Higher Dimensional Manifolds Research Articles

Pressure and exponential rate over periodic orbits

For a measure μ preserved by a C1+α0 (α0>0) diffeomorphism f and a continuous function ϕ on the manifold, we study the relationship between the exponential growth rate of ∑eSnϕ(x) over the orbits of some periodic points and the free energy hμ(f)+∫ϕdμ (in certain cases, it is equal to the measure theoretic pressure) or the topological pressure. When μ is an ergodic hyperbolic measure, we prove that the exponential growth rate coincides with the free energy (measure theoretic pressure). And we also verify the equality of the exponential growth rate and the topological pressure when the manifold is 2-dimensional. However, for the higher-dimensional manifold, we show an inequality between the exponential growth rate and the topological pressure. For an ergodic hyperbolic measure ω, we also prove that there is a ω-full measured set Λ˜ such that for every f-invariant measure supported on Λ˜, the exponential growth rate equals to the free energy. And moreover, we prove that there is another ω-full measured set Δ˜ such that for every f-invariant measure supported on Δ˜, the exponential growth rate equals to the topological pressure.

Morphometric Gaussian Process for Landmarking on Grey Matter Tetrahedral Models.

High-dimensional manifold modeling increases the precision and performance of cortical morphometry analysis by densely sampling on the grey matters. But this also brings redundant information and increased computational burden. Gaussian process regression has been used to tackle this problem by learning a mapping to a low-dimensional subspace. However, current methods may not take relevant morphometric properties, usually measured by geometric features, into account, and as a result, may generate morphometrically insignificant selections. In this paper, we propose a morphometric Gaussian process (M-GP) as a novel Bayesian model on the gray matter tetrahedral meshes. We also implement an M-GP regression landmarking algorithm as a manifold learning method for non-linear dimensionality reduction. The definition of M-GP involves a scale-invariant wave kernel signature distance map measuring the local similarities of geometric features, and a heat flow entropy which implicitly embeds the global curvature flow. With such a design, the prior knowledge fully encodes the geometric information so that a posterior predictive inference is morphometrically significant. In experiments, we use 518 grey matter tetrahedral meshes generated from structural magnetic resonance images of a publicly available Alzheimer's disease imaging cohort to empirically and numerically evaluate our method. The results verify that our method is theoretically and experimentally valid in selecting a representative subset from the original massive data. Our work may benefit any studies involving large-scale or iterative computations on extensive manifold-valued data, including morphometry analyses and general medical data processing.

Metric curvatures and their applications 2: metric Ricci curvature and flow

In this second part of our overview of the different metric curvatures and their various applications, we concentrate on the Ricci curvature and flow for polyhedral surfaces and higher dimensional manifolds, and we largely review our previous studies on the subject, based upon Wald's curvature. In addition to our previous metric approaches to the discretization of Ricci curvature, we consider yet another one, based on the Haantjes curvature, interpreted as a geodesic curvature. We also try to understand the mathematical reasons behind the recent proliferation of discretizations of Ricci curvature. Furthermore, we propose another approach to the metrization of Ricci curvature, based on Forman's discretization, and in particular we propose on that uses our graph version of Forman's Ricci curvature.

Strong Solution Existence for a Class of Degenerate Stochastic Differential Equations

Existence and uniqueness results for stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coefficient is nondegenerate. Here, existence and uniqueness of a strong solution is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solution of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coefficient is exactly half that of the state. In addition to the degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential and the resulting solutions to the dequantized Schrodinger equation. The first type consists of discontinuities that may occur on a possibly high-dimensional manifold (up to codimension one). The second consists of singularities that may occur on a lower-dimensional manifold (up to codimension two).

The General Non-Abelian Kuramoto Model on the 3-sphere

We introduce non-Abelian Kuramoto model on $ S^3 $ in the most general form. Following an analogy with the classical Kuramoto model (on the circle $ S^1 $), we study some interesting variations of the model on $ S^3 $ that are obtained for particular coupling functions. As a partial case, by choosing 'standard' coupling function one obtains a previously known model, that is referred to as Kuramoto-Lohe model on $ S^3 $.We briefly address two particular models: Kuramoto models on $ S^3 $ with frustration and with external forcing. These models on higher dimensional manifolds have not been studied so far. By choosing suitable values of parameters we observe different nontrivial dynamical regimes even in the simplest setup of globally coupled homogeneous population.Although non-Abelian Kuramoto models can be introduced on various symmetric spaces, we restrict our analysis to the case when underlying manifold is the 3-sphere. Due to geometric and algebraic properties of this specific manifold, variations of this model are meaningful and geometrically well justified.

