Convergence and Stability of Locally ℝ N -Invariant Solutions of Ricci Flow

Valuable models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally $\mathcal{G}Valuable models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally G\mathcal{G} -invariant solutions on bundles G^N\hookrightarrowM \oversetp? Bⁿ\mathcal{G}^{N}\hookrightarrow\mathcal{M}\,\overset{\pi }{\mathcal{\longrightarrow}}\,\mathcal{B}^{n} , with G\mathcal{G} a nilpotent Lie group. In this paper, we establish convergence and asymptotic stability, modulo smooth finite-dimensional center manifolds, of certain ℝ ^N -invariant model solutions. In case N+n=3, our results are relevant to work of Lott classifying the asymptotic behavior of all 3-dimensional Ricci flow solutions whose sectional curvatures and diameters are respectively O(t^-1)\mathcal{O}(t^{-1}) and O(t^1/2)\mathcal{O}(t^{1/2}) as t→∞.     

Triangulating Smooth Submanifolds with Light Scaffolding

We propose an algorithm to sample and mesh a k-submanifold M{\mathcal{M}} of positive reach embedded in \mathbbR^d{\mathbb{R}^{d}} . The algorithm first constructs a crude sample of M{\mathcal{M}} . It then refines the sample according to a prescribed parameter e{\varepsilon} , and builds a mesh that approximates M{\mathcal{M}} . Differently from most algorithms that have been developed for meshing surfaces of \mathbbR ³{\mathbb{R} ^3} , the refinement phase does not rely on a subdivision of \mathbbR ^d{\mathbb{R} ^d} (such as a grid or a triangulation of the sample points) since the size of such scaffoldings depends exponentially on the ambient dimension d. Instead, we only compute local stars consisting of k-dimensional simplices around each sample point. By refining the sample, we can ensure that all stars become coherent leading to a k-dimensional triangulated manifold [^(M)]{\hat{\mathcal{M}}} . The algorithm uses only simple numerical operations. We show that the size of the sample is O(e^-k){O(\varepsilon ^{-k})} and that [^(M)]{\hat{\mathcal{M}}} is a good triangulation of M{\mathcal{M}} . More specifically, we show that M{\mathcal{M}} and [^(M)]{\hat{\mathcal{M}}} are isotopic, that their Hausdorff distance is O(e²){O(\varepsilon ^{2})} and that the maximum angle between their tangent bundles is O(e){O(\varepsilon )} . The asymptotic complexity of the algorithm is T(e) = O(e^-k2-k){T(\varepsilon) = O(\varepsilon ^{-k^2-k})} (for fixed M, d{\mathcal{M}, d} and k).     

On the Unital C*-Algebras Generated by Certain Subnormal Tuples

We consider an important class of subnormal operator m-tuples M _p (p = m,m + 1, . . .) that is associated with a class of reproducing kernel Hilbert spaces H_p{{\mathcal H}_p} (with M _m being the multiplication tuple on the Hardy space of the open unit ball \mathbb B^2m{{\mathbb B}^{2m}} in \mathbb C^m{{\mathbb C}^m} and M _m+1 being the multiplication tuple on the Bergman space of \mathbb B^2m{{\mathbb B}^{2m}}). Given any two C*-algebras A{\mathcal A} and B{\mathcal B} from the collection {C^*(M_p), C^*([(M)\tilde]_p): p 3 m}{\{C^*({M}_p), C^*({\tilde M}_p): p \geq m\}} , where C*(M _p ) is the unital C*-algebra generated by M _p and C^*([(M)\tilde]_p){C^*({\tilde M}_p)} the unital C*-algebra generated by the dual [(M)\tilde]_p{{\tilde M}_p} of M _p , we verify that A{\mathcal A} and B{\mathcal B} are either *-isomorphic or that there is no homotopy equivalence between A{\mathcal A} and B{\mathcal B} . For example, while C*(M _m ) and C*(M _m+1) are well-known to be *-isomorphic, we find that C^*([(M)\tilde]_m){C^*({\tilde M}_m)} and C^*([(M)\tilde]_m+1){C^*({\tilde M}_{m+1})} are not even homotopy equivalent; on the other hand, C*(M _m ) and C^*([(M)\tilde]_m){C^*({\tilde M}_{m})} are indeed *-isomorphic. Our arguments rely on the BDF-theory and K-theory.     

