a characterization of ideal polyhedra in hyperbolic 3 space the goals of this paper are to provide a characterization of dihedral angles of convex ideal those with all vertices on the sphere at infinity polyhedra in h3 and also of those convex polyhedra with some vertices on the sphere at infinity and some in the finite part of h3 these characterizations are given in respectively theorems 0 1 and 10 5 the first theorem is proved in detail while the proof of the second which is similar is only outlined the results of this paper grow out of the general framework of the author s doctoral dissertation 13 as published in 20 a lot of the language and some of the auxilary results come from there as well so familiarity with the latter reference is very helpful the necessity of the conditions postulated in theorem 0 1 has been shown in 14 hence only the sufficiency direction will be shown in the current paper in 1832 jakob steiner asked for a combinatorial characterization of convex polyhedra inscribed in the sphere this was considered intractable it took almost a hundred years to find a single example of an uninscribable combinatorial type by steinitz 24 however theorem 0 1 gives such a characterization section 11 is dedicated to a very brief historical survey and some purely graph theoretic and computational geometric consequences of theorem 0 1 in order to state this theorem suppose that a convex ideal polyhedron p in h3 is given let p denote the poincar dual of p and assign to each edge e of p a weight w e equal to the exterior dihedral angle at the corresponding edge e of p then the following result holds

fixed points koebe uniformization and circle packings a domain in the riemann sphere hat mathbb c is called a circle domain if every connected component of its boundary is either a circle or a point in 1908 p koebe ko1 posed the following conjecture known as koebe s kreisnormierungsproblem any plane domain is conformally homeomorphic to a circle domain in hat mathbb c when the domain is simply connected this is the content of the riemann mapping theorem

computational design and additive manufacturing of periodic conformal metasurfaces by synthesizing topology optimization with conformal mapping abstract in this paper we present a computational framework for computational design and additive manufacturing of free form periodic metasurfaces the proposed scheme rests on the level set based topology approach and the conformal mapping theory a metamaterial with pre specified performance is created using a level set based topology optimization method the achieved unit cell is further mapped to the 3d quad meshes on a free form surface by applying the conformal mapping method which can preserve the local shape and angle during the mapping with embedded geometric information the proposed level set based optimization methods not only can act as a motivator for design synthesis but also can be seamlessly hooked with additive manufacturing without the need of cad reconstructions the current computational framework provides a solution to increasing applications involving innovative metamaterial designs on free form surfaces for different fields of interests the performance of the proposed scheme is illustrated through two benchmark examples where a negative poisson s ratio unit cell pattern and a stiff and light inner structure are mapped to 3d free form surfaces and fabricated through additive manufacturing

multiperiod mean absolute deviation fuzzy portfolio selection model with risk control and cardinality constraints abstract this paper considers a multiperiod fuzzy portfolio selection problem maximizing the terminal wealth imposed by risk control in which the returns of assets are characterized by possibilistic mean values a possibilistic absolute deviation is defined as the risk control of portfolio a new multiperiod mean absolute deviation fuzzy portfolio selection model with transaction cost borrowing constraints threshold constraints and cardinality constraints is proposed based on the theory of possibility measure the proposed model is transformed into a crisp nonlinear programming problem because of the transaction cost the multiperiod portfolio selection is a dynamic optimization problem with path dependence the discrete approximate iteration method is designed to obtain the optimal portfolio strategy and is proved convergent finally an example is given to illustrate the behavior of the proposed model and the designed algorithm using real data from the shanghai stock exchange

rigidity of inversive distance circle packings revisited abstract inversive distance circle packing metric was introduced by p bowers and k stephenson 7 as a generalization of thurston s circle packing metric 34 they conjectured that the inversive distance circle packings are rigid for nonnegative inversive distance guo 22 proved the infinitesimal rigidity and then luo 27 proved the global rigidity in this paper based on an observation of zhou 37 we prove this conjecture for inversive distance in 1 by variational principles we also study the global rigidity of a combinatorial curvature introduced in 14 16 19 with respect to the inversive distance circle packing metrics where the inversive distance is in 1