More examples of pseudo-collars on high-dimensional manifolds

In a previous paper, we developed general techniques for constructing a variety of pseudo-collars, as defined by Guilbault and Tinsley, with roots in earlier work by Chapman and Siebenmann. As an application of our techniques, we exhibited an uncountable collection of pseudo-collars, all with the same boundary and similar fundamental group systems at infinity. Construction of that family was very specific; it relied on properties of Thompson's group V. In this paper, we provide a more general approach to constructing similar collections of examples. Instead of using Thompson's group V, we base our new examples on a broader and more common collection of groups, in particular, fundamental groups of certain hyperbolic manifolds.

What have I discovered from ocean phenomena

Almost everything I have done in the last twenty years is based on what I have discovered from studying ocean phenomena. Specifically, the waves on the surface of the ocean and their dynamics. Two items stand out: The first one is from studying the evolution of the waves from ripples to fully developed giant ocean waves. To quantify the changes, I found that frequency is very different from what had been defined traditionally through Fourier analysis. Ture frequency should be determined by differentiation of the phase function defined by an adaptive method rather than by any form of integral transform. Therefore, it should have ever changing instantaneous values. The second discovery is from studying wave turbulence interactions. To reveal the detailed dynamics, I found that spectral representation should not be a single line on the frequency and energy plane. True spectrum should be a high dimensional manifold to cover all the possible dynamic interactions: additive and multiplicative ones. These discoveries have enabled me to develop a set of nonlinear and nonstationary data analysis tools to study many other phenomena, from turbulence to brain waves. They have led me to new wonders I have never even contemplated before.

Negatively curved bundles in the Igusa stable range

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in the homotopy groups of the classifying space for smooth M-bundles, provided the dimension of M is large enough.

On a topological version of Pach's overlap theorem

Pach showed that every d + 1 sets of points Q 1 , … , Q d + 1 ⊂ R d contain linearly sized subsets P i ⊂ Q i such that all the transversal simplices that they span intersect. We show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size C ( log n ) 1 / ( d − 1 ) . We show that this is tight in dimension 2, for all surfaces other than S 2 . Surprisingly, the optimal bound for S 2 in the topological version of Pach's theorem is of the order ( log n ) 1 / 2 . We conjecture that, among higher dimensional manifolds, spheres are similarly distinguished. This improves upon the results of Bárány, Meshulam, Nevo and Tancer.

On analysis of inputs triggering large nonlinear neural responses Slow-fast dynamics in the Wendling neural mass model

Many applications in neuroscience, such as electrical and magnetic stimulation, can be modelled as short transient input to non-linear dynamical systems. In excitable systems, small input yields more or less linear responses, while for increasing stimulation strength large non-linear responses may show up suddenly. A challenging task is to determine the transition between the two different response types.In this work we consider such a transition between normal and pathological responses in a model of coupled Wendling neural masses as we encountered in a previous study. First, the different timescales of inhibition in this model allow a slow-fast analysis. This reveals two different dynamical regimes for the systems’ response. Second, the two response types are separated by a high-dimensional stable manifold of a saddle slow manifold. Large pathological responses appear if the fast subsystem escapes from this manifold to another attractor. The typical fast oscillations seen during the pathological responses are explained by the bifurcation diagram of the fast subsystem. Under normal conditions these oscillations are suppressed by slow inhibition. External stimulation temporarily releases the fast subsystem from this slow inhibition. The critical response can be formulated as a boundary value problem with one free parameter and can be used to study the dependency of the transition between the two response types upon the system parameters.

Bounded distortion tetrahedral metric interpolation

We present a method for volumetric shape interpolation with unique shape preserving features. The input to our algorithm are two or more 3-manifolds, immersed into R 3 and discretized as tetrahedral meshes with shared connectivity. The output is a continuum of shapes that naturally blends the input shapes, while striving to preserve the geometric character of the input. The basis of our approach relies on the fact that the space of metrics with bounded isometric and angular distortion is convex [Chien et al. 2016b]. We show that for high dimensional manifolds, the bounded distortion metrics form a positive semidefinite cone product space. Our method can be seen as a generalization of the bounded distortion interpolation technique of [Chen et al. 2013] from planar shapes immersed in R 2 to solids in R 3 . The convexity of the space implies that a linear blend of the (squared) edge lengths of the input tetrahedral meshes is a simple yet powerful-and-natural choice. Linearly blending flat metrics results in a new metric which is, in general, not flat, and cannot be immersed into three-dimensional space. Nonetheless, the amount of curvature that is introduced in the process tends to be very low in practical settings. We further design an extremely robust nonconvex optimization procedure that efficiently flattens the metric. The flattening procedure strives to preserve the low distortion exhibited in the blended metric while guaranteeing the validity of the metric, resulting in a locally injective map with bounded distortion. Our method leads to volumetric interpolation with superb quality, demonstrating significant improvement over the state-of-the-art and qualitative properties which were obtained so far only in interpolating manifolds of lower dimensions.

