Loop Space Research Articles

MKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere

We investigate geometric evolution equations for Legendrian curves in the 3-sphere which are invariant under the action of the unitary group ${\rm U}(2)$. We define a natural symplectic structure on the space of Legendrian loops and show that the modified Korteweg-de Vries equation, along with its associated hierarchy, are realized as curvature evolutions induced by a sequence of Hamiltonian flows. For the flow among these that induces the mKdV equation, we investigate the geometry of solutions which evolve by rigid motions in ${\rm U}(2)$. Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed.

Chairside digital design and manufacturing method for children's band and loop space maintainers.

This study proposes a chairside digital design and manufacturing method for band and loop space maintainers and preliminarily validates its clinical feasibility. Clinical cases of 10 children requiring space maintenance caused by premature loss of primary teeth were collected. Intraoral scan data of the affected children were also collected to establish digital models of the missing teeth. Using a pediatric band and loop space maintainer design software developed by our research team, a rapid personalized design of band and loop structures was achieved, and a digital model of an integrated band and loop space maintainer was ultimately generated. A chairside space maintainer was manufactured through metal computer numerical control machining for the experimental group, whereas metal 3D printing in the dental laboratory was used for the control group. A model fitting assessment was conducted for the space maintainers of both groups, and senior pediatric dental experts were invited to evaluate the clinical feasibility of the space maintainers with regard to fit and stability using the visual analogue scale scoring system. Statistical analysis was also performed. The time spent in designing and manufacturing the 10 space maintainers of the experimental group was all less than 1 h. Statistical analysis of expert ratings showed that the experimental group outperformed the control group with regard to fit and stability. Both types of space maintainers met clinical requirements. The chairside digital design and manufacturing method for pediatric band and loop space maintainers proposed in this study can achieve same-day fitting of space maintainers at the first appointment, demonstrating good clinical feasibility and significant potential for clinical application.

An FFT based approach to account for elastic interactions in OkMC: Application to dislocation loops in iron

Object kinetic Monte Carlo (OkMC) is a fundamental tool for modeling defect evolution in volumes and times far beyond atomistic models. The elastic interaction between defects is classically considered using a dipolar approximation but this approach is limited to simple cases and can be inaccurate for large and close interacting defects. In this work a novel framework is proposed to include exact elastic interactions between defects in OkMC valid for any type of defect and anisotropic media. In this method, the elastic interaction energy of a defect is computed by volume integration of its elastic strain multiplied by the stress created by all the other defects, being both fields obtained numerically using a FFT solver. The resulting interaction energies reproduce analytical elastic solutions and show the limited accuracy of dipole approaches for close and large defects.The OkMC framework proposed is used to simulate the evolution in space and time of self-interstitial atoms and dislocation loops in iron. It is found that including the anisotropy has a quantitative effect in the evolution of all the type of defects studied. Regarding dislocation loops, it is observed that using the exact interaction energy results in higher interactions than using the dipole approximation for close loops.

The Clifford Algebra Bundle on Loop Space

We construct a Clifford algebra bundle formed from the tangent bundle of the smooth loop space of a Riemannian manifold, which is a bundle of super von Neumann algebras on the loop space. We show that this bundle is in general non-trivial, more precisely that its triviality is obstructed by the transgressions of the second Stiefel-Whitney class and the first (fractional) Pontrjagin class of the manifold.

Upper bounds for the critical values of homology classes of loops

In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of compact manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a compact and simply-connected n-dimensional manifold of positive Ricci curvature Ric≥n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Ric}\\ge n-1$$\\end{document} has length ≤nπ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le n \\pi .$$\\end{document} This improves the bound 8π(n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi (n-1)$$\\end{document} given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022).

Large N master field optimization: the quantum mechanics of two Yang-Mills coupled matrices

We study the large N dynamics of two massless Yang-Mills coupled matrix quantum mechanics, by minimization of a loop truncated Jevicki-Sakita effective collective field Hamiltonian. The loop space constraints are handled by the use of master variables. The method is successfully applied directly in the massless limit for a range of values of the Yang-Mills coupling constant, and the scaling behaviour of different physical quantities derived from their dimensions are obtained with a high level of precision. We consider both planar properties of the theory, such as the large N ground state energy and multi-matrix correlator expectation values, and also the spectrum of the theory. For the spectrum, we establish that the U(N) traced fundamental constituents remain massless and decoupled from other states, and that bound states develop well defined mass gaps, with the mass of the two degenerate lowest lying bound states being determined with a particularly high degree of accuracy. In order to confirm, numerically, the physical interpretation of the spectrum properties of the U(N) traced constituents, we add masses to the system and show that, indeed, the U(N) traced fundamental constituents retain their “bare masses”. For this system, we draw comparisons with planar results available in the literature.

