A remark on K-theory and S-categories

It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (Invent. Math. 150 (2002) 111). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: “To which extent K-theory is not an invariant of triangulated derived categories? ”     

Schnyder woods for higher genus triangulated surfaces (abstract)

We study a well known characterization of planar graphs, also called Schnyder wood or Schnyder labelling, which yields a decomposition into vertex spanning trees. The goal is to extend previous algorithms and characterizations designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. We define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and colouration of the edges that extends the one proposed by Schnyder for the planar case. As a by-product we show how to characterize our edge coloration in terms of genus g maps.     

On the approximation of a smooth surface with a triangulated mesh

We approximate the normals and the area of a smooth surface with the normals and the area of a triangulated mesh whose vertices belong to the smooth surface. Both approximations only depend on the triangulated mesh (which is supposed to be known), on an upper bound on the smooth surface's curvature, on an upper bound on its reach (which is linked to the local feature size) and on an upper bound on the Hausdorff distance between both surfaces.

We show in particular that the upper bound on the error of the normals is better when triangles are right-angled (even if there are small angles). We do not need every angle to be quite large. We just need each triangle of the triangulated mesh to contain at least one angle whose sinus is large enough.     

An invariant of triangulated links from the quantum dilogarithm

An invariant of triangulated links in triangulated three-dimensional manifolds is constructed by means of the cyclic quantum dilogarithm. Apparently, it is an isotopic invariant of nonoriented links in an oriented closed three-dimensional manifold. Bibliography: 13 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 208–214. Translated by B. Bekker.     

Orlov spectra: bounds and gaps

The Orlov spectrum is a new invariant of a triangulated category. It was introduced by D. Orlov, building on work of A. Bondal-M. Van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singularities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a generator, by closing the object under a certain monodromy action, and uniformly bound this generator’s generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one.     

On a proper acute triangulation of a polyhedral surface

Let Σ be a polyhedral surface in R³ with n edges. Let L be the length of the longest edge in Σ, δ be the minimum value of the geodesic distance from a vertex to an edge that is not incident to the vertex, and θ be the measure of the smallest face angle in Σ. We prove that Σ can be triangulated into at most CLn/(δθ) planar and rectilinear acute triangles, where C is an absolute constant.     

Generating weakly triangulated graphs

We show that a graph is weakly triangulated, or weakly chordal, if and only if it can be generated by starting with a graph with no edges, and repeatedly adding an edge, so that the new edge is not the middle edge of any chordless path with four vertices. This is a corollary of results due to Sritharan and Spinrad, and Hayward, Hoång and Maffray, and a natural analog of a theorem due to Fulkerson and Gross, which states that a graph is triangulated, or chordal, if and only if it can be generated by starting with a graph with no vertices, and repeatedly adding a vertex, so that the new vertex is not the middle vertex of any chordless path with three vertices. Our result answers the question of whether there exists a composition scheme that generates exactly the class of weakly triangulated graphs. © 1996 John Wiley & Sons, Inc.     

Spherical subcategories in representation theory

Mathematische Zeitschrift - We introduce a new invariant for triangulated categories: the poset of spherical subcategories ordered by inclusion. This yields several numerical invariants, like the...     

Invariants of the stable equivalence of symmetric special biserial algebras

The present paper is one in a series of papers devoted to the classification of some classes of tame algebras up to stable category equivalence. In this paper, we study symmetric algebras (their stable categories have a structure of triangulated categories) and the simplest class of tame algebras-the class of special biserial algebras (SB-algebras). In the paper, we give a relevant version of the “diagrammatic method” and study the structure of the triangulated category “in a neighborhood” of the periodic part (with respect to Ω) of the stable category. Thus we prove the invariance of the collection of lengths of G-cycles under equivalence of stable categories (see Theorem 2.12). Then we use the invariance stated above, together with some properties of the Cartan matrix of a symmetric SB-algebra, to prove that the number of A-cycles (but not their lengths!) is also an invariant of stable equivalence. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 5–28.     