on the deformation of inversive distance circle packings ii abstract we show that the results in 8 are still true in hyperbolic background geometry setting that is the solution to chow luo s combinatorial ricci flow can always be extended to a solution that exists for all time furthermore the extended solution converges exponentially fast if and only if there exists a metric with zero curvature we also give some results about the range of discrete gaussian curvatures which generalize andreev thurston s theorem to some extent

combinatorial p th calabi flows on surfaces abstract for triangulated surfaces and any p 1 we introduce the combinatorial p th calabi flows which precisely equal the combinatorial calabi flows first introduced in h ge s thesis 9 or see h ge 13 when p 2 the difficulties for the generalizations come from the nonlinearity of the p th flow equation when p 2 adopting different approaches we show that the solution to the combinatorial p th calabi flow exists for all time and converges if and only if there exists a circle packing metric of constant zero resp curvature in euclidean hyperbolic resp background geometry our results generalize the work of h ge 13 ge xu 19 and ge hua 14 on the combinatorial calabi flow from p 2 to any p 1

portfolio selection problem with value at risk constraints under non extensive statistical mechanics the optimal portfolio selection problem is a major issue in the financial field in which the process of asset prices is usually modeled by a wiener process that is the return distribution of the asset is normal however several empirical results have shown that the return distribution of the asset has the characteristics of fat tails and aiguilles and is not normal in this work we propose an optimal portfolio selection model with a value at risk var constraint in which the process of asset prices is modeled by the non extensive statistical mechanics instead of the classical wiener process the model can describe the characteristics of fat tails and aiguilles of returns using the dynamic programming principle we derive a hamilton jacobi bellman hjb equation then employing the lagrange multiplier method we obtain closed form solutions for the case of logarithmic utility moreover the empirical results show that the price process can more accurately fit the empirical distribution of returns than the familiar wiener process in addition as the time increases the constraint becomes binding that is to control the risk the agent reduces the proportion of the wealth invested in the risky asset furthermore at the same confidence level the agent reduces the proportion of the wealth invested in the risky asset more quickly under our model than under the model based on the wiener process this can give investors a good decision making reference

variational principles for branched circle patterns abstract we are concerned with variational principles for branched circle patterns we use critical point theory to establish existence and uniqueness furthermore we apply the ricci flow method to give an approximation scheme which converges exponentially to the branched circle pattern metric

a variational principle for weighted delaunay triangulations and hyperideal polyhedra we use a variational principle to prove an existence and uniqueness theorem for planar weighted delaunay triangulations with non intersecting site circles with prescribed combinatorial type and circle intersection angles such weighted delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles more generally we consider weighted delaunay triangulations in piecewise flat surfaces allowing cone singularities with prescribed cone angles in the vertices the material presented here extends work by rivin on delaunay triangulations and ideal polyhedra

a new fuzzy programming approach for multi period portfolio optimization with return demand and risk control abstract this paper considers a multi period portfolio selection problem imposed by return demand and risk control in a fuzzy investment environment in which the returns of assets are characterized by fuzzy numbers a fuzzy lower semi deviation is originally defined as the risk control of portfolio a proportion entropy constraint is added as the divergence measure of portfolio based on the theories of possibility and necessity measures a new multi period portfolio optimization model with return demand and risk control is proposed and then the proposed model is transformed into a crisp nonlinear programming problem by using fuzzy programming approach furthermore an improved differential evolution algorithm is designed for obtaining the optimal strategy finally a numerical example is given to illustrate the practicality and efficiency of the proposed model and the corresponding algorithm

multi period semi variance portfolio selection model and numerical solution variance is substituted by semi variance in markowitz s portfolio selection model moreover one period portfolio selection is extended to multi period in this paper a class of multi period semi variance model is formulated originally besides a hybrid genetic algorithm ga which makes use of the position displacement strategy of the particle swarm optimizer pso as a mutation operation is applied to solve the multi period semi variance model for this class of portfolio model numerical results show that the hybrid ga with pso is effective and feasible

duality structures and discrete conformal variations of piecewise constant curvature surfaces abstract a piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating the simplex has an isometric embedding into a constant curvature background geometry euclidean hyperbolic or spherical with the specified edge lengths additional geometric structure leads to a notion of discrete conformal structure generalizing circle packings and their generalizations as studied by thurston and others we analyze discrete conformal variations of piecewise constant curvature 2 manifolds giving particular attention to the variation of angles formulas are derived for the derivatives of angles in each background geometry which yield formulas for the derivatives of curvatures and to curvature functionals finally we provide a complete classification of possible definitions of discrete conformal structures in each of the background geometries

