We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, with the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai-Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern-Weil map. We give interesting cohomology classes in the new theory that have no classical analogues.

The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian G -manifold is a stratified symplectic space. Suppose $1\ra A\ra G\ra T\ra 1$ is an exact sequence of compact Lie groups and T is a torus. Then the reduction of a Hamiltonian G -manifold with respect to A yields a Hamiltonian T -space. We show that if the A -moment map is proper, then the convexity theorem holds for such a Hamiltonian T -space, even when it is singular. We also prove that if, furthermore, the T -space has dimension 2dimT and T acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case.

For any vertex algebra V and any subalgebra A of V, there is a new subalgebra of V known as the commutant of A in V. This construction was introduced by Frenkel-Zhu, and is a generalization of an earlier construction due to Kac-Peterson and Goddard-Kent-Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (ie, a pair of mutual commutants) in the vertex algebra setting.

This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of O(sg) algebras. The new cohomology theory can be viewed as a kind of "chiralization'' of the classical equivariant cohomology, the latter being defined on the category of $G^*$ algebras a la H. Cartan. In this paper, we further develop the chiral theory by first extending it to allow a much larger class of algebras which we call sg[t] algebras. In the geometrical setting, our principal example of an O(sg) algebra is the chiral de Rham complex Q(M) of a G manifold M. There is an interesting subalgebra of Q(M) which does not admit a full O(sg) algebra structure but retains the structure of an sg[t] algebra, enough for us to define its chiral equivariant cohomology. The latter then turns out to have many surprising features that allows us to delineate a number of interesting geometric aspects of the G manifold M, sometimes in ways that are quite different from the classical theory.

This project was sponsored through the Schiff Fellowship program of Brandeis University. This project involved using the power series method to construct a third order nonlinear ordinary differential equation, a Schwarzian equation, for each of the "genus zero" modular functions, described in the Conway-Norton paper. We first use the Borcherd recursion formuli to generate, in each case, a modular function up to whatever degree we desire, and then use the fact that there is a Schwarzian equation, determined by a single rational function we call a Q-value. By similar power series methods, we compute the coefficients of our rational function, and hence have all the necessary data to create a Schwarzian differential equation for each modular function. This equation can, in turn, be used to recover the modular function itself.

This is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra, and contains and generalizes the classical equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In this paper, we compute this cohomology for spheres and show that for any simple connected group G, there is a sphere with infinitely many actions of G which have distinct chiral equivariant cohomology, but identical classical equivariant cohomology. Unlike the classical case, the description of the chiral equivariant cohomology of spheres requires a substantial amount of new structural theory, which we fully develop in this paper. This includes a quasi-conformal structure, equivariant homotopy invariance, and the values of this cohomology on homogeneous spaces. These results rely on crucial features of the underlying vertex algebra valued complex that have no classical analogues. ,

We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a toric Calabi-Yau hypersurface, and deriving a general formula for the d -exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for Abel-Jacobi map, which has played an important role in recent study of D-branes [25]. Using the variation formalism, we prove that the relative periods of toric B-branes on a toric Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature [6], and discuss the relations between the works [25] [21] [6] in recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.

We study period integrals of CY hypersurfaces in a partial flag variety. We construct a holonomic system of differential equations which govern the period integrals. By means of representation theory, a set of generators of the system can be described explicitly. The results are also generalized to CY complete intersections. The construction of these new systems of differential equations have lead us to the notion of a tautological system.

We develop a global Poincar\'e residue formula to study period integrals of families of complex manifolds. For any compact complex manifold X equipped with a linear system V ∗ of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on X . Two important ingredients of our construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize our construction to CY and general type complete intersections. When X is an algebraic manifold having a sufficiently large automorphism group G and V ∗ is a linear representation of G , we construct a holonomic D-module that governs the period integrals. The construction is based in part on the theory of tautological systems we have developed in the paper \cite{LSY1}, joint with R. Song. The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular. In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety.

A tautological system, introduced in \cite{LSY}\cite{LY}, arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold X , equipped with a suitable Lie group action. In this article, we introduce two formulas -- one purely algebraic, the other geometric -- to compute the rank of the solution sheaf of such a system for CY hypersurfaces in a generalized flag variety. The algebraic version gives the local solution space as a Lie algebra homology group, while the geometric one as the middle de Rham cohomology of the complement of a hyperplane section in X . We use both formulas to find certain degenerate points for which the rank of the solution sheaf becomes 1. These rank 1 points appear to be good candidates for the so-called large complex structure limits in mirror symmetry. The formulas are also used to prove a conjecture of Hosono, Lian and Yau on the completeness of the extended GKZ system when X is $\P^n$.

