General Manifolds Research Articles

Poincaré duality for a class of posets

We prove the Poincaré duality theorem for bi-cellular posets X, that is, both X and Xop are cellular, in terms of cap product for finite posets which will be introduced. Moreover, we show that our results include face posets of h-regular homology manifolds.

A general center manifold theorem on fields of Banach spaces

A general center manifold theorem on fields of Banach spaces

Mechanistic insights on C(acyl)-N functionalisation mediated by late transition metals.

The carboxamide functional group has a privileged role in organic and biological chemistry due to its prevalence and utility across synthetic and natural products. Due to nN → π*CO delocalisation, amides and related functional groups are typically kinetically resistant to degradation. Nonetheless, over the past decade, transition metal catalysis has transformed our ability to utilise molecules featuring C(acyl)-N units as reactants. Alongside the burgeoning catalytic applications ranging from COx utilisation to small molecule synthesis, elucidation of the underlying mechanisms remains a critical ongoing effort. Herein, we aggregate and analyse current understanding of the mechanisms for C(acyl)-N functionalisation of amides and related functional groups with a focus on recent developments involving mechanisms unique to the late transition metals. Discussion is organized around three general mechanistic manifolds: redox-neutral mechanisms, 2e- redox-cycling mechanisms, and mechanisms involving 1e- redox steps. For each class, we focus on reactions that directly involve a transition metal mediator/catalyst in the C(acyl)-N cleavage step. We conclude with an outlook on the outstanding ambiguities and opportunities for innovation.

Stability of isometric immersions of hypersurfaces

Abstract We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$ -perturbations of their fundamental forms: For a manifold ${\mathcal M}^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$ , whose pullback metrics and shape operators are arbitrary close in $L^p$ to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold ${\mathcal N}$ , removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

Angle-of-Arrival Estimation With Practical Phone Antenna Configurations

With the advances of the Internet of Things and mobile connectivity, location-based services are becoming increasingly popular and continue to enhance our experience. Multiple antennas have been pivotal in providing reliable wireless communications and high-resolution localization. If the antennas of the array are isotropic, then the simplified array manifold determined by the array geometry can be used to estimate the angle-of-arrival (AOA). However, in the real world, mobile handsets tend to have very limited space, where the practical antennas are equipped on the same ground plane, and the array geometry hardly obeys the rule of half-wavelength spacing. Therefore, a practical antenna couples signals from other antennas, causing a mutual coupling effect. Complex array manifolds are produced on an antenna even if the received signal is propagated through a single path channel. In addition, the irregular radiation pattern of each antenna further impairs the AOA estimation capability. Given the above effects, the simplified array manifold determined by the array geometry can no longer provide precise localization. In this paper, we propose a generic array manifold model for both isotropic and practical antennas. We also present an efficient algorithm to enable AOA estimation on practical antennas on the basis of the proposed model and implement it on a 5G phone at a mid-band spectrum with a 100MHz channel bandwidth. Results reveal the promising performance of the proposed model, with the AOA estimation errors lower than 10∘ in over 90% of the scenarios.

Generalization of instanton-induced inflation and dynamical compactification

It was shown that Yang-Mills instantons on an internal space can trigger the expansion of our four-dimensional universe as well as the dynamical compactification of the internal space. We generalize the instanton-induced inflation and dynamical compactification to general Einstein manifolds with positive curvature and also to the FLRW metric with spatial curvature. We explicitly construct Yang-Mills instantons on all Einstein manifolds under consideration and find that the homogeneous and isotropic universe is allowed only if the internal space is homogeneous. We then consider the FLRW metric with spatial curvature as a solution of the eight-dimensional Einstein-Yang-Mills theory. We find that open universe (k = −1) admits bouncing solutions unlike the other cases (k = 0, +1).

Controlled Mather–Thurston theorems

Classical results of Milnor, Wood, Mather, and Thurston produce flat connections in surprising places. The Milnor–Wood inequality is for circle bundles over surfaces, whereas the Mather–Thurston theorem is about cobording general manifold bundles to ones admitting a flat connection. The surprise comes from the close encounter with obstructions from Chern–Weil theory and other smooth obstructions such as the Bott classes and the Godbillion–Vey invariant. Contradiction is avoided because the structure groups for the positive results are larger than required for the obstructions, e.g., \operatorname{PSL}(2,\mathbb{R}) versus \operatorname{U}(1) in the former case and C^1 versus C^2 in the latter. This paper adds two types of control strengthening the positive results: In many cases we are able to (1) refine the Mather–Thurston cobordism to a semi- s -cobordism (ssc), and (2) provide detail about how, and to what extent, transition functions must wander from an initial, small, structure group into a larger one. The motivation is to lay mathematical foundations for a physical program. The philosophy is that living in the IR we cannot expect to know, for a given bundle, if it has curvature or is flat, because we cannot resolve the fine scale topology which may be present in the base, introduced by an ssc, nor minute symmetry violating distortions of the fiber. Small scale, UV, “distortions“ of the base topology and structure group allow flat connections to simulate curvature at larger scales. The goal is to find a duality under which curvature terms, such as Maxwell's F \wedge F^\ast and Hilbert's \int R\, d\mathrm{vol} are replaced by an action which measures such “distortions“. In this view, curvature results from renormalizing a discrete, group theoretic, structure.