piecewise linear manifolds einstein metrics and ricci flows this article provides an attempt to extend concepts from the theory of riemannian manifolds to piecewise linear p l spaces in particular we propose an analogue of the ricci tensor which we give the name of an einstein vector field on a given set of p l spaces we define and discuss normalized einstein flows p l einstein metrics are defined and examples are provided criteria for flows to approach einstein metrics are formulated second variations of the total scalar curvature at a specific einstein space are calculated

gauss images of hyperbolic cusps with convex polyhedral boundary we prove that a 3 dimensional hyperbolic cusp with convex poly hedral boundary is uniquely determined by its gauss image furthermore any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than 2 is the metric of the gauss image of some convex polyhedral cusp this result is an analog of the rivin hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their gauss images the proof uses a variational method namely a cusp with a given gauss image is identified with a critical point of a functional on the space of cusps with cone type singularities along a family of half lines the functional is shown to be concave and to attain maximum at an interior point of its domain as a byproduct we prove rigidity statements with respect to the gauss image for cusps with or without cone type singularities in a special case our theorem is equivalent to existence of a circle pattern on the torus with prescribed combinatorics and intersection angles this is the genus one case of a theorem by thurston in fact our theorem extends thurston s theorem in the same way as rivin hodgson s theorem extends an dreev s theorem on compact convex polyhedra with non obtuse dihedral angles the functional used in the proof is the sum of a volume term and curvature term we show that in the situation of thurston s theorem it is the potential for the combinatorial ricci flow considered by chow and luo our theorem represents the last special case of a general statement about isometric immersions of compact surfaces

on serre s uniformity conjecture for semistable elliptic curves over totally real fields let k be a totally real field and let s be a finite set of non archimedean places of k it follows from the work of merel momose and david that there is a constant b k s so that if e is an elliptic curve defined over k semistable outside s then for all p b k s the representation overline rho e p is irreducible we combine this with modularity and level lowering to show the existence of an effectively computable constant c k s and an effectively computable set of elliptic curves over k with cm e 1 cdots e n such that the following holds if e is an elliptic curve over k semistable outside s and p c k s is prime then either overline rho e p is surjective or overline rho e p sim overline rho e i p for some i 1 dots n

brain surface conformal parameterization with the ricci flow in brain mapping research parameterized 3 d surface models are of great interest for statistical comparisons of anatomy surface based registration and signal processing here we introduce the theories of continuous and discrete surface ricci flow which can create riemannian metrics on surfaces with arbitrary topologies with user defined gaussian curvatures the resulting conformal parameterizations have no singularities and they are intrinsic and stable first we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves secondly we conformally parameterize each cortical surface to a parameter domain with a user designed gaussian curvature arrangement in the parameter domain a shape index based on conformal invariants is computed and inter subject cortical surface matching is performed by solving a constrained harmonic map we illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data to map statistical differences in cortical morphometry we studied brain asymmetry in 14 healthy control subjects we used a manifold version of hotelling s t2 test applied to the jacobian matrices of the surface parameterizations a permutation test along with the cumulative distribution of p values were used to estimate the overall statistical significance of differences the results show our algorithm s power to detect subtle group differences in cortical surfaces

discrete yamabe flows with r curvature revisited abstract using ge jiang shen s extension method 8 we extend ge xu s discrete yamabe flow with r curvature 11 we show the solution to the extended flow is unique we give a picture of the singularities in finite time that the degeneration is at least codimension two we show the solution exists for all time t 0 if all vertex degree are no less than 23 which partly confirm the conjecture 1 in 11 we also give two sufficient conditions the energy gap condition and the big regular triangulation condition for the sub convergence to a constant r curvature metric

combinatorial ricci flows on surfaces we show that the analogue of hamilton s ricci flow in the combinatorial setting produces solutions which converge exponentially fast to thurston s circle packing on surfaces as a consequence a new proof of thurston s existence of circle packing theorem is obtained as another consequence ricci flow suggests a new algorithm to find circle packings