A tautological system, introduced in [16][17], arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold X , equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [4], and was verified for the case of projective homogeneous space under an assumption. In this paper, we prove this conjecture in full generality. By means of the Riemann-Hilbert correspondence and Fourier transforms, we also generalize the rank formula to an arbitrary projective manifold with a group action.

A CY bundle on a compact complex manifold X was a crucial ingredient in constructing differential systems for period integrals in \cite{LY}, by lifting line bundles from the base X to the total space. A question was therefore raised as to whether there exists such a bundle that supports the liftings of all line bundles from X , simultaneously. This was a key step for giving a uniform construction of differential systems for arbitrary complete intersections in X . In this paper, we answer the existence question in the affirmative if X is assumed to be K\"ahler, and also in general if the Picard group of X is assumed to be free. Furthermore, we prove a rigidity property of CY bundles if the principal group is an algebraic torus, showing that such a CY bundle is essentially determined by its character map.

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system , arising from the universal family of Calabi-Yau hypersurfaces $Y_a$ in a G-variety $X$ of dimension $n$. First, we construct a natural topological correspondence between relative cycles in $H_n (X − Y_a, \cup D − Y_a)$ bounded by the union of G-invariant divisors $\cup D$ in $X$ to the solution sheaf of $\tau$, in the form of chain integrals. Applying this to a toric variety with torus action, we show that in addition to the period integrals over cycles in $Y_a$, the new chain integrals generate the full solution sheaf of a GKZ system. This extends an earlier result for hypersurfaces in a projective homogeneous variety, whereby the chains are cycles [3, 7]. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of $H_n(X − Ya, \cup D − Y_a)$. In addition, we generalize the result on chain integral solutions to the case of general type hypersurfaces. This chain integral correspondence can also be seen as the Riemann-Hilbert correspondence in one homological degree. Finally, we consider interesting cases in which the chain integral correspondence possibly fails to be bijective.

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Period mappings were introduced in the sixties [G] to study variation of complex structures of families of algebraic varieties. The theory of tautological systems was introduced recently [LSY,LY] to understand period integrals of algebraic manifolds. In this paper, we give an explicit construction of a tautological system for each component of a period mapping.

Finding correspondences between two surfaces is a fundamental operation in various applications in computer graphics and related fields. Candidate correspondences can be found by matching local signatures, but as they only consider local geometry, many are globally inconsistent. We provide a novel algorithm to prune a set of candidate correspondences to those most likely to be globally consistent. Our approach can handle articulated surfaces, and ones related by a deformation which is globally non-isometric, provided that the deformation is locally approximately isometric. Our approach uses an efficient diffusion framework, and only requires geodesic distance calculations in small neighbourhoods, unlike many existing techniques which require computation of global geodesic distances. We demonstrate that, for typical examples, our approach provides significant improvements in accuracy, yet also reduces time and memory costs by a factor of several hundred compared to existing pruning techniques. Our method is furthermore insensitive to holes, unlike many other methods.

Identifying multiple deformable parts on meshes and establishing dense correspondences between them are tasks of fundamental importance to computer graphics, with applications to e.g. geometric edit propagation and texture transfer. Much research has considered establishing correspondences between non-rigid surfaces, but little work can both identify similar multiple deformable parts and handle partial shape correspondences. This paper addresses two related problems, treating them as a whole: (i) identifying similar deformable parts on a mesh, related by a non-rigid transformation to a given query part, and (ii) establishing dense point correspondences automatically between such parts. We show that simple and efficient techniques can be developed if we make the assumption that these parts locally undergo isometric deformation. Our insight is that similar deformable parts are suggested by large clusters of point correspondences that are isometrically consistent. Once such parts are identified, dense point correspondences can be obtained by an iterative propagation process. Our techniques are applicable to models with arbitrary topology. Various examples demonstrate the effectiveness of our techniques.

Centroidal Voronoi tessellations and optimal Delaunay triangulations can be approximated eciently by non-linear optimisation algorithms. This paper demonstrates that the point distribution used to initialise the optimisation algorithms is important. Compared to conventional random initialisation, certain low-discrepancy point distributions help convergence towards more spatially regular results and require fewer iterations for planar and volumetric tessellations.

