average requires at least Ω(2n/4poly(n)) queries to C, but not more than O(2n/3) queries to C, if C is either a Haar-random n-qubit unitary, or a canonical state preparation oracle for a Haar-random n-qubit state. We also show that when C samples from the Fourier distribution of a random Boolean function, the naive algorithm that samples from C is the optimal 1-query algorithm for maximizing |⟨z|C|0n⟩|2 on average.

$G$-homotopy invariance of the analytic signature of proper co-compact $G$-manifolds and equivariant Novikov conjecture

The main result of this paper is the $G$-homotopy invariance of the $G$-index of signature operator of proper co-compact $G$-manifolds. If proper co-compact $G$ manifolds $X$ and $Y$ are $G$-homotopy equivalent, then we prove that the images of their signature operators by the $G$-index map are the same in the $K$-theory of the $C^{*}$-algebra of the group $G$. Neither discreteness of the locally compact group $G$ nor freeness of the action of $G$ on $X$ are required, so this is a generalization of the classical case of closed manifolds. Using this result we can deduce the equivariant version of Novikov conjecture for proper co-compact $G$-manifolds from the Strong Novikov conjecture for $G$.

Points of small height on semiabelian varieties

The equidistribution conjecture is proved for general semiabelian varieties over number fields. Previously, this conjecture was only known in the special case of almost split semiabelian varieties through work of Chambert-Loir. The general case has remained intractable so far because the height of a semiabelian variety is negative unless it is almost split. In fact, this places the conjecture outside the scope of Yuan’s equidistribution theorem on algebraic dynamical systems. To overcome this, an asymptotic adaption of the equidistribution technique invented by Szpiro, Ullmo, and Zhang is used here. It also allows a new proof of the Bogomolov conjecture and hence a self-contained proof of the strong equidistribution conjecture in the same general setting.

Strong positivity for quantum theta bases of quantum cluster algebras

We construct “quantum theta bases,” extending the set of quantum cluster monomials, for various versions of skew-symmetric quantum cluster algebras. These bases consist precisely of the indecomposable universally positive elements of the algebras they generate, and the structure constants for their multiplication are Laurent polynomials in the quantum parameter with non-negative integer coefficients, proving the quantum strong cluster positivity conjecture for these algebras. The classical limits recover the theta bases considered by Gross–Hacking–Keel–Kontsevich (J Am Math Soc 31(2):497–608, 2018). Our approach combines the scattering diagram techniques used in loc. cit. with the Donaldson–Thomas theory of quivers.

Serre weights and the Breuil–Mézard conjecture for modular forms

Serre’s strong conjecture, now a theorem of Khare and Wintenberger, states that every two-dimensional continuous, odd, irreducible mod p Galois representation ρ arises from a modular form of a specific minimal weight k(ρ), level N(ρ) and character ϵ(ρ). In this short paper we show that the minimal weight k(ρ) is equal to a notion of minimal weight inspired by the recipe for weights introduced by Buzzard, Diamond and Jarvis in [4]. Moreover, using the Breuil–Mézard conjecture we show that both weight recipes are equal to the smallest k≥2 such that ρ has a crystalline lift of Hodge–Tate type (0,k-1).

Diving inside a hairy black hole

We investigate the interior of the Einstein-Gauss-Bonnet charged black-hole with scalar hair. We find a variety of dynamical epochs, with the particular important feature that the Cauchy horizon is not present. This makes the violation of the no-hair theorem a possible tool to understand how might the strong cosmic censorship conjecture work.

Grundy Domination of Forests and the Strong Product Conjecture

A maximum sequence $S$ of vertices in a graph $G$, so that every vertex in $S$ has a neighbor which is independent, or is itself independent, from all previous vertices in $S$, is called a Grundy dominating sequence. The Grundy domination number, $\gamma_{gr}(G)$, is the length of $S$. We show that for any forest $F$, $\gamma_{gr}(F)=|V(T)|-|\mathcal{P}|$ where $\mathcal{P}$ is a minimum partition of the non-isolate vertices of $F$ into caterpillars in which if two caterpillars of $\mathcal{P}$ have an edge between them in $F$, then such an edge must be incident to a non-leaf vertex in at least one of the caterpillars. We use this result to show the strong product conjecture of B. Brešar, Cs. Bujtás, T. Gologranc, S. Klavžar, G. Košmrlj, B.~Patkós, Zs. Tuza, and M. Vizer, Dominating sequences in grid-like and toroidal graphs, Electron. J. Combin. 23(4): P4.34 (2016), for all forests. Namely, we show that for any forest $G$ and graph $H$, $\gamma_{gr}(G \boxtimes H) = \gamma_{gr}(G) \gamma_{gr}(H)$. We also show that every connected graph $G$ has a spanning tree $T$ so that $\gamma_{gr}(G)\le \gamma_{gr}(T)$ and that every non-complete connected graph contains a Grundy dominating set $S$ so that the induced subgraph of $S$ contains no isolated vertices.

Strong cosmic censorship in two dimensions

We study the behaviour of a free massive scalar wave-packet near the Cauchy horizon of an $\mathrm{AdS}_2$ black hole and find that it becomes infinitely differentiable for smooth initial data, independently of the parameters describing the spacetime or the scalar. This indicates a violation of the strong cosmic censorship conjecture at the classical level. We discuss our result in connection with some recent observations of violation of the conjecture for certain nearly extremal black holes.