Invariant surfaces in the homogeneous space Sol with constant curvature

A surface in homogeneous space is said to be an invariant surface if it is invariant under some of the two 1‐parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classify invariant surfaces with constant mean curvature and constant Gaussian curvature. Also, we characterize invariant surfaces that satisfy a linear Weingarten relation.     

Triangulated Quotient Categories

A notion of mutation of subcategories in a right triangulated category is defined in this article. When (𝒵, 𝒵) is a 𝒟-mutation pair in a right triangulated category 𝒞, the quotient category 𝒵/𝒟 carries naturally a right triangulated structure. Moreover, if the right triangulated category satisfies some reasonable conditions, then the right triangulated quotient category 𝒵/𝒟 becomes a triangulated category. When 𝒞 is triangulated, our result unifies the constructions of the quotient triangulated categories by Iyama-Yoshino and by Jørgensen, respectively.     

Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers for self-maps to an equivariant K-homology class. We compute the Lefschetz invariants for self-maps of finite-dimensional simplicial complexes and smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in this case. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Lück and Rosenberg.     

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph ℋ appears within the resolution of its edge ideal ℐ(ℋ). We discuss when recursive formulas to compute the graded Betti numbers of ℐ(ℋ) in terms of its sub-hypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405–425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are “well behaved.” For such a hypergraph ℋ (and thus, for any simple graph), we give a lower bound for the regularity of ℐ(ℋ) via combinatorial information describing ℋ and an upper bound for the regularity when ℋ=G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When ℋ is a triangulated hypergraph, we explicitly compute the regularity of ℐ(ℋ) and show that the graded Betti numbers of ℐ(ℋ) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs. Dedicated to Anthony V. Geramita on the occasion of his 65th birthday.     

Affine Invariant Detection: Edge Maps, Anisotropic Diffusion, and Active Contours

In this paper we undertake a systematic investigation of affine invariant object detection and image denoising. Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets. In this case, affine invariant maps are derived as a weighted difference of images at different scales. We then introduce the affine gradient as an affine invariant differential function of lowest possible order with qualitative behavior similar to the Euclidean gradient magnitude. These edge detectors are the basis for the extension of the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration. The active contours are obtained as a gradient flow in a conformally Euclidean space defined by the image on which the object is to be detected. That is, we show that objects can be segmented in an affine invariant manner by computing a path of minimal weighted affine distance, the weight being given by functions of the affine edge detectors. The gradient path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology. Based on the same theory of affine invariant gradient flows we show that the affine geometric heat flow is minimizing, in an affine invariant form, the area enclosed by the curve.     

Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of R^d. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.     

On the integral invariants of a closed ruled surface

Summary In this paper, it is shown that the dual integral invariant of a closed ruled surface, the dual angle of pitch, corresponds to the dual spherical surface area described by the dual spherical indicatrix of the closed ruled surface. So, new geometric interpretations of the real angle of pitch and the real pitch of a closed ruled surface, and some results are given.     

Conformal cochains

In this paper we define holomorphic cochains and an associated period matrix for triangulated closed topological surfaces. We use the combinatorial Hodge star operator introduced in the author's paper of 2007, which depends on the choice of an inner product on the simplicial 1-cochains.

We prove that for a triangulated Riemannian 2-manifold (or a Riemann surface), and a particularly nice choice of inner product, the combinatorial period matrix converges to the (conformal) Riemann period matrix as the mesh of the triangulation tends to zero.

     

Recognizing triangulated Cartesian graph products

Computational meshes for numerical simulation frequently show—at least locally—a structure resembling a triangulated grid. Our goal is to recognize product-like structures in triangular meshes. We define triangulated Cartesian products of graphs and analyze their structural properties. We show how to recognize and factorize graphs that are triangulated products of two factors, when the factors are triangle-free graphs. We also discuss properties of products with more than two factors.     

Surface group representations with maximal Toledo invariant

We study representations of compact surface groups on Hermitian symmetric spaces and characterize those with maximal Toledo invariant. To cite this article: M. Burger et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds

We present a fast enumeration algorithm for combinatorial 2- and 3-manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3-manifolds with 11 vertices. We further determine all equivelar polyhedral maps on the non-orientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the non-orientable surfaces of genus 5 and 6.