Families of relatively exact Lagrangians, free loop spaces and generalised homology

AbstractWe prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy $$\psi ^1$$ ψ 1 of a symplectic manifold $$(M, \omega )$$ ( M , ω ) fixing a relatively exact Lagrangian L setwise must act trivially on $$R_*(L)$$ R ∗ ( L ) , where $$R_*$$ R ∗ is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over $${\mathbb {Z}}/2$$ Z / 2 and over $${\mathbb {Z}}$$ Z under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, $$\psi ^1|_L$$ ψ 1 | L is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that $$\psi ^1|_L$$ ψ 1 | L acts trivially on $$R_*({\mathcal {L}}L)$$ R ∗ ( L L ) , where $${\mathcal {L}}L$$ L L is the free loop space of L. From this we deduce that when L is a surface or a $$K(\pi , 1)$$ K ( π , 1 ) , $$\psi ^1|_L$$ ψ 1 | L is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over $${\mathbb {Z}}/2$$ Z / 2 .

The string topology coproduct on complex and quaternionic projective space

On the free loop space of compact symmetric spaces Ziller introduced explicit cycles generating the homology of the free loop space. We use these explicit cycles to compute the string topology coproduct on complex and quaternionic projective space. The behavior of the Goresky-Hingston product for these spaces then follows directly.

Adams’ cobar construction as a monoidal -coalgebra model of the based loop space

Abstract We prove that the classical map comparing Adams’ cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal $E_\infty $ -coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams’ map preserves monoidal coalgebra structures.

Stress induced on permanent mandible first molar and space maintainer under normal masticatory forces: a finite element study.

The band and loop space maintainer is used to maintain the missing space of deciduous molars which are lost early. When the second deciduous molar is lost prematurely, the stress on the first permanent molar during different degrees of development may vary when it is the abutment. The design and use of the space maintainer may also lead to damage of the loop. The purpose of this article is to use the finite element method to study the stress on the first permanent molar and the loop with or without occlusal contact, with the first permanent molar of four different degrees of development serving as the abutment. We aimed to guide the clinical design and use of the space maintainer. We developed finite element models of the mandibular first permanent molar and the band and loop space maintainer, and simulated alveolar bone, periodontal ligament (PDL), enamel and dentin. The four developmental stages were 1/2 (I), 2/3 (II), 3/4 (III) and full development (IV). Ansys Workbench was used to analyze the effects of root development and occlusal contact between the loop and the opposite jaw on abutment teeth and the loop. Abutment teeth were statically loaded vertically and obliquely with a force of 70 N. The loop was statically loaded vertically with a force of 14 N. The stress on all structures and the displacement trends of the loop were calculated. The stress on enamel, dentin, PDL and alveolar bone were similar, and the concentration was consistent. But if there was occlusal contact, the loop produced maximum displacement at the near middle edge of contact with the anterior teeth. When the loop was in occlusal contact with the opposing occlusal tooth, the peak value of the equivalent stress on the space maintainer under vertical load was: group I > group IV > group III > group II, and the maximum principal stress peak change was: group I > group III > group II > group IV. The change of the equivalent stress peak value of the loop under oblique load was: group I > group III > group IV > group II, and the maximum principal stress peak change was: group III > group I > group II > group IV. When the loop was not in occlusal contact with the opposing occlusal tooth, the peak value of the equivalent stress on the space maintainer under vertical load was: group IV > group I > group II > group III, and the maximum principal stress peak change was: group IV > group I > group II > group III. The change of the equivalent stress peak value of the space maintainer under oblique load was: group I > group IV > group II > group III, and the maximum principal stress peak change was: group I > group IV > group II > group III. Our results suggested that whenever possible, choosing the teeth with nearly complete root development as the abutment of the space maintainer is advisable. The design and use of the band and loop space maintainer should avoid occlusal contact with the occlusal teeth to prevent deformation of the loop.

Arm design of band and loop space maintainer affects its longevity: a patient-specific finite element study.