Open strings in IIB orientifold reductions

We consider type IIB compactifications on a general 4D group manifold with different types of possible spacetime filling O-planes and the corresponding D-branes parallel to them. Once fluxes allowed by the associated orientifold projection are included, a 6D \U0001d4a9 = (1, 1) gauged supergravity is obtained. In this paper we show how the consistent coupling to dynamical open strings living on the spacetime filling D-branes may be captured by the inclusion of extra vector multiplets and extra embedding tensor deformations on the gauged supergravity side. As a result, the quadratic constraints on the embedding tensor consistently reproduce the source corrected 10D Bianchi identities. Furthermore, the field strength modifications induced by the open string sector could potentially be understood as U-dual versions of the Green-Schwarz terms. Finally, the entire scalar potential of the theory exactly matches the one obtained from reduction of the bulk action plus the source contributions.

Casimir energy and modularity in higher-dimensional conformal field theories

An important problem in Quantum Field Theory (QFT) is to understand the structures of observables on spacetime manifolds of nontrivial topology. Such observables arise naturally when studying physical systems at finite temperature and/or finite volume and encode subtle properties of the underlying microscopic theory that are often obscure on the flat spacetime. Locality of the QFT implies that these observables can be constructed from more basic building blocks by cutting-and-gluing along a spatial slice, where a crucial ingredient is the Hilbert space on the spatial manifold. In Conformal Field Theory (CFT), thanks to the operator-state correspondence, we have a non-perturbative understanding of the Hilbert space on a spatial sphere. However it remains a challenge to consider more general spatial manifolds. Here we study CFTs in spacetime dimensions d > 2 on the spatial manifold T2 × ℝd−3 which is one of the simplest manifolds beyond the spherical topology. We focus on the ground state in this Hilbert space and analyze universal properties of the ground state energy, also commonly known as the Casimir energy, which is a nontrivial function of the complex structure moduli τ of the torus. The Casimir energy is subject to constraints from modular invariance on the torus which we spell out using PSL(2, ℤ) spectral theory. Moreover we derive a simple universal formula for the Casimir energy in the thin torus limit using the effective field theory (EFT) from Kaluza-Klein reduction of the CFT, with exponentially small corrections from worldline instantons. We illustrate our formula with explicit examples from well-known CFTs including the critical O(N) model in d = 3 and holographic CFTs in d ≥ 3.

Reduction theory for singular symplectic manifolds and singular forms on moduli spaces

The investigation of symmetries of b-symplectic manifolds and folded-symplectic manifolds is well-understood when the group under consideration is a torus (see, for instance, [25,21,28] for b-symplectic manifolds and [12,14] for folded symplectic manifolds). However, reduction theory has not been set in this realm in full generality. This is fundamental, among other reasons, to advance in the ‘‘quantization commutes with reduction” programme for these manifolds initiated in [29,30]. In this article, we fill in this gap and investigate the Marsden-Weinstein reduction theory under general symmetries for general bm-symplectic manifolds and other singular symplectic manifolds, including certain folded symplectic manifolds. In this new framework, the set of admissible Hamiltonian functions is larger than the category of smooth functions as it takes the singularities of the differential forms into account. The quasi-Hamiltonian set-up is also considered and brand-new constructions of (singular) quasi-Hamiltonian spaces are obtained via a reduction procedure and the fusion product.

Poincaré duality for posets

The main goal of this paper is to prove a Poincaré duality theorem in the context of finite posets. This result will be established for the class of homologically bi-admissible finite posets, which includes the well-known subclass of finite closed homology manifolds.

Removable sets and [formula omitted]-uniqueness on manifolds and metric measure spaces

We study symmetric diffusion operators on metric measure spaces. Our main question is whether essential self-adjointness or Lp-uniqueness are preserved under the removal of a small closed set from the space. We provide characterizations of the critical size of removed sets in terms of capacities and Hausdorff dimension without any further assumption on removed sets. As a key tool we prove a non-linear truncation result for potentials of nonnegative functions. Our results are robust enough to be applied to Laplace operators on general Riemannian manifolds as well as sub-Riemannian manifolds and metric measure spaces satisfying curvature-dimension conditions. For non-collapsing Ricci limit spaces with two-sided Ricci curvature bounds we observe that the self-adjoint Laplacian is already fully determined by the classical Laplacian on the regular part.