skewness of fuzzy numbers and its applications in portfolio selection a fuzzy number is a normal and convex fuzzy subset of the real line in this paper based on membership function we redefine the concepts of mean and variance for fuzzy numbers furthermore we propose the concept of skewness and prove some desirable properties a fuzzy mean variance skewness portfolio selection model is formulated and two variations are given which are transformed to nonlinear optimization models with polynomial objective and constraint functions such that they can be solved analytically finally we present some numerical examples to demonstrate the effectiveness of the proposed models

on combinatorial calabi flow with hyperbolic circle patterns abstract in this paper we prove the long time existence of the combinatorial calabi flow in hyperbolic background geometry and the convergence of the flow to a smooth hyperbolic surface under thurston s combinatorial condition in full generality which improves the result in 14 this flow provides a natural algorithm to find thurston s hyperbolic circle patterns

automorphic lifts of prescribed types we prove a variety of results on the existence of automorphic galois representations lifting a residual automorphic galois representation we prove a result on the structure of deformation rings of local galois representations and deduce from this and the method of khare and wintenberger a result on the existence of modular lifts of specified type for galois representations corresponding to hilbert modular forms of parallel weight 2 we discuss some conjectures on the weights of n dimensional mod p galois representations finally we use recent work of taylor to prove level raising and lowering results for n dimensional automorphic galois representations

rigidity of infinite disk patterns let p be a locally flnite disk pattern on the complex plane c whose combinatorics are described by the one skeleton g of a triangulation of the open topological disk and whose dihedral angles are equal to a function e 0 2 on the set of edges let p be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function we show that p and p difier only by a euclidean similarity in particular when the dihedral angle function is identically zero this yields the rigidity theorems of b rodin and d sullivan and of o schramm whose arguments rely essentially on the pairwise disjointness of the interiors of the disks the approach here is analytical and uses the maximum principle the concept of vertex extremal length and the recurrency of a family of electrical networks obtained by placing resistors on the edges in the contact graph of the pattern a similar rigidity property holds for locally flnite disk patterns in the hyperbolic plane where the proof follows by a simple use of the maximum principle also we have a uniformization result for disk patterns in a future paper the techniques of this paper will be extended to the case when 0 in particular we will show a rigidity property for a class of inflnite convex polyhedra in the 3 dimensional hyperbolic space

computing surface uniformization using discrete beltrami flow in this paper we propose a novel algorithm for computing surface uniformization for surfaces with arbitrary topology according to the celebrated uniformization theorem all riemann surfaces can be classified as elliptic parabolic or hyperbolic our algorithm is able to work on all these cases by first constructing an initial map onto an appropriate domain such as a sphere or a polygon in the plane mathbb r 2 or the hyperbolic disk d and then morphing the diffeomorphism based on the discrete beltrami flow algorithm for high genus surfaces both the final mapping and the target domain are unknown which presents a challenge in general each such surface can be conformally mapped onto d modulo a discrete subgroup of all fractional linear transforms on d a conformal copy of the surface which is also its uniformization domain can be visualized as a fundamental polygon in d corresponding to the discrete subgroup where the generators of the discrete subgroup map each side of the fundament

rigidity of polyhedral surfaces ii we study the rigidity of polyhedral surfaces using variational principles the action functionals are derived from the cosine laws the main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics several of these cosine laws were first discovered and used by fenchel and nielsen by studying the derivative of the cosine laws we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary as a consequence the work of penner bobenko and springborn and thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context

semi global existence and convergence of solutions of the robinson trautman 2 dimensional calabi equation it is shown that for smooth initial data solutions of the robinson trautman equation also known as the two dimensional calabi equation exist for all positive times and asymptotically converge to a constant curvature metric

fermat type equations of signature 13 13 p via hilbert cuspforms in this paper we prove that for p 13649 equations of the form x 13 y 13 cz p have no non trivial primitive solutions a b c such that 13 not mid c for an infinite family of values for c our method consists on relating a solution a b c to the previous equation to a solution a b c 1 of another diophantine equation with coefficients in mathbb q sqrt 13 then we attach to a b c 1 a frey curve e a b defined over mathbb q sqrt 13 that is not a mathbb q curve we prove a modularity result of independent interest for certain elliptic curves over totally real abelian number fields satisfying some local conditions at 3 this theorem in particular implies modularity of e a b this enables us to use level lowering results and apply the modular approach via hilbert cuspforms over mathbb q sqrt 13 to prove the non existence of a b c 1 and consequently of a b c