Remeshing aims to produce a more regular mesh from a given input mesh, while representing the original geometry as accurately as possible. Many existing remeshing methods focus on where to place new mesh vertices; these samples are placed exactly on the input mesh. However, considering the output mesh as a piecewise linear approximation of some geometry, this simple scheme leads to significant systematic error in non-planar regions. Here, we use parameterised meshes and the recent mathematical development of orthogonal approximation using Sobolev-type inner products to develop a novel sampling scheme which allows vertices to lie in space near the input surface, rather than exactly on it. The algorithm requires little extra computational effort and can be readily incorporated into many remeshing approaches. Experimental results show that on average, approximation error can be reduced by 40% with the same number of vertices.

Non-rigid registration of 3D shapes is an essential task of increasing importance as commodity depth sensors become more widely available for scanning dynamic scenes. Non-rigid registration is much more challenging than rigid registration as it estimates a set of local transformations instead of a single global transformation, and hence is prone to the overfitting issue due to underdetermination. The common wisdom in previous methods is to impose an l2-norm regularization on the local transformation differences. However, the l2-norm regularization tends to bias the solution towards outliers and noise with heavy-tailed distribution, which is verified by the poor goodness- of-fit of the Gaussian distribution over transformation differences. On the contrary, Laplacian distribution fits well with the transformation differences, suggesting the use of a sparsity prior. We propose a sparse non-rigid registration (SNR) method with an l1-norm regularized model for transformation estimation, which is effectively solved by an alternate direction method (ADM) under the augmented Lagrangian framework. We also devise a multi-resolution scheme for robust and progressive registration. Results on both public datasets and our scanned datasets show the superiority of our method, particularly in handling large-scale deformations as well as outliers and noise.

Shape registration is fundamental to 3D object acquisition; it is used to fuse scans from multiple views. Existing algorithms mainly utilize geometric information to determine alignment, but this typically results in noticeable misalignment of textures (i.e. surface colors) when using RGB-depth cameras. We address this problem using a novel approach to color-aware registration, which takes both color and geometry into consideration simultaneously. Color information is exploited throughout the pipeline to provide more effective sampling, correspondence and alignment, in particular for surfaces with detailed textures. Our method can furthermore tackle both rigid and non-rigid registration problems (arising, for example, due to small changes in the object during scanning, or camera distortions). We demonstrate that our approach produces significantly better results than previous methods.

This paper deals with the merging problem, i.e. to approximate two adjacent Bézier curves by a single Bézier curve. A novel approach for approximate merging is introduced in the paper by using the constrained optimization method. The basic idea of this method is to find conditions for the precise merging of Bézier curves first, and then compute the constrained optimization solution by moving the control points. “Discrete” coefficient norm in L2 sense and “squared difference integral” norm are used in our method. Continuity at the endpoints of curves are considered in the merging process, and approximate merging with points constraints are also discussed. Further, it is shown that the degree elevation of original Bézier curves will reduce the merging error.

In this paper, we propose the generalized B divided difference, with which the $(k−1)$th derivative of B-spline curves of order $k$ can be obtained directly without the need to compute the first $(k−2)$ derivatives as before. Based on the generalized B divided difference, the necessary and sufficient condition for degree-reducible B-spline curves is presented. Algorithms for degree reduction of B-spline curves are proposed using the constrained optimization methods.

The ability to directly manipulate an embedded object in the free-form deformation (FFD) method improves controllability. However, the existing solution to this problem involves a pseudo-inverse matrix that requires complicated calculations. This paper solves the problem using a constrained optimization method. We derive the explicit solutions for deforming an object which is to pass through a given target point. For constraints with multiple target points, the proposed solution also involves simple calculations, only requiring solving a system of linear equations.We show that the direct manipulations exhibit the commutative group property, namely commutative, associative, and invertible properties, which further enhance the controllability of FFD. In addition, we show that multiple point constraints can be decomposed into separate manipulations of single point constraints, thus providing the user the freedom of specifying the constraints in any appropriate order.

In this paper, two explicit conversion formulae between triangular and rectangular Bézier patches are derived. Using the formulae, one triangular Bézier patch of degree n can be converted into one rectangular Bézier patch of degree $n \times n$. And one rectangular Bézier patch of degree m × n can be converted into two triangular Bézier patches of degree $m + n$ . Besides, two stable recursive algorithms corresponding to the two conversion formulae are given. Using the algorithms, when converting triangular Bézier patches to rectangular Bézier patches, we can computer the relations between the control points of the two types of patches for any $n\ge2$ based on the relationships for $n=1$. When converting rectangular Bézier patches to triangular Bézier patches, we can computer the relations between the control points of the two types of patches for any $m\ge 2, n \subset N^+ $ and $n \ge 2, m \subset N^+$ based on the relationships for $m=n=1$.