Combinatorial Boundary Tracking of a 3D Lattice Point Set

Boundary tracking and surface generation are ones of main topological topics for three-dimensional digital image analysis. However, there is no adequate theory to make relations between these different topological properties in a completely discrete way. In this paper, we present a new boundary tracking algorithm which gives not only a set of border points but also the surface structures by using the concepts of combinatorial/algebraic topologies. We also show that our boundary becomes a triangulation of border points (in the sense of general topology), that is, we clarify relations between border points and their surface structures

Counting tropical elliptic plane curves with fixed j-invariant

In complex algebraic geometry, the problem of enumerating plane elliptic curves of given degree with fixed complex structure has been solved by R.Pandharipande using Gromov-Witten theory. In this article we treat the tropical analogue of this problem, the determination of the number of tropical elliptic plane curves of degree d and fixed ``tropical j-invariant'' interpolating an appropriate number of points in general position. We show that this number is independent of the position of the points and the value of the j-invariant and that it coincides with the number of complex elliptic curves. The result can be used to simplify Mikhalkin's algorithm to count curves via lattice paths in the case of rational plane curves.Comment: 34 pages; minor changes to match the published versio

Gorenstein toric Fano varieties

We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.Comment: AMS-LaTeX, 29 pages with 5 figure

Combinatorial Continuous Maximal Flows

Maximum flow (and minimum cut) algorithms have had a strong impact on computer vision. In particular, graph cuts algorithms provide a mechanism for the discrete optimization of an energy functional which has been used in a variety of applications such as image segmentation, stereo, image stitching and texture synthesis. Algorithms based on the classical formulation of max-flow defined on a graph are known to exhibit metrication artefacts in the solution. Therefore, a recent trend has been to instead employ a spatially continuous maximum flow (or the dual min-cut problem) in these same applications to produce solutions with no metrication errors. However, known fast continuous max-flow algorithms have no stopping criteria or have not been proved to converge. In this work, we revisit the continuous max-flow problem and show that the analogous discrete formulation is different from the classical max-flow problem. We then apply an appropriate combinatorial optimization technique to this combinatorial continuous max-flow CCMF problem to find a null-divergence solution that exhibits no metrication artefacts and may be solved exactly by a fast, efficient algorithm with provable convergence. Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the fact, already proved by Nozawa in the continuous setting, that the max-flow and the total variation problems are not always equivalent.Comment: 26 page

Three-dimensional simplicial gravity and combinatorics of group presentations

We demonstrate how some problems arising in simplicial quantum gravity can be successfully addressed within the framework of combinatorial group theory. In particular, we argue that the number of simplicial 3-manifolds having a fixed homology type grows exponentially with the number of tetrahedra they are made of. We propose a model of 3D gravity interacting with scalar fermions, some restriction of which gives the 2-dimensional self-avoiding-loop-gas matrix model. We propose a qualitative picture of the phase structure of 3D simplicial gravity compatible with the numerical experiments and available analytical results.Comment: 24 page

Ideal webs, moduli spaces of local systems, and 3d Calabi-Yau categories

A decorated surface S is an oriented surface with punctures and a finite set of marked points on the boundary, such that each boundary component has a marked point. We introduce ideal bipartite graphs on S. Each of them is related to a group G of type A, and gives rise to cluster coordinate systems on certain spaces of G-local systems on S. These coordinate systems generalize the ones assigned to ideal triangulations of S. A bipartite graph on S gives rise to a quiver with a canonical potential. The latter determines a triangulated 3d CY category with a cluster collection of spherical objects. Given an ideal bipartite graph on S, we define an extension of the mapping class group of S which acts by symmetries of the category. There is a family of open CY 3-folds over the universal Hitchin base, whose intermediate Jacobians describe the Hitchin system. We conjecture that the 3d CY category with cluster collection is equivalent to a full subcategory of the Fukaya category of a generic threefold of the family, equipped with a cluster collection of special Lagrangian spheres. For SL(2) a substantial part of the story is already known thanks to Bridgeland, Keller, Labardini-Fragoso, Nagao, Smith, and others. We hope that ideal bipartite graphs provide special examples of the Gaiotto-Moore-Neitzke spectral networks.Comment: 60 page

Non-commutative quantum geometric data in group field theories

We review briefly the motivations for introducing additional group-theoretic data in tensor models, leading to the richer framework of group field theories, themselves a field theory formulation of loop quantum gravity. We discuss how these data give to the GFT amplitudes the structure of lattice gauge theories and simplicial gravity path integrals, and make their quantum geometry manifest. We focus in particular on the non-commutative flux/algebra representation of these models.Comment: 10 pages; to appear in the proceedings of the workshop "Non-commutative field theory and gravity", Corfu', Greece, EU, September 201