Curved Space Research Articles

The Soul Conjecture in Alexandrov geometry in dimension 4

In this paper, we prove the Soul Conjecture in Alexandrov geometry in dimension 4, i.e. if X is a complete non-compact 4-dimensional Alexandrov space of non-negative curvature and positive curvature around one point, then a soul of X is a point.

Minimal Rational Curves and 1-Flat Irreducible G-Structures

1-Flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. We propose to study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification result of Merkulov and Schwachhöfer on irreducible affine holonomy, the problem is reduced to the case when the VMRT at a general point of the uniruled projective manifold is isomorphic to a subadjoint variety. In the latter situation, we prove a stronger result that, without the assumption of 1-flatness, the structure arising from VMRT is always locally flat. The proof employs the method of Cartan connections. An interesting feature is that Cartan connections are considered not for the G-structures themselves, but for certain geometric structures on the spaces of minimal rational curves.

Matroid psi classes.

Motivated by the intersection theory of moduli spaces of curves, we introduce psi classes in matroid Chow rings and prove a number of properties that naturally generalize properties of psi classes in Chow rings of Losev-Manin spaces. We use these properties of matroid psi classes to give new proofs of (1) a Chow-theoretic interpretation for the coefficients of the reduced characteristic polynomials of matroids, (2) explicit formulas for the volume polynomials of matroids, and (3) Poincaré duality for matroid Chow rings.

On the Goulden–Jackson–Vakil conjecture for double Hurwitz numbers

Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of CP1 with prescribed ramification profile over ∞, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers.In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first involves Hodge classes on moduli spaces of stable maps to classifying spaces; the second involves Chiodo classes on moduli spaces of r-spin curves; and the third involves tautological classes on moduli spaces of stable curves. We proceed to discuss the merits of these formulas against a list of desired properties enunciated by Goulden, Jackson and Vakil. Our formulas lead to non-trivial relations between tautological intersection numbers on moduli spaces of stable curves and hints at further structure underlying Chiodo classes. The paper concludes with generalisations of our results to the context of r-spin Hurwitz numbers.

Real Moduli Spaces And Density Of Non-simple Real Abelian Varieties

Abstract For fixed k < g and a family of polarized abelian varieties of dimension g over ${{\mathbb{R}}}$, we give a criterion for the density in the parameter space of those abelian varieties over ${{\mathbb{R}}}$ containing a k-dimensional abelian subvariety over ${{\mathbb{R}}}$. As application, we prove density of such a set in the moduli space of polarized real abelian varieties of dimension g and density of real algebraic curves mapping non-trivially to real k-dimensional abelian varieties in the moduli space of real algebraic curves as well as in the space of real plane curves. This extends to the real setting results by Colombo and Pirola [10]. We then consider the real locus of an algebraic stack over $\mathbb{R}$, attaching a topological space to it. For a real moduli stack, this defines a real moduli space. We show that for ${{\mathcal{M}}}_g$ and ${{\mathcal{A}}}_g$, the real moduli spaces that arise in this way coincide with the moduli spaces of Gross–Harris [13] and Seppälä–Silhol [23].

Computational investment in generation of elliptic curves randomly over large prime fields

AbstractRandomly computed elliptic curves over prime fields are desirable for implementation in strategic or military grade cryptosystems. A randomly computed elliptic curve eliminates the possibility of exhibiting any pre‐studied value or special structure of the curve parameters which may be doubted to have intentionally non‐disclosed vulnerability. However, computing elliptic curves randomly over large prime fields for cryptography is an intricate task which requires reasonably high computational resources and time. These computational resources are considered in terms of the number of CPU clock cycles and the number of attempts or searches made in the security parameter space of elliptic curve. The estimate of the number of CPU clock cycles helps in determining processor requirements whereas the number of attempts or searches help to decide the number of CPU cores for speeding up the curve generation process. We present efficient statistical estimates of computational resources to randomly compute cryptographically safe elliptic curve over desired prime field size using a standard procedure.

Classification of the Orthogonal Separable Webs for the Hamilton-Jacobi and Klein-Gordon Equations on 3-Dimensional Minkowski Space

We review a new theory of orthogonal separation of variables on pseudo-Riemannian spaces of constant zero curvature via concircular tensors and warped products. We then apply this theory to three-dimensional Minkowski space, obtaining an invariant classification of the forty-five orthogonal separable webs modulo the action of the isometry group. The eighty-eight inequivalent coordinate charts adapted to the webs are also determined and listed. We find a number of separable webs which do not appear in previous works in the literature. Further, the method used seems to be more efficient and concise than those employed in earlier works.

