This paper develops a novel computational technique to define and construct manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to an arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite its earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (ie,| 2 g 2| for any genus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their

Motivated by mirror symmetry, we study certain integral representations of solutions to the Gelfand-Kapranov-Zelevinsky (GKZ) hypergeometric system. Some of these solutions arise as period integrals for Calabi-Yau manifolds in mirror symmetry. We prove that for a suitable compactification of the parameter space, there exist certain special boundary points, which we called maximal degeneracy points, at which all solutions but one become singular.

Abstract Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+ 1 and 3+ 1 dimensions are explored. Many of our TQFTs are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of DijkgraafWitten twisted discrete gauge theories. Other bosonic TQFTs beyond the DijkgraafWitten description and all fermionic TQFTs (namely the spin TQFTs) are either higher-form gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gauging the fermionic Symmetry-Protected Topological states (SPTs). We analytically calculate both the Abelian and non-Abelian braiding statistics data of anyonic particle and string excitations in these theories, where the statistics data can one-to-one characterize the underlying

Let (V, E) denote a graph with vertex set V= V (T) and edge set E= E (). Suppose a group TL acts on V such that:(i) for all g Ti,{gu, gv} GE if and only if {u, v} E,(ii) for any two vertices and v, there is a g T such that guv. Then we say is a homogeneous graph with the associated group Ti. In other words, is vertex-transitive under the action of Ti and We can identify V with the coset space Ti/X where X={g E Ti: gvv}, for a fixed vertex v, denotes the isotropy group. We note that the Cayley graph is a special case of homogeneous graphs with X trivial. The edge set of a homogeneous graph can be described by an (edge) generating set ii H such that each edge of is of the form {v, gv} for some v V, and g . In this paper we require the generating set to be symmetric, ie, g if and only if g^ 1 K. We say that a homogeneous graph is invariant if for every vertex a K, we have aKa-1= K.

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of\cite {sy1} to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension 1 or 1 the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.