Fixed space maintainers (FSMs) are commonly utilized in pediatric dentistry to prevent space loss following premature tooth extraction. Although previous studies have examined the survival rates and causes of FSM failure, the impact of arm design on failure has not been investigated. This study aimed to investigate the tensile and compressive stresses related to FSMs with different arm designs and evaluate the effect of arm designs on FSM failure. Cone beam computed tomography images of a child who experienced premature loss of a primary mandibular left second molar tooth were retrieved from our database, then processed and simulated using the Rhinoceros software. Finite element analysis was performed to evaluate the stresses on four distinct FSM arm designs under simulated chewing forces. The results showed that the straight-arm FSM design exhibited the highest von Mises principal stress, while FSMs with curved arms and surrounding primary mandibular left first molar in the mesial area demonstrated the lowest von Mises stress accumulation. Intense stress accumulation on the distal surface of tooth 74 was observed in the test models due to the transmitted forces by the FSM. The maximum principal stresses accumulated at the base of the alveolar socket of the mesial root of tooth 36, while the minimum principal stresses were identified at the mesio-marginal area of the alveolar crest. The arm design played a crucial role in enabling the appliance to effectively withstand the stresses accumulating on the Space maintainer (SM) and orthodontic band. Bending the SM arms to match the surrounding profile with curvature increased the stress absorption capacity by increasing the arm length.

Cogroupoid structures on the circle and the Hodge degeneration

Abstract We exhibit the Hodge degeneration from nonabelian Hodge theory as a $2$ -fold delooping of the filtered loop space $E_2$ -groupoid in formal moduli problems. This is an iterated groupoid object which in degree $1$ recovers the filtered circle $S^1_{fil}$ of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an $E_2$ -cogroupoid object in the $\infty $ -category of spaces. We relate this cogroupoid structure with the more commonly studied ‘pinch map’ on $S^1$ , as well as the Todd class of the Lie algebroid $\mathbb {T}_{X}$ ; this is an invariant of a smooth and proper scheme X that arises, for example, in the Grothendieck-Riemann-Roch theorem. In particular, we relate the existence of nontrivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally, we record some consequences of this bit of structure at the level of Hochschild cohomology.

A cocyclic construction of $S^1$-equivariant homology and application to string topology

Given a space with a circle action, we study certain cocyclic chain complexes and prove a theorem relating cyclic homology to S^1 -equivariant homology, in the spirit of celebrated work of Jones. As an application, we describe a chain level refinement of the gravity algebra structure on the (negative) S^1 -equivariant homology of the free loop space of a closed oriented smooth manifold, based on work of Irie on chain level string topology and work of Ward on an S^1 -equivariant version of operadic Deligne’s conjecture.

Range decreasing group homomorphisms and holomorphic maps between generalized loop spaces

Range decreasing group homomorphisms and holomorphic maps between generalized loop spaces

Ramipril therapy in integrin α1-null, autosomal recessive Alport mice triples lifespan: mechanistic clues from RNA-seq analysis.

The standard of care for patients with Alport syndrome (AS) is angiotensin-converting enzyme (ACE) inhibitors. In autosomal recessive Alport (ARAS) mice, ACE inhibitors double lifespan. We previously showed that deletion of Itga1 in Alport mice [double-knockout (DKO) mice] increased lifespan by 50%. This effect seemed dependent on the prevention of laminin 211-mediated podocyte injury. Here, we treated DKO mice with vehicle or ramipril starting at 4 weeks of age. Proteinuria and glomerular filtration rates were measured at 5-week intervals. Glomeruli were analyzed for laminin 211 deposition in the glomerular basement membrane (GBM) and GBM ultrastructure was analyzed using transmission electron microscopy (TEM). RNA sequencing (RNA-seq) was performed on isolated glomeruli at all time points and the results were compared with cultured podocytes overlaid (or not) with recombinant laminin 211. Glomerular filtration rate declined in ramipril-treated DKO mice between 30 and 35 weeks. Proteinuria followed these same patterns with normalization of foot process architecture in ramipril-treated DKO mice. RNA-seq revealed a decline in the expression of Foxc2, nephrin (Nphs1), and podocin (Nphs2) mRNAs, which was delayed in the ramipril-treated DKO mice. GBM accumulation of laminin 211 was delayed in ramipril-treated DKO mice, likely due to a role for α1β1 integrin in CDC42 activation in Alport mesangial cells, which is required for mesangial filopodial invasion of the subendothelial spaces of the glomerular capillary loops. Ramipril synergized with Itga1 knockout, tripling lifespan compared with untreated ARAS mice. © 2023 The Pathological Society of Great Britain and Ireland. Published by John Wiley & Sons, Ltd.

