Curved Space Research Articles

A GENERALISATION OF WITTEN’S CONJECTURE FOR THE PIXTON CLASS AND THE NONCOMMUTATIVE KDV HIERARCHY

AbstractIn this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative Korteweg-de Vries hierarchy, which we introduced in a previous work. The specialisation of this conjecture to the top degree part of Pixton’s class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we prove that this conjecture follows from the double ramification/Dubrovin–Zhang equivalence conjecture. We also provide several independent computational checks in support of it.

Setting Bias Specifications Based on Qualitative Assays With a Quantitative Cutoff Using COVID-19 as a Disease Model.

ObjectivesAutomated qualitative serology assays often measure quantitative signals that are compared against a manufacturer-defined cutoff for qualitative (positive/negative) interpretation. The current general practice of assessing serology assay performance by overall concordance in a qualitative manner may not detect the presence of analytical shift/drift that could affect disease classifications.MethodsWe describe an approach to defining bias specifications for qualitative serology assays that considers minimum positive predictive values (PPVs) and negative predictive values (NPVs). Desirable minimum PPVs and NPVs for a given disease prevalence are projected as equi-PPV and equi-NPV lines into the receiver operator characteristic curve space of coronavirus disease 2019 serology assays, and the boundaries define the allowable area of performance (AAP).ResultsMore stringent predictive values produce smaller AAPs. When higher NPVs are required, there is lower tolerance for negative biases. Conversely, when higher PPVs are required, there is less tolerance for positive biases. As prevalence increases, so too does the allowable positive bias, although the allowable negative bias decreases. The bias specification may be asymmetric for positive and negative direction and should be method specific.ConclusionsThe described approach allows setting bias specifications in a way that considers clinical requirements for qualitative assays that measure signal intensity (eg, serology and polymerase chain reaction).

Topological invariants of groups and Koszul modules

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K⊆∧2V as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, and the degree of growth and virtual nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves, (2) nilpotent fundamental groups of compact Kähler manifolds, and (3) the Torelli group of a free group.

Superintegrability on the Dunkl oscillator model in three-dimensional spaces of constant curvature

In this paper, three-dimensional Dunkl oscillator models are studied in a generalized form of superintegrable Euclidean Hamiltonian systems to curved ones. These models are defined based on curved Hamiltonians, which depend on a deformation parameter of underlying space and involve reflection operators. The corresponding symmetries are obtained by the Jordan–Schwinger representations in the family of the Cayley–Klein orthogonal algebras using the creation and annihilation operators of the dynamical sl−1(2) algebra of the one-dimensional Dunkl oscillator. The resulting algebra is a deformation of soκ1κ2(4) with reflections, which is known as the Jordan–Schwinger–Dunkl algebra jsdκ1κ2(4). Hence, it is shown that this model is maximally superintegrable. On the other hand, the superintegrability of the three-dimensional Dunkl oscillator model is studied from the viewpoint of the factorization approach. The spectrum of this system is derived through the separation of variables in geodesic polar coordinates, and the resulting eigenfunctions are algebraically given in terms of Jacobi polynomials.

Surface bundles and the section conjecture

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of this conjecture. In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over p-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classes o_1 and o_2, which obstruct the existence of pi_1-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves M_g studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs pi_1-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus g>2, and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).

On close-packings of spheres in spaces of constant curvature

On close-packings of spheres in spaces of constant curvature

Chow rings of low-degree Hurwitz spaces

Abstract While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space ℋ k , g {\mathcal{H}_{k,g}} parametrizing smooth degree k, genus g covers of ℙ 1 {\mathbb{P}^{1}} . Let k = 3 , 4 , 5 {k=3,4,5} . We prove that the rational Chow rings of ℋ k , g {\mathcal{H}_{k,g}} stabilize in a suitable sense as g tends to infinity. In the case k = 3 {k=3} , we completely determine the Chow rings for all g. We also prove that the rational Chow groups of the simply branched Hurwitz space ℋ k , g s ⊂ ℋ k , g {\mathcal{H}^{s}_{k,g}\subset\mathcal{H}_{k,g}} are zero in codimension up to roughly g k {\frac{g}{k}} . In [S. Canning and H. Larson, The Chow rings of the moduli spaces of curves of genus 7 , 8 {7,8} and 9, preprint 2021, arXiv:2104.05820], results developed in this paper are used to prove that the Chow rings of ℳ 7 , ℳ 8 , {\mathcal{M}_{7},\mathcal{M}_{8},} and ℳ 9 {\mathcal{M}_{9}} are tautological.

Gromov–Witten theory and invariants of matroids

We use techniques from Gromov–Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov–Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement’s wonderful model is a combinatorial invariant, i.e., it depends only on the matroid. When the boundary divisor is maximal, we use toric intersection theory to convert the virtual fundamental class into a balanced weighted fan in a vector space, having the expected dimension. We explain how the associated Gromov–Witten theory is completely encoded by intersections with this weighted fan. We include a number of questions whose positive answers would lead to a well-defined Gromov–Witten theory of non-realizable matroids.