the ricci flow on the 2 sphere the classical uniformization theorem interpreted differential geomet rically states that any riemannian metric on a 2 dimensional surface ispointwise conformal to a constant curvature metric thus one can con sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met ric the primary interest in this question is not so much to give a newproof of the uniformization theorem but rather to understand nonlinearparabolic equations better especially those arising in differential geome try a sufficiently deep understanding of parabolic equations should yieldimportant new results in riemannian geometry the question in the preceding paragraph has been studied by richardhamilton 3 and brad osgood ralph phillips and peter sarnak 6 in 3 hamilton studied the following equation which we refer to as

characterizations of circle patterns and finite convex polyhedra in hyperbolic 3 space the aim of this paper is to study finite convex polyhedra in three dimensional hyperbolic space mathbb h 3 we characterize the quasiconformal deformation space of each finite convex polyhedron as a corollary we obtain some results on finite circle patterns in the riemann sphere with dihedral angle 0 le theta pi that is for any circle pattern on hat mathbb c its quasiconformal deformation space can be naturally identified with the product of the teichmuller spaces of its interstices

variational principles for circle patterns and koebe s theorem we prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces our method is based on two new functionals one for the euclidean and one for the hyperbolic case we show how colin de verdiere s bragger s and rivin s functionals can be derived from ours

computing quasiconformal maps using an auxiliary metric and discrete curvature flow surface mapping plays an important role in geometric processing which induces both area and angular distortions if the angular distortion is bounded the mapping is called a quasiconformal mapping qc mapping many surface mappings in our physical world are quasiconformal the angular distortion of a qc mapping can be represented by the beltrami differentials according to qc teichmuller theory there is a one to one correspondence between the set of beltrami differentials and the set of qc surface mappings under normalization conditions therefore every qc surface mapping can be fully determined by the beltrami differential and reconstructed by solving the so called beltrami equation in this work we propose an effective method to solve the beltrami equation on general riemann surfaces the solution is a qc mapping associated with the prescribed beltrami differential the main strategy is to define an auxiliary metric am on the domain surface such that the original qc mapping becomes conformal under the auxiliary metric the desired qc mapping can then be obtained by using the conventional conformal mapping method in this paper we first formulate a discrete analogue of qc mappings on triangular meshes then we propose an algorithm to compute discrete qc mappings using the discrete yamabe flow method to the best of our knowledge it is the first work to compute the discrete qc mappings for general riemann surfaces especially with different topologies numerically the discrete qc mapping converges to the continuous solution as the mesh grid size approaches to 0 we tested our algorithm on surfaces scanned from real life with different topologies experimental results demonstrate the generality and accuracy of our auxiliary metric method

discrete quasi einstein metrics and combinatorial curvature flows in 3 dimension abstract motivated by the definition of combinatorial scalar curvature given by cooper and rivin we introduce a new combinatorial scalar curvature then we define the discrete quasi einstein metric which is a combinatorial analogue of the constant scalar curvature metric in smooth case we find that discrete quasi einstein metric is critical point of both the combinatorial yamabe functional and the quadratic energy functional we defined on triangulated 3 manifolds we introduce combinatorial curvature flows including a new type of combinatorial yamabe flow to study the discrete quasi einstein metrics and prove that the flows produce solutions converging to discrete quasi einstein metrics if the initial normalized quadratic energy is small enough as a corollary we prove that nonsingular solution of the combinatorial yamabe flow with nonpositive initial curvatures converges to discrete quasi einstein metric the proof relies on a careful analysis of the discrete dual laplacian which we interpret as the jacobian matrix of curvature map

