Combinatorial models for topology-based geometric modeling

Many combinatorial (topological) models have been proposed in geometric modeling, computational geometry, image processing or analysis, for representing subdivided geometric objects, i.e. partitionned into cells of different dimensions: vertices, edges, faces, volumes, etc. We can distinguish among models according to the type of cells (regular or not regular ones), the type of assembly ("manifold" or "non manifold"), the type of representation (incidence graphs or ordered models), etc

Quantising on a category

We review the problem of finding a general framework within which one can construct quantum theories of non-standard models for space, or space-time. The starting point is the observation that entities of this type can typically be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose `configuration space' (or history-theory analogue) is the set of objects \Ob\Q in a category \Q. We develop a scheme based on constructing an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold $Q\simeq G/H$, where $G$ and $H$ are Lie groups. In particular, we choose as the analogue of $G$ the monoid of `arrow fields' on \Q. Physically, this means that an arrow between two objects in the category is viewed as an analogue of momentum. After finding the `category quantisation monoid', we show how suitable representations can be constructed using a bundle (or, more precisely, presheaf) of Hilbert spaces over \Ob\Q. For the example of a category of finite sets, we construct an explicit representation structure of this type.Comment: To appear in a volume dedicated to the memory of James Cushin

Representing and Understanding Non-Manifold Objects

Solid Modeling is a well-established field. The significance of the contributions of this field is visible in the availability of abundant commercial and free modeling tools for the applications of CAD, animation, visualization etc. There are various approaches to modeling shapes. A common problem to all of them however, is the handling of non-manifold shapes. Manifold shapes are shapes with the property of topological ``smoothness'' at the local neighbourhood of every point. Objects that contain one or more points that lack this smoothness are all considered non-manifold. Non-manifold objects form a huge catagory of shapes. In the field of solid modeling, solutions typically limit the application domain to manifold shapes. Where the occurrence of non-manifold shapes is inevitable, they are often processed at a high cost. The lack of understanding on the nature of non-manifold shapes is the main cause of it. There is a tremendous gap between the well-established mathematical theories in topology and the materialization of such knowledge in the discrete combinatorial domain of computer science and engineering. The motivation of this research is to bridge this gap between the two. We present a characterization of non-manifoldness in 3D simplicial shapes. Based on this characterization, we propose data structures to address the applicational needs for the representation of 3D simplicial complexes with mixed dimensions and non-manifold connectivities, which is an area that is greatly lacking in the literature. The availability of a suitable data structure makes the structural analysis of non-manifold shapes feasible. We address the problem of non-manifold shape understanding through a structural analysis that is based on decomposition

From hyperbolic Dehn filling to surgeries in representation varieties

Hyperbolic Dehn surgery and the bending procedure provide two ways which can be used to describe hyperbolic deformations of a complete hyperbolic structure on a 3-manifold. Moreover, one can obtain examples of non-Haken manifolds without the use of Thurston's Uniformization Theorem. We review these gluing techniques and present a logical continuity between these ideas and gluing methods for Higgs bundles. We demonstrate how one can construct certain model objects in representation varieties $\text{Hom} \left( \pi_{1} \left( \Sigma \right), G \right)$ for a topological surface $\Sigma$ and a semisimple Lie group $G$. Explicit examples are produced in the case of $\Theta$-positive representations lying in the smooth connected components of the $\text{SO} \left(p,p+1 \right)$-representation variety