The super Mumford form and Sato Grassmannian

We describe a supersymmetric generalization of the construction of Kontsevich and Arbarello, De Concini, Kac, and Procesi, which utilizes a relation between the moduli space of curves with the infinite-dimensional Sato Grassmannian. Our main result is the existence of a flat holomorphic connection on the line bundle λ3/2⊗λ1/2−5 on the moduli space of triples: a super Riemann surface, a Neveu-Schwarz puncture, and a formal coordinate system. We also prove a superconformal Noether normalization lemma for families of super Riemann surfaces.

Subelliptic Harmonic Maps with Values in Metric Spaces of Nonpositive Curvature

Subelliptic Harmonic Maps with Values in Metric Spaces of Nonpositive Curvature

THE BOUNDARY OF THE p-RANK STRATUM OF THE MODULI SPACE OF CYCLIC COVERS OF THE PROJECTIVE LINE

AbstractWe study the p-rank stratification of the moduli space of cyclic degree $\ell $ covers of the projective line in characteristic p for distinct primes p and $\ell $ . The main result is about the intersection of the p-rank $0$ stratum with the boundary of the moduli space of curves. When $\ell =3$ and $p \equiv 2 \bmod 3$ is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type $(r,s)$ , and p-rank f satisfying the clear necessary conditions.

Globally generated vector bundles on and the unirationality of

Chang and Ran proved, in 1984, the unirationality of the moduli spaces of curves of genus 11, 12, and 13. They consider a family U of nonsingular space curves. If U is non-empty, it dominates the moduli space. Considering certain vector bundles related to the curves, one shows that U is unirational. Finally, Chang and Ran prove that U is non-empty by constructing smoothable reducible curves. We prove, instead, the existence of (almost) globally generated vector bundles of the above kind. The dependency locus of (rank − 1) general global sections of such a bundle is a nonsingular curve from U.

Curvature conditions for spatial isotropy

In the context of mathematical cosmology, the study of necessary and sufficient conditions for a semi-Riemannian manifold to be a (generalized) Robertson-Walker space-time is important. In particular, it is a requirement for the development of initial data to reproduce or approximate the standard cosmological model. Usually these conditions involve the Einstein field equations, which change if one considers alternative theories of gravity or if the coupling matter fields change. Therefore, the derivation of conditions which do not depend on the field equations is an advantage. In this work we present a geometric derivation of such a condition. We require the existence of a unit vector field to distinguish at each point of space two (non-equal) sectional curvatures. This is equivalent for the Riemann tensor to adopt a specific form. Our geometrical approach yields a local isometry between the space and a Robertson-Walker space of the same dimension, curvature and metric tensor sign (the dimension of the largest subspace on which the metric tensor is negative definite). Remarkably, if the space is simply-connected, the isometry is global. Our result generalizes to a class of spaces of non-constant curvature the theorem that spaces of the same constant curvature, dimension and metric tensor sign must be locally isometric. Because we do not make any assumptions regarding field equations, matter fields or metric tensor sign, one can readily use this result to study cosmological models within alternative theories of gravity or with different matter fields.

On Submanifolds with a Parallel Normal Vector Field in Spaces of Constant Curvature

In this paper, we describe normal vector fields of a special form along geodesic lines on n-dimensional submanifolds of (n + p)-dimensional spaces of constant curvature, in particular, fields of normal curvature and normal torsion of a submanifold at a point in a given direction. We study submanifolds such that these normal vector fields are parallel in the normal connection along their geodesic lines.

Prediction of the trabecular iris angle after posterior chamber phakic intraocular lens implantation.

To create an equation for predicting the trabecular iris angle (TIA) and to verify its accuracy after implantable collamer lens (ICL) implantation. Nagoya Eye Clinic, Nagoya, Japan. Retrospective evaluation of a screening approach. 174 eyes (174 patients) that underwent ICL implantation were included. Patients were randomly assigned to the prediction equation group (116 eyes) or verification group (58 eyes). Anterior segment optical coherence tomography (AS-OCT) (CASIA2 TOMEY) was performed before and 3 months after ICL surgery. For the prediction group, a prediction equation was created with the preoperative AS-OCT parameters and ICL size as independent variables and the postoperative anterior chamber depth (ACD) as the dependent variable. Then, by applying the predicted post-ACD and preoperative AS-OCT parameters as independent variables and TIA after ICL surgery as the dependent variable, a prediction equation was created to predict the postoperative TIA (post-TIA) after ICL surgery. Each prediction equation was created using stepwise multiple regression analysis, and its accuracy was verified by a Bland-Altman plot in the verification group. The explanatory variables (standardized partial regression coefficient) selected in the post-TIA prediction equation were post-ACD (0.629), TIA750 (0.563), iris curvature (0.353), pupil diameter (-0.281), iris area (-0.249), and trabecular iris space area 250 (-0.171) (R2 = 0.646) (n = 116). There were no clinically significant systematic errors between measured and predictive post-TIA values in the verification group (n = 58). The mean absolute prediction error was 3.43 ± 2.22 degrees. Post-TIA was accurately predicted from the predicted post-ACD and other preoperative AS-OCT parameters.