Higher-Order Averaging, Formal Series and Numerical Integration I: B-series

We show how B-series may be used to derive in a systematic way the analytical expressions of the high-order stroboscopic averaged equations that approximate the slow dynamics of highly oscillatory systems. For first-order systems we give explicitly the form of the averaged systems with O(e^j)\mathcal{O}(\epsilon^{j}) errors, j=1,2,3 (2π ε denotes the period of the fast oscillations). For second-order systems with large O(e^-1)\mathcal{O}(\epsilon^{-1}) forces, we give the explicit form of the averaged systems with O(e^j)\mathcal{O}(\epsilon^{j}) errors, j=1,2. A variant of the Fermi–Pasta–Ulam model and the inverted Kapitsa pendulum are used as illustrations. For the former it is shown that our approach establishes the adiabatic invariance of the oscillatory energy. Finally we use B-series to analyze multiscale numerical integrators that implement the method of averaging. We construct integrators that are able to approximate not only the simplest, lowest-order averaged equation but also its high-order counterparts.     

Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

Let (M,[(g)\tilde]){(\mathcal {M},\tilde{g})} be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation

FFTs on the Rotation Group

We discuss an implementation of an efficient algorithm for the numerical computation of Fourier transforms of bandlimited functions defined on the rotation group SO(3). The implementation is freely available on the web. The algorithm described herein uses O(B ⁴) operations to compute the Fourier coefficients of a function whose Fourier expansion uses only (the O(B ³)) spherical harmonics of degree at most B. This compares very favorably with the direct O(B ⁶) algorithm derived from a basic quadrature rule on O(B ³) sample points. The efficient Fourier transform also makes possible the efficient calculation of convolution over SO(3) which has been used as the analytic engine for some new approaches to searching 3D databases (Funkhouser et al., ACM Trans. Graph. 83–105, [2003]; Kazhdan et al., Eurographics Symposium in Geometry Processing, pp. 167–175, [2003]). Our implementation is based on the “Separation of Variables” technique (see, e.g., Maslen and Rockmore, Proceedings of the DIMACS Workshop on Groups and Computation, pp. 183–237, [1997]). In conjunction with techniques developed for the efficient computation of orthogonal polynomial expansions (Driscoll et al., SIAM J. Comput. 26(4):1066–1099, [1997]), our fast SO(3) algorithm can be improved to give an algorithm of complexity O(B ³log ² B), but at a cost in numerical reliability. Numerical and empirical results are presented establishing the empirical stability of the basic algorithm. Examples of applications are presented as well. First author was supported by NSF ITR award; second author was supported by NSF Grant 0219717 and the Santa Fe Institute.     

The Moduli Space of Points in the Boundary of Complex Hyperbolic Space

We consider the space M(n,m)\mathcal{M}(n,m) of ordered m-tuples of distinct points in the boundary of complex hyperbolic n-space, H_\mathbbCⁿ\mathbf{H}_{\mathbb{C}}^{n}, up to its holomorphic isometry group PU(n,1). An important problem in complex hyperbolic geometry is to construct and describe the moduli space for M(n,m)\mathcal{M}(n,m). In particular, this is motivated by the study of the deformation space of complex hyperbolic groups generated by loxodromic elements. In the present paper, we give the complete solution to this problem.     

The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3

In the moduli space M \mathcal{M} _g of genus-g Riemann surfaces, consider the locus RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} of Riemann surfaces whose Jacobians have real multiplication by the order O \mathcal{O} in a totally real number field F of degree g. If g = 3, we compute the closure of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} in the Deligne–Mumford compactification of M \mathcal{M} _g and the closure of the locus of eigenforms over RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} . Boundary strata of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} are parameterized by configurations of elements of the field F satisfying a strong geometry of numbers type restriction.     