Bi-Hamiltonian Recursion, Liu\u2013Pandharipande Relations, and Vanishing Terms of the Second Dubrovin\u2013Zhang Bracket

The Dubrovin–Zhang hierarchy is a Hamiltonian infinite-dimensional integrable system associated to a semi-simple cohomological field theory or, alternatively, to a semi-simple Dubrovin–Frobenius manifold. Under an extra assumption of homogeneity, Dubrovin and Zhang conjectured that there exists a second Poisson bracket that endows their hierarchy with a bi-Hamiltonian structure. More precisely, they gave a construction for the second bracket, but the polynomiality of its coefficients in the dispersion parameter expansion is yet to be proved. In this paper we use the bi-Hamiltonian recursion and a set of relations in the tautological rings of the moduli spaces of curves derived by Liu and Pandharipande in order to analyze the second Poisson bracket of Dubrovin and Zhang. We give a new proof of a theorem of Dubrovin and Zhang that the coefficients of the dispersion parameter expansion of the second bracket are rational functions with prescribed singularities. We also prove that all terms in the expansion of the second bracket in the dispersion parameter that cannot be realized by polynomials because they have negative degree do vanish, thus partly confirming the conjecture of Dubrovin and Zhang.

Asymptotic growth of Betti numbers of ordered configuration spaces of an elliptic curve

We construct a dga to computing the cohomology of ordered configuration spaces on an algebraic variety with vanishing Euler characteristic. It follows that the $k$-th Betti number of $Conf(C,n)$ ($C$ is an elliptic curve) grows as a polynomial of degree exactly $2k-2$. We also compute $H^k(Conf(C,n))$ for $k \leq 5$ and arbitrary $n$.

Traveling Waves for the Generalized Sinh-Gordon Equation with Variable Coefficients

The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed.

Automorphic Bloch theorems for hyperbolic lattices

Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of two-dimensional (2D) hyperbolic space, a non-Euclidean space of uniform negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface. Three important questions were left unanswered: whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; whether a formal Bloch theorem can be rigorously established; and whether hyperbolic band theory can describe finite lattices realized in an experiment. In the present work, we address all three questions simultaneously. By formulating periodic boundary conditions for finite but arbitrarily large lattices, we show that a generalized Bloch theorem can be rigorously proved but may or may not involve higher-dimensional irreducible representations (irreps) of the nonabelian translation group, depending on the lattice geometry. Higher-dimensional irreps correspond to points in a moduli space of higher-rank stable holomorphic vector bundles, which further generalizes the notion of Brillouin zone beyond the Jacobian. For a large class of finite lattices, only 1D irreps appear, and the hyperbolic band theory previously developed becomes exact.

Manual therapy combined with posterior percutaneous endoscopic cervical decompression for the treatment of cervical spondylotic radiculopathy

To explore the clinical efficacy and safety of manual therapy combined with posterior percutaneous endoscopic cervical decompression(PECD) in the treatment of intractable cervical spondylotic radiculopathy. From May 2016 to May 2018, 23 CSR patients who responded poorly to conservative treatment for at least 6 weeks underwent the combination management. Firstly, the patients received the posterior percutaneous endoscopic cervical decompression routine care for the following 4 weeks and manual therapy for another 4 weeks. A total of 23 patients were followed up, including 14 males and 9 females, the age ranged from 29 to 78 years old with an average of (50.30±12.28) years, the course of disease was 3 to 24 months with an average of (9.74±5.76) months. The lesion segment involved C4,5 in 4 cases, C5,6 in 13 cases, C6,7 in 6 cases. The visual analogue scale (VAS), neck disability index (NDI), changes of cervical physiological curvature and interbody stability, adverse events were observed before and after operation. The follow-up time points were before operation, 1 day after operation and 1, 3 and 6 months after operation. All patients successfully completed the operation and manual treatment for 4 to 8 times. Among the 29 cases, 23 patients were followed up for more than 6 months. There was no spinal cord and nerve root injury during the treatment and follow-up. Operation time was from 80 to 120 min with a median of 90 min;intraoperative blood loss was from 35 to 80 ml with a median of 50 ml. NDI, VAS of neck, shoulder and arm each period after operation were significantly lower than those before PECD(P<0.05), while there were no significant improvement in cervical physiological curvature and target segment intervertebral space height(P>0.05);there was no significant change in interbody stability (P>0.05). After received the manual therapy, NDI significantly decreased (P<0.05), however, there was no significant difference in VAS of neck, shoulder and arm, physiological curvature of cervical spine and intervertebral space height of target segment compared with that before manual treatment (P>0.05);there was no significant change in interbody stability (P>0.05). Manual therapy combined with PECD in the treatment of intractable cervical spondylotic radiculopathy can not only quickly improve the symptoms, but also alleviate the residual symptoms after PECD safely and effectively, and can not cause obvious signs of accelerated instability of cervical adjacent segments in the short term.