A variational approach to SKT and balanced metrics

We investigate compact complex manifolds endowed with SKT or balanced metrics. In each case, we define a new functional whose critical points are proved to be precisely the Kähler metrics, if any, on the manifold. As general manifolds of either type need not admit Kähler metrics, this provides an approach to new obstructions to Kählerianity within these two families of metrics.

FIXED POINT APPROXIMATION USING UNIFIED ITERATION SCHEME FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN CAT(0) SPACES

In the present paper, we focus on reexamination of unified three step iteration scheme in more general infinite-dimensional manifolds i.e. in geodesic CAT(0) spaces for asymptotically non-expansive mappings. The findings hold true for both asymptotically non-expansive type mappings and asymptotically quasi nonexpansive mappings. Since, numerous iteration schemes have been introducing for so long and also claimed new and different from other which shows huge lacking of existing iteration based literature. It is to be noted that there are several iteration schemes which are claimed to be different and unique but is special case of some existing scheme. Our results improve the existing iteration scheme based literature.

On the effective action for scalars in a general manifold to any loop order

Functional methods and a derivative expansion are employed for laying out a procedure to compute the effective action to any loop order, for scalar fields parametrising an arbitrary Riemannian manifold, while maintaining explicit field-space covariance. These results are of use in the characterization of effective field theories for electroweak symmetry breaking and extend a geometric perspective in field space beyond one loop.

The Plateau-Douglas Problem for Singular Configurations and in General Metric Spaces

Assume you are given a finite configuration Gamma of disjoint rectifiable Jordan curves in {mathbb {R}}^n. The Plateau-Douglas problem asks whether there exists a minimizer of area among all compact surfaces of genus of at most p which span Gamma . While the solution to this problem is well-known, the classical approaches break down if one allows for singular configurations Gamma , where the curves are potentially non-disjoint or self-intersecting. Our main result solves the Plateau-Douglas problem for such potentially singular configurations. Moreover, our proof works not only in {mathbb {R}}^n but in general proper metric spaces. In particular, the existence of an area minimizer is new for disjoint configurations of Jordan curves in general complete Riemannian manifolds. A minimal surface of fixed genus p bounding a given configuration Gamma need not always exist, even in the most regular settings. Concerning this problem, we also generalize the approach for singular configurations via minimal sequences satisfying conditions of cohesion and adhesion to the setting of metric spaces.

A Nonlinear MPC Scheme for Output Tracking Without Terminal Ingredients

In this paper, we propose a novel nonlinear model predictive control (MPC) scheme without terminal ingredients for output tracking. The control objective is to steer the output of the controlled system towards some desired set, or towards the best reachable set, in case of unreachable targets. The considered MPC formulation combines a stabilizing MPC scheme without terminal ingredients with an output tracking formulation using an artificial steady-state. In addition, we propose an offset-cost that is applicable to general non-convex steady-state manifolds. The resulting MPC scheme is simple to implement since it does not require the design of terminal ingredients (i.e., terminal controller, terminal cost, and terminal region) and is applicable to general (possibly unreachable) output specifications. We show the benefits of the proposed approach by means of a numerical example where we compare it to the state of art tracking MPC formulation.

Dimensional Reduction of Cobordism and K-theory

It has been proposed that cobordism and K-theory groups, which can be mathematically related in certain cases, are physically associated to generalised higher-form symmetries. As a consequence, they should be broken or gauged in any consistent theory of quantum gravity, in accordance with swampland conjectures. We provide further support to this idea by showing that cobordism and K-theory groups of a general manifold X reproduce the pattern of symmetries expected from the dimensional reduction of the theory on X, as well as their breaking and gauging. To this end, we employ the Atiyah-Hirzebruch spectral sequence to compute such groups for common choices of X in string compactifications.

Quantum entanglement and the growth of Laplacian eigenfunctions

We study the growth of Laplacian eigenfunctions on compact manifolds (M, g). Hörmander proved sharp polynomial bounds on which are attained on the sphere. On a “generic” manifold, the behavior seems to be different: both numerics and Berry’s random wave model suggest as the typical behavior. We propose a mechanism, centered around an analog of the spectral projector, for explaining the slow growth in the generic case: for to be large, it is necessary that either (1) several of the first n eigenfunctions were large in x 0 or (2) that is strongly correlated with a suitable linear combination of the first n eigenfunctions on most of the manifold or (3) both. An interesting byproduct is quantum entanglement for Laplacian eigenfunctions: the existence of two distinct points such that the sequences and do not behave like independent random variables. The existence of such points is not to be expected for generic manifolds but common for the classical manifolds and subtly intertwined with eigenfunction concentration.