Projective MV-algebras and rational polyhedra

Let M be an n-generator projective MV-algebra. Then there is a rational polyhedron P in the n-cube [0, 1] ⁿ such that M is isomorphic to the MV-algebra M(P){{\rm{\mathcal {M}}}(P)} of restrictions to P of the McNaughton functions of the free n-generator MV-algebra. P necessarily contains a vertex vP of the n-cube. We characterize those polyhedra contained in the n-cube such that M(P){{\mathcal {M}}(P)} is projective. In particular, if the rational polyhedron P is a union of segments originating at some fixed vertex vP of the n-cube, then M(P){{\mathcal {M}}(P)} is projective. Using this result, we prove that if A = M(P){A = {\mathcal {M}}(P)} and B = M(Q){B = {\mathcal {M}}(Q)} are projective, then so is the subalgebra of A × B given by {(f, g) | f(v _P ) = g(v _Q ), and so is the free product A \coprod B{A \coprod B} .     

Foliated Lie and Courant Algebroids

If A is a Lie algebroid over a foliated manifold (M, F){(M, {\mathcal {F}})}, a foliation of A is a Lie subalgebroid B with anchor image TF{T{\mathcal {F}}} and such that A/B is locally equivalent with Lie algebroids over the slice manifolds of F{\mathcal F}. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure and Vaintrob's supervector field, cohomology and deformations of the foliation, integration to a Lie groupoid. In the last section, we define a corresponding notion of a foliation of a Courant algebroid A as a bracket–closed, isotropic subbundle B with anchor image TF{T{\mathcal {F}}} and such that B ^{^} /B{B^{ \bot } /B} is locally equivalent with Courant algebroids over the slice manifolds of F{\mathcal F}. Examples that motivate the definition are given.     

Vertex algebras and the formal loop space

We construct a certain algebro-geometric version L(X)\mathcal{L}(X) of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme L⁰(X)\mathcal{L}^{0}(X) of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on L(X)\mathcal{L}(X) supported in L⁰(X)\mathcal{L}^{0}(X) . We also show that L(X)\mathcal{L}(X) possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that all linear constructions applied to the free loop space produce vertex algebras.     

Slices of the unitary spread

We prove that slices of the unitary spread of Q⁺(7,q)\mathcal{Q}^{+}(7,q), q≡2 (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under the action of the subgroup of PΓO ⁺(8,q) fixing the unitary spread. When q is even, there is a connection between spreads of Q⁺(7,q)\mathcal{Q}^{+}(7,q) and symplectic 2-spreads of PG(5,q) (see Dillon, Ph.D. thesis, 1974 and Dye, Ann. Mat. Pura Appl. (4) 114, 173–194, 1977). As a consequence of the above result we determine all the possible non-equivalent symplectic 2-spreads arising from the unitary spread of Q⁺(7,q)\mathcal{Q}^{+}(7,q), q=2^2h+1. Some of these already appeared in Kantor, SIAM J. Algebr. Discrete Methods 3(2), 151–165, 1982. When q=3 ^h , we classify, up to the action of the stabilizer in PΓO(7,q) of the unitary spread of Q(6,q), those among its slices producing spreads of the elliptic quadric Q^-(5,q)\mathcal{Q}^{-}(5,q).     

Sampling Sets and Quadrature Formulae on the Rotation Group

In this paper, we construct sampling sets over the rotation group SO(3). The proposed construction is based on a parameterization, which reflects the product nature ² × ¹ of SO(3) very well, and leads to a spherical Pythagorean-like formula in the parameter domain. We prove that by using uniformly distributed points on ² and ¹, we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on ² and ¹ lead to quadratures on SO(3), as well. For scattered data on SO(3), we give a necessary condition on the mesh norm such that the sampling nodes possess nonnegative quadrature weights. We propose an algorithm for computing the quadrature weights for scattered data on SO(3) based on fast algorithms. We confirm our theoretical results with examples and numerical tests.     

Quasi-interpolation operators based on the trivariate seven-direction <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript> quartic box spline

This paper investigates the space S(X)\mathcal{S}(X) generated by the integer translates of the trivariate C ² quartic box spline B defined by a set X of seven directions that forms a regular partition of the space into tetrahedra.     