ricci flow embedding for rectifying non euclidean dissimilarity data pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition one of the problems encountered in the analysis of such data is that the dissimilarities are rarely euclidean while statistical learning algorithms often rely on euclidean dissimilarities such non euclidean dissimilarities are often corrected or a consistent euclidean geometry is imposed on them via embedding this paper commences by reviewing the available algorithms for analysing non euclidean dissimilarity data the novel contribution is to show how the ricci flow can be used to embed and rectify non euclidean dissimilarity data according to our representation the data is distributed over a manifold consisting of patches each patch has a locally uniform curvature and this curvature is iteratively modified by the ricci flow the raw dissimilarities are the geodesic distances on the manifold rectified euclidean dissimilarities are obtained using the ricci flow to flatten the curved manifold by modifying the individual patch curvatures we use two algorithmic components to implement this idea firstly we apply the ricci flow independently to a set of surface patches that cover the manifold second we use curvature regularisation to impose consistency on the curvatures of the arrangement of different surface patches we perform experiments on three real world datasets and use these to determine the importance of the different algorithmic components i e ricci flow and curvature regularisation we conclude that curvature regularisation is an essential step needed to control the stability of the piecewise arrangement of patches under the ricci flow

on a combinatorial curvature for surfaces with inversive distance circle packing metrics abstract in this paper we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics then we prove that this combinatorial curvature has global rigidity to study the yamabe problem of the new curvature we introduce a combinatorial ricci flow along which the curvature evolves almost in the same way as that of scalar curvature along the surface ricci flow obtained by hamilton 20 then we study the long time behavior of the combinatorial ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergence of the flow on triangulated surfaces with nonpositive euler number we further generalize the combinatorial curvature to curvature and prove that it is also globally rigid which is in fact a generalized bowers stephenson conjecture 6 we also use the combinatorial ricci flow to study the corresponding yamabe problem

super ricci flows for weighted graphs abstract we present a notion of super ricci flow for time dependent finite weighted graphs a challenging feature is that these flows typically encounter singularities where the underlying graph structure changes our notion is robust enough to allow the flow to continue past these singularities as a crucial tool for this purpose we study the heat flow on such singular time dependent weighted graphs with changing graph structure we then give several equivalent characterizations of super ricci flows in terms of a discrete dynamic bochner inequality gradient and transport estimates for the heat flow and dynamic convexity of the entropy along discrete optimal transport paths the latter property can be used to show that our notion of super ricci flow is consistent with classical super ricci flows for manifolds or metric measure spaces in a discrete to continuum limit

ricci flow for 3d shape analysis ricci flow is a powerful curvature flow method which is invariant to rigid motion scaling isometric and conformal deformations we present the first application of surface ricci flow in computer vision previous methods based on conformal geometry which only handle 3d shapes with simple topology are subsumed by the ricci flow based method which handles surfaces with arbitrary topology we present a general framework for the computation of ricci flow which can design any riemannian metric by user defined curvature the solution to ricci flow is unique and robust to noise we provide implementation details for ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry our ricci flow based method can convert all 3d problems into 2d domains and offers a general framework for 3d shape analysis we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems such as 3d shape matching and registration and shape indexing surfaces with large nonrigid anisotropic deformations can be registered using ricci flow with constraints of feature points and curves we show how conformal equivalence can be used to index shapes in a 3d surface shape space with the use of teichmuller space coordinates experimental results are shown on 3d face data sets with large expression deformations and on dynamic heart data

rigidity of infinite circle packings the nerve of a packing is a graph that encodes its combinatorics the vertices of the nerve correspond to the packed sets and an edge occurs between two vertices in the nerve precisely when the corresponding sets of the packing intersect

rigidity of circle domains whose boundary hasσ finite linear measure let be a circle domain in the riemann sphere whose boundary has finite linear measure we show that is rigid in the sense that any conformal homeomorphism of onto any other circle domain is equal to the restriction of a mobius transformation previously kaufman and bishop have independently found examples of non rigid circle domains whose boundary is a cantor set of hausdorff dimension one

moduli of finite flat group schemes and modularity we prove that under some mild conditions a two dimensional p adic galois representation which is residually modular and potentially barsotti tate at p is modular this provides a more conceptual way of establishing the shimura taniyama weil conjecture especially for elliptic curves which acquire good reduction over a wildly ramified extension of 3 the main ingredient is a new technique for analyzing flat deformation rings it involves resolving them by spaces which parametrize finite flat group scheme models of galois representations introduction