R-N black holes can be transformed into K-N black holes

If there is no strong cosmic supervision conjecture, there may be naked singularities. The decrease in black hole entropy indicates that there is periodicity inside the black hole.This article points out that when the entropy of a black hole decreases, its enthalpy will increase, and the black hole charge will be converted into angular momentum through a certain channel. This article hereby asserts that a non-rotating charged black hole will transform into a rotating charged black hole under the effect of superradiation under preset boundary conditions.

Coloring linear hypergraphs: the Erdős–Faber–Lovász conjecture and the Combinatorial Nullstellensatz

The long-standing Erdős–Faber–Lovász conjecture states that every n-uniform linear hypergaph with n edges has a proper vertex-coloring using n colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős–Faber–Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations, which verifies a necessary condition for our algebraic approach to work.

On the existence of Killing fields in smooth spacetimes with a compact Cauchy horizon

We prove that the surface gravity of a compact non-degenerate Cauchy horizon in a smooth vacuum spacetime, can be normalized to a non-zero constant. This result, combined with a recent result by Oliver Petersen and István Rácz, end up proving the Isenberg–Moncrief conjecture on the existence of Killing fields, in the smooth differentiability class. The well known corollary of this, in accordance with the strong cosmic censorship conjecture, is that the presence of compact Cauchy horizons is a non-generic phenomenon. Though we work in 3 + 1, the result is valid line by line in any n + 1-dimensions.

Strictness of the log-concavity of generating polynomials of matroids

Recently, it was proved by Anari–Oveis Gharan–Vinzant, Anari–Liu–Oveis Gharan–Vinzant and Brändén–Huh that, for any matroid M, its basis generating polynomial and its independent set generating polynomial are log-concave on the positive orthant. Using these, they obtain some combinatorial inequalities on matroids including a solution of strong Mason's conjecture. In this paper, we study the strictness of the log-concavity of these polynomials and determine when equality holds in these combinatorial inequalities. We also consider a generalization of our result to morphisms of matroids.

Global solvability of the vacuum Einstein equation and the strong cosmic censorship in four dimensions

Let M be a connected, simply connected, oriented, closed, smooth four-manifold which is spin (or equivalently having even intersection form) and put M×≔M∖{point}. In this paper we prove that if X× is a smooth four-manifold homeomorphic but not necessarily diffeomorphic to M× (more precisely, it carries a smooth structure à la Gompf) then X× can be equipped with a complete Ricci-flat Riemannian metric. As a byproduct of the construction it follows that this metric is self-dual as well consequently X× with this metric is in fact a hyper-Kähler manifold. In particular we find that the largest member of the Gompf–Taubes radial family of large exotic R4’s admits a complete Ricci-flat metric (and in fact it is a hyper-Kähler manifold).These Riemannian solutions are then converted into Ricci-flat Lorentzian ones thereby exhibiting lot of new vacuum solutions which are not accessible by the initial value formulation. A natural physical interpretation of them in the context of the strong cosmic censorship conjecture and topology change is discussed.

Non-ridge-chordal complexes whose clique complex has shellable Alexander dual

A recent conjecture that appeared in three papers by Bigdeli–Faridi, Dochtermann, and Nikseresht, is that every simplicial complex whose clique complex has shellable Alexander dual, is ridge-chordal. This strengthens the long-standing Simon's conjecture that the k-skeleton of the simplex is extendably shellable, for any k. We show that the stronger conjecture has a negative answer, by exhibiting an infinite family of counterexamples.

Strong Novikov conjecture for low degree cohomology and exotic group C*-algebras

We strengthen a result of Hanke-Schick about the strong Novikov conjecture for low degree cohomology by showing that their non-vanishing result for the maximal group C*-algebra holds for many other exotic group C*-algebras, in particular the one associated to the smallest strongly Morita compatible and exact crossed product functor used in the new version of the Baum-Connes conjecture. To achieve this we provide a Fell absorption principle for certain exotic crossed product functors.

The Strong Maximal Rank conjecture and higher rank Brill–Noether theory

In this paper, we compute the cohomology class of certain "special maximal-rank loci" originally defined by Aprodu and Farkas. By showing that such classes are nonzero, we are able to verify the non-emptiness portion of the Strong Maximal Rank Conjecture in a wide range of cases. As an application, we obtain new evidence for the existence portion of a well-known conjecture due to Bertram, Feinberg and independently Mukai in higher-rank Brill--Noether theory.

Scalar perturbations of black holes in Jackiw-Teitelboim gravity

We study linear scalar perturbations of black holes in two-dimensional (2D) gravity models with a particular emphasis on Jackiw-Teitelboim (JT) gravity. We obtain an exact expression of the quasinormal mode frequencies for single horizon black holes in JT gravity and then verify it numerically using the Horowitz-Hubeny method. For a 2D Reissner-Nordstr\"om like solution, we find that the massless scalar wave equation reduces to the confluent Heun equation using which we calculate the Hawking spectra. Finally, we consider the dimensionally reduced Ba\~nados-Teitelboim-Zanelli (BTZ) black hole and obtain the exterior and interior quasinormal modes. The dynamics of a scalar field near the Cauchy horizon mimics the behavior of the same for the usual BTZ black hole, indicating a possible violation of the strong cosmic censorship conjecture in the near extreme limit. However, quantum effects seem to rescue strong cosmic censorship.