On a new construction in group theory

This paper continues the investigation of the groups RF(G)\mathcal{RF}(G) introduced and studied in [I.M. Chiswell and T.W. Müller, A class of groups with canonical ℝ-tree action, Springer LNM, to appear]. Two new concepts, that of a test function, and that of a pair of locally incompatible (test) functions are introduced, and their theory is developed. As application, we obtain a number of new quantitative as well as structural results concerning RF(G)\mathcal{RF}(G) and its quotient RF(G)/E(G)\mathcal{RF}(G)/E(G) modulo the subgroup E(G) generated by the elliptic elements. Among other things, the cardinality of RF(G)\mathcal{RF}(G) is determined, and it is shown that both RF(G)\mathcal{RF}(G) and RF(G)/E(G)\mathcal{RF}(G)/E(G) contain large free subgroups, and that their abelianizations both contain a large ℚ-vector space as direct summand.     

Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups

Let G(O_S)\mathbf{G}(\mathcal{O}_{S}) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S)\mathbf{G}(\mathcal{O}_{S}) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O_S)\mathbf{G}(\mathcal{O}_{S}) established in an earlier paper is sharp in this case.     

Tension spline solution of nonlinear sine-Gordon equation

The sine-Gordon equation plays an important role in modern physics. By using spline function approximation, two implicit finite difference schemes are developed for the numerical solution of one-dimensional sine-Gordon equation. Stability analysis of the method has been given. It has been shown that by choosing the parameters suitably, we can obtain two schemes of orders O(k²+k²h²+h²)\mathcal{O}(k^{2}+k^{2}h^{2}+h^{2}) and O(k²+k²h²+h⁴)\mathcal{O}(k^{2}+k^{2}h^{2}+h^{4}). At the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.     

On realcompactifications defined by bornologies

For each bornology B\mathcal{B} on a Tychonoff space X we can construct a realcompactification u_B(X)\upsilon_{\mathcal{B}}(X) of X with the property that it contains all elements of B\mathcal{B} as relatively compact subsets. Here we give various characterizations of the realcompactifications of a space X that are defined in this way.     

Spectral Concentration of Positive Functions on?Compact Groups

The problem of understanding the Fourier-analytic structure of the cone of positive functions on a group has a long history. In this article, we develop the first quantitative spectral concentration results for such functions over arbitrary compact groups. Specifically, we describe a family of finite, positive quadrature rules for the Fourier coefficients of band-limited functions on compact groups. We apply these quadrature rules to establish a spectral concentration result for positive functions: given appropriately nested band limits A ì B ì [^(G)]\mathcal {A}\subset \mathcal {B} \subset\widehat{G}, we prove a lower bound on the fraction of L ²-mass that any B\mathcal {B}-band-limited positive function has in A\mathcal {A}. Our bounds are explicit and depend only on elementary properties of A\mathcal {A} and B\mathcal {B}; they are the first such bounds that apply to arbitrary compact groups. They apply to finite groups as a special case, where the quadrature rule is given by the Fourier transform on the smallest quotient whose dual contains the Fourier support of the function.     

On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms

Let M ⁿ be a Riemannian n-manifold. Denote by S(p) and [`(Ric)](p)\overline {Ric}(p) the Ricci tensor and the maximum Ricci curvature on M <sup>n</sup> at a point p ? M<sup>n</sup>p\in M^n, respectively. First we show that every isotropic submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies S ￡ ((n-1)c+ [(n<sup>2</sup>)/4] H<sup>2</sup>)gS\leq ((n-1)c+ {n^2 \over 4} H^2)g, where H<sup>2</sup> and g are the squared mean curvature function and the metric tensor on M <sup>n</sup>, respectively. The equality case of the above inequality holds identically if and only if either M <sup>n</sup> is totally geodesic submanifold or n = 2 and M <sup>n</sup> is a totally umbilical submanifold. Then we prove that if a Lagrangian submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies [`(Ric)] = (n-1)c+ [(n²)/4] H²\overline {Ric}= (n-1)c+ {n^2 \over 4} H^2 identically, then it is a minimal submanifold. Finally, we describe the geometry of Lagrangian submanifolds which satisfy the equality under the condition that the dimension of the kernel of second fundamental form is constant.