Tamely ramified covers of the projective line with alternating and symmetric monodromy

Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite \'{e}tale covers of $X$ that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime $p\ge 5$, there are families of tamely ramified covers with monodromy the symmetric group $S_n$ or alternating group $A_n$ for infinitely many $n$. These covers come from the moduli spaces of elliptic curves with $PSL_2(\mathbb{F}_\ell)$-structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder, about Markoff triples modulo $\ell$.

Anabelian geometry of certain types of hyperbolic polycurves

In this paper, we study higher dimensional anabelian geometry. We prove the Grothendieck conjecture for certain types of hyperbolic polycurves (i.e., algebraic varieties in the form of successive fibrations by hyperbolic curves) over a sub-p-adic field (i.e., a field isomorphic to a subfield of a field finitely generated over the p-adic field Qp) for some prime number p. Among other things, we prove the Grothendieck conjecture for hyperbolic polycurves obtained as certain types of finite étale covers of the moduli space of curves of genus two with ordered r marked points over a sub-p-adic field.

Construction of a linear K-system in Hamiltonian Floer theory

The notion of linear K-system was introduced by the present authors as an abstract model arising from the structure of compactified moduli spaces of solutions to Floer’s equation in the book (Fukaya et al. in Springer monographs in mathematics, Springer, Berlin, 2020). The purpose of the present article is to provide a geometric realization of the linear K-system associated with solutions to Floer’s equation in the Morse–Bott setting. Immediate consequences [when combined with the abstract theory from Fukaya et al. (Springer monographs in mathematics, Springer, Berlin, 2020)] are the construction of Floer cohomology for periodic Hamiltonian systems on general compact symplectic manifolds without any restriction, and the construction of an isomorphism over the Novikov ring between the Floer cohomology and the singular cohomology of the underlying symplectic manifold. The present article utilizes various analytical results on pseudoholomorphic curves established in our earlier papers and books. However, the paper itself is geometric in nature, and does not presume much prior knowledge of Kuranishi structures and their construction but assumes only the elementary part thereof, and results from Fukaya et al. (Surv Differ Geom 22:133–190, 2018) and Fukaya et al. (Exponential decay estimate and smoothness of the moduli space of pseudoholomorphic curves) on their construction, and the standard knowledge on Hamiltonian Floer theory. We explain the general procedure of the construction of a linear K-system by explaining in detail the inductive steps of ensuring the compatibility conditions for the system of Kuranishi structures leading to a linear K-system for the case of Hamiltonian Floer theory.

Quantum Gravity If Non-Locality Is Fundamental.

I take non-locality to be the Michelson–Morley experiment of the early 21st century, assume its universal validity, and try to derive its consequences. Spacetime, with its locality, cannot be fundamental, but must somehow be emergent from entangled coherent quantum variables and their behaviors. There are, then, two immediate consequences: (i). if we start with non-locality, we need not explain non-locality. We must instead explain an emergence of locality and spacetime. (ii). There can be no emergence of spacetime without matter. These propositions flatly contradict General Relativity, which is foundationally local, can be formulated without matter, and in which there is no “emergence” of spacetime. If these be true, then quantum gravity cannot be a minor alteration of General Relativity but must demand its deep reformulation. This will almost inevitably lead to: matter not only curves spacetime, but “creates” spacetime. We will see independent grounds for the assertion that matter both curves and creates spacetime that may invite a new union of quantum gravity and General Relativity. This quantum creation of spacetime consists of: (i) fully non-local entangled coherent quantum variables. (ii) The onset of locality via decoherence. (iii) A metric in Hilbert space among entangled quantum variables by the sub-additive von Neumann entropy between pairs of variables. (iv) Mapping from metric distances in Hilbert space to metric distances in classical spacetime by episodic actualization events. (v) Discrete spacetime is the relations among these discrete actualization events. (vi) “Now” is the shared moment of actualization of one among the entangled variables when the amplitudes of the remaining entangled variables change instantaneously. (vii) The discrete, successive, episodic, irreversible actualization events constitute a quantum arrow of time. (viii) The arrow of time history of these events is recorded in the very structure of the spacetime constructed. (ix) Actual Time is a succession of two or more actual events. The theory inevitably yields a UV cutoff of a new type. The cutoff is a phase transition between continuous spacetime before the transition and discontinuous spacetime beyond the phase transition. This quantum creation of spacetime modifies General Relativity and may account for Dark Energy, Dark Matter, and the possible elimination of the singularities of General Relativity. Relations to Causal Set Theory, faithful Lorentzian manifolds, and past and future light cones joined at “Actual Now” are discussed. Possible observational and experimental tests based on: (i). the existence of Sub- Planckian photons, (ii). knee and ankle discontinuities in the high-energy gamma ray spectrum, and (iii). possible experiments to detect a creation of spacetime in the Casimir system are discussed. A quantum actualization enhancement of repulsive Casimir effect would be anti-gravitational and of possible practical use. The ideas and concepts discussed here are not yet a theory, but at most the start of a framework that may be useful.

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.