Linear series on a curve of compact type bridged by a chain of elliptic curves

In the present paper we investigate conditions for the non-existence of a smoothable limit linear series on a curve of compact type such that two smooth curves are bridged by a chain of two elliptic curves. Combining this work with results on the existence of a smoothable limit linear series on such a curve, we show relations among Brill–Noether loci of codimension at most two in the moduli space of complex curves. Specifically, Brill–Noether loci of codimension two have mutually distinct supports.

Calculating the Euler Characteristic of the Moduli Space of Curves

The orbifold Euler characteristic of the moduli space \({{\cal M}_{g;1}}\) of genus g smooth curves with one marked point (g ≥ 1) was calculated by Harer and Zagier: \(\chi ({{\cal M}_{g;1}}) = \zeta (1 - 2g) = - {B_{2g}}/(2g)\), where ζ is the Riemann zeta function and B2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics.

Boundary complexes of moduli spaces of curves in higher genus

Given a collection of boundary divisors in the moduli space M ¯ 0 , n \overline {\mathcal {M}}_{0,n} of stable genus-zero n n -pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs ( g , n ) (g,n) for which the analogous statement holds in M ¯ g , n \overline {\mathcal {M}}_{g,n} .

Variational symmetries and superintegrability in multifield cosmology

We consider a spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker background space with an ideal gas and a multifield Lagrangian consisting of two minimally coupled scalar fields which evolve in a field space of constant curvature. For this cosmological model we classify the potential function for the scalar fields such that variational point symmetries exist. The corresponding conservation laws are calculated. Finally, analytic solutions are presented for specific functional forms of the scalar field potential in which the cosmological field equations are characterized as a Liouville integrable system by point symmetries. The free parameters of the cosmological model are constrained in order to describe analytic solutions for an inflationary epoch. Finally, stability properties of exact closed-form solutions are investigated. These solutions are scaling solutions with important physical properties for the cosmological model.

Tropical moduli spaces as symmetric Δ$\Delta$‐complexes

We develop techniques for studying fundamental groups and integral singular homology of symmetric Delta-complexes, and apply these techniques to study moduli spaces of stable tropical curves of unit volume, with and without marked points. As one application, we show that Delta_g and Delta_{g,n} are simply connected, for positive g. We also show that Delta_3 is homotopy equivalent to the 5-sphere, and that Delta_4 has 3-torsion in H_5.

Cohomogeneity one manifolds and homogeneous spaces of positive scalar curvature

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.

Bergman kernel and period map for curves

As for any symmetric space the tangent space to Siegel upper-half space is endowed with an operation coming from the Lie bracket on the Lie algebra. We consider the pull-back of this operation to the moduli space of curves via the Torelli map. We characterize it in terms of the geometry of the curve, using the Bergman kernel form associated to the curve. It is known that the second fundamental form of the Torelli map outside the hyperelliptic locus can be seen as the multiplication by a certain meromorphic form. Our second result says that the Bergman kernel form is the harmonic representative—in a suitable sense—of this meromorphic form.

The moduli space of tropical curves with fixed Newton polygon

Abstract Given a lattice polygon, we study the moduli space of all tropical plane curves with that Newton polygon. We determine a formula for the dimension of this space in terms of combinatorial properties of that polygon. If this polygon is nonhyperelliptic or maximal and hyperelliptic, then this formula matches the dimension of the moduli space of nondegenerate algebraic curves with the given Newton polygon.