MathSciDoc: An Archive for Mathematician ∫

Here we propose a novel method to compute Teichmller shape space based shape index to study brain morphometry. Such a shape index is intrinsic, and invariant under conformal transformations, rigid motions and scaling. We conformally map a genus-zero open boundary surface to the Poincar disk with the Yamabe flow method. The shape indices that we compute are the lengths of a special set of geodesics under hyperbolic metric. Tests on longitudinal brain imaging data were used to demonstrate the stability of the derived feature vectors. In leave-one-out validation tests, we achieved 100% accurate classification (versus only 68% accuracy for volume measures) in distinguishing 11 HIV/AIDS individuals from 8 healthy control subjects, based on Teichmller coordinates for lateral ventricular surfaces extracted from their 3D MRI scans.

The chapter discusses the equivariant loop theorem for three-dimensional manifolds that is needed in settling the Smith conjecture and reviews the existence theorems for minimal surfaces. The equivariant version of the loop theorem says that there are a finite number of properly embedded disks in <i>M</i> that satisfy the required properties. The loop theorem respects the action of the group <i>G</i> in a suitable manner. The chapter puts a metric on <i>M</i> so that the group <i>G</i> acts isometrically and so that <i>M</i> is convex with respect to the outward normal. Then with respect to this metric, the existence of an immersed disk <i>D</i>₁ is demonstrated, in <i>M</i> whose boundary <i> D₁</i>, represents a nontrivial element in ₁(<i>S</i>) and whose area is minimal among all such disks. The chapter describes Morrey's solution for the plateau problem in a general Riemannian manifold. The existence theorem for manifolds with boundary

Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators. The possible applications include condensed matter and quantum cosmology. We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain topological quantum field theories (TQFTs), in any dimension. This information can be related to the counting of extended operators of bosonic/fermionic TQFTs. On the lattice scale, anyonic particles/strings live at the ends of line/surface operators. Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry-protected topological states (with fermion parity \mathbb{Z}_2^f) of symmetry groups \mathbb{Z}_2^f and \mathbb{Z}_2^f in

Detailed studies of four dimensional N= 2 superconformal field theories (SCFT) defined by isolated complete intersection singularities are performed: we compute the Coulomb branch spectrum, Seiberg-Witten solutions and central charges. Most of our theories have exactly marginal deformations and we identify the weakly coupled gauge theory descriptions for many of them, which involve (affine) D and E shaped quiver gauge theories and theories formed from Argyres-Douglas matters. These investigations provide strong evidence for the singularity approach in classifying 4d N= 2 SCFTs.

Automatic computation of surface correspondence via harmonic map is an active research field in computer vision, computer graphics and computational geometry. It may help document and understand physical and biological phenomena and also has broad applications in biometrics, medical imaging and motion capture industries. Although numerous studies have been devoted to harmonic map research, limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints. This work conquers this problem by changing the Riemannian metric on the target surface to a hyperbolic metric so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints. The computational algorithms are based on Ricci flow and nonlinear heat diffusion methods. The approach is general and robust. We employ our algorithm to study the

We consider N= 2 superconformal field theory with following properties: a) Coulomb branch operators have fractional scaling dimensions, b) there are exact marginal deformations. The weakly coupled gauge theory descriptions are found by decomposing 3d mirror into different components, and different decompositions correspond to different duality frames. The gauge theory is formed by gauging Argyres-Douglas matter, and we write down all duality frames for several classes with infinite sequence of theories.

In this paper, we formulate a problem of simultaneous redundant via insertion and line end extension for via yield optimization. Our problem is more general than previous works in the sense that more than one type of line end extension is considered and the objective function to be optimized directly accounts for via yield. We present a zero-one integer linear program based approach, that is equipped with two speedup techniques, to solve the addressed problem optimally. In addition, we describe how to modify our approach to exactly solve a previous work. Extensive experimental results are shown to demonstrate the effectiveness and efficiency of our approaches.

Redundant via insertion is highly recommended for improving chip yield and reliability. In this paper, we study the problem of double via insertion with wire bending (DVI/WB) in a post-routing stage, where a single via can have at most one redundant via inserted next to it. Aside from this, we are allowed to bend existing signal wires for enhancing the insertion rate of double vias. The goal of DVI/WB is to primarily insert as many double vias as possible and to minimize the amount of layout perturbation as the secondary objective. We formulate the DVI/WB problem as that of finding a minimum-weight maximum independent set (mWMIS) on an enhanced conflict graph. We propose algorithms to perform wire bending and to construct the enhanced conflict graph from a given design. Moreover, we also propose a zero-one integer linear program (0-1 ILP) based approach to solve mWMIS. Experimental results show that

In this paper we study the birational geometry of HyperKaehler manifolds by combining the method of minimal model program and the traditional approach of symplectic geometry.

We study SYZ mirror symmetry in the context of non-Khler CalabiYau manifolds. In particular, we study the six-dimensional Type II supersymmetric <i>SU</i>(3) systems with RamondRamond fluxes, and generalize them to higher dimensions. We show that FourierMukai transform provides the mirror map between these Type IIA and Type IIB supersymmetric systems in the semi-flat setting. This is concretely exhibited by nilmanifolds.