Clinical application of a digital semi-rigid bridge space maintainer fabricated from polyetheretherketone for premature loss of primary molars

BackgroundPremature loss of primary molars can be treated with a band loop space maintainer (SM). However, fabricating a conventional band loop SM requires multiple clinical and laboratory procedures, which can potentially affect the accuracy of the SM. Moreover, the conventional SM is unable to fully restore masticatory function and maintain the vertical dimension of the edentulous space. In this current study, a fully digital workflow to fabricate a semi-rigid bridge SM made from polyetheretherketone (PEEK) has been described and evaluated for its clinical effectiveness.MethodsA total of 15 children (eight males and seven females) between the ages of 4–8 years, who experienced the premature loss of a single primary molar, were included in this study. Digital impressions were taken using the CEREC CAD/CAM chair system and imported into CAD software to design the semi-rigid bridge SM, which was fabricated using PEEK block as the maintainer material. The digital SM was tried-in and bonded to the abutment with resin cement. The edentulous space was measured immediately after bonding (T0) and 1 month (T1), 3 months (T2), and 6 months (T3) after treatment. The periodontal condition and mobility of the SM and abutment were also examined.ResultsThe use of digital impressions resulted in a decreased occurrence of the pharyngeal reflex. The digital semi-rigid bridge SM, fabricated with PEEK, was both convenient and aesthetically pleasing, and successfully restored the anatomy and masticatory function of the missing primary molar. None of the 15 semi-rigid bridge SMs or abutments became loose or fell off during the study, and only one child presented with gingivitis. Furthermore, the difference in the edentulous space at T0, T1, T2, and T3 was not statistically significant (all P > 0.05).ConclusionsThe digital semi-rigid bridge SM fabricated with PEEK was clinically effective in maintaining the missing space and had advantages over the traditional band/crown loop SM.

Genuine versus naïve symmetric monoidal $G$-categories

We prove that through the eyes of equivariant weak equivalences the genuine symmetric monoidal G -categories of Guillou and May [Algebr. Geom. Topol. 17 (2017), no. 6, 3259–3339] are equivalent to just ordinary symmetric monoidal categories with G -action. Along the way, we give an operadic model of global infinite loop spaces and provide an equivalence between the equivariant category theory of genuine symmetric monoidal G -categories and the G -parsummable categories studied by Schwede [J. Topol. 15 (2022), no. 3, 1325–1454] and the author [New York J. Math. 29 (2023), 635–686].

A Chern-Simons transgression formula for supersymmetric path integrals on spin manifolds

Earlier results show that the N=1/2 supersymmetric path integral Jg on a closed even dimensional Riemannian spin manifold (X,g) can be constructed in a mathematically rigorous way via Chen differential forms and techniques from noncommutative geometry, if one considers Jg as a current on the loop space LX, that is, as a linear form on differential forms on LX. This construction admits a Duistermaat-Heckman localization formula. In this note, fixing a topological spin structure on X, we prove that any smooth family g•=(gt)t∈[0,1] of Riemannian metrics on X canonically induces a Chern-Simons current Cg• which fits into a transgression formula for the supersymmetric path integral. In particular, this result entails that the supersymmetric path integral induces a differential topological invariant on X, which essentially stems from the Aˆ-genus of X.

A natural pseudometric on homotopy groups of metric spaces

Abstract For a path-connected metric space $(X,d)$ , the $n$ -th homotopy group $\pi _n(X)$ inherits a natural pseudometric from the $n$ -th iterated loop space with the uniform metric. This pseudometric gives $\pi _n(X)$ the structure of a topological group, and when $X$ is compact, the induced pseudometric topology is independent of the metric $d$ . In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on $\pi _n(X)$ . Our main result is that the pseudometric topology agrees with the shape topology on $\pi _n(X)$ if $X$ is compact and $LC^{n-1}$ or if $X$ is an inverse limit of finite polyhedra with retraction bonding maps.

On loop spaces of simplicial sets with marking

On loop spaces of simplicial sets with marking