Geometric Models for Lie–Hamilton Systems on ℝ2

This paper provides a geometric description for Lie–Hamilton systems on R 2 with locally transitive Vessiot–Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie–Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie–Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give a natural framework for the analysis of Lie–Hamilton systems on R 2 while retrieving known results in a natural manner. Our methods may be extended to study Lie–Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

Homotopy of gauge groups over non-simply-connected five-dimensional manifolds

Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper, we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity. Our treatments here are also very effective for rational gauge groups in the general context, and applicable for higher dimensional manifolds.

Shake genus and slice genus

An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-handle to the 4-ball along a knot $K$ in $S^3$. The $0$-shake genus of $K$ records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of $X_0(K)$ and is clearly bounded above by the slice genus of $K$. We prove that slice genus is not an invariant of $X_0(K)$, and thereby provide infinitely many examples of knots with $0$-shake genus strictly less than slice genus. This resolves Problem 1.41 of [Kir97]. As corollaries we show that Rasmussen's $s$ invariant is not a $0$-trace invariant and we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but do not preserve slice genus. These corollaries resolve some questions from [4MKC16].

Homological stability of topological moduli spaces

Given a graded $E_1$-module over an $E_2$-algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for the graded pieces of the module with respect to constant and abelian coefficients. We furthermore introduce a notion of coefficient systems of finite degree in this context and show that, without further assumptions, the corresponding twisted homology groups stabilize as well. This generalizes a framework of Randal-Williams and Wahl for families of discrete groups. In many examples, the canonical resolution recovers geometric resolutions with known connectivity bounds. As a consequence, we derive new twisted homological stability results for e.g. moduli spaces of high-dimensional manifolds, unordered configuration spaces of manifolds with labels in a fibration, and moduli spaces of manifolds equipped with unordered embedded discs. This in turn implies representation stability for the ordered variants of the latter examples.

Momentum Sections in Hamiltonian Mechanics and Sigma Models

We show a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently introduced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic manifold by considering gauged sigma models on a higher dimensional manifold.

On some examples and constructions of contact manifolds

The first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties. Our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and the fact that every closed contact 3-manifold, which is not (smoothly) a rational homology sphere, contact-embeds with trivial normal bundle inside a hypertight closed contact 5-manifold. This uses known construction procedures by Bourgeois (on products with tori) and Geiges (on branched covering spaces). We pass from these procedures to definitions; this allows to prove a uniqueness statement in the case of contact branched coverings, and to study the global properties (such as tightness and fillability) of the results of both constructions without relying on any auxiliary choice in the procedures. A second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book, as introduced by Giroux, and of contact fiber bundle, as introduced by Lerman. For instance, we give a definition of Bourgeois contact structures on flat contact fiber bundles which is local, (strictly) includes the results of the Bourgeois construction, and allows to recover an isotopy class of supporting open books on the fibers. This last point relies on a reinterpretation, inspired by an idea by Giroux, of supporting open books in terms of pairs of contact vector fields.

Topologie

The talks covered advances in algebraic K -theory and topological cyclic homology, geometric group theory, low dimensional topology relying on a mixture of combinatorial and analytic methods, classification of high-dimensional manifolds and more. Special emphasis was given to a recent breakthrough on the question of triangulability of high-dimensional manifolds.

An obstruction to the existence of real projective structures

In this short note, we give an obstruction to obtain examples of higher dimensional manifolds with infinite fundamental groups, including the infinite cyclic group Z, admitting no real projective structure.

Heteroclinic and homoclinic connections in a Kolmogorov-like flow.

Recent studies suggest that unstable recurrent solutions of the Navier-Stokes equation provide new insights into dynamics of turbulent flows. In this study, we compute an extensive network of dynamical connections between such solutions in a weakly turbulent quasi-two-dimensional Kolmogorov flow that lies in the inversion-symmetric subspace. In particular, we find numerous isolated heteroclinic connections between different types of solutions-equilibria, periodic, and quasiperiodic orbits-as well as continua of connections forming higher-dimensional connecting manifolds. We also compute a homoclinic connection of a periodic orbit and provide strong evidence that the associated homoclinic tangle forms the chaotic repeller that underpins transient turbulence in the symmetric subspace.