The Minkowski product can be viewed as a higher dimensional version of interval arithmetic. We discuss a collection of geometric constructions based on the Minkowski product and on one of its natural generalizations, the quaternion action. We also will present some topological facts about these products, and discuss the applications of these constructions to computer aided geometric design.

Algebraic geometry is the study of solutions in polynomial equations using objects and shapes. Differential geometry is based on surfaces, curves, and dimensions of shapes and applying calculus and algebra. Desingularizing the singularities of a variety plays an important role in research in algebraic and differential geometry. Toroidal Embedding is one of the tools used in desingularization. Therefore, Toroidal Embedding and desingularization will be the main focus of my project. In this paper, we first provide a brief introduction on Toroidal Embedding, then show an explicit construction on how to smooth a variety with singularity through Toroidal Embeddings.

acase@tulane.edu / We study the problem of computing the minimum homotopy area of a planar normal curve. The area of a homotopy is the area swept by the homotopy on the plane. First, we consider a specific class of curves, namely self-overlapping curves, and show that the minimum homotopy area of a self-overlapping curve is equal to its winding area. For an arbitrary normal curve, we show that there is a decomposition of the curve into the self-overlapping subcurves such that the minimum homotopy area can be computed as the sum of winding areas of each self-overlapping subcurve in the decomposition. / 1 / Karakoc, Selcuk

We begin with a rapid introduction to the theory of contact topology, first spending more time than you would probably want on developing the notion of contact manifold before launching right into the thick of the theory. The tools of characteristic foliations and convex surfaces are introduced next, concluding with an overview of Legendrian knots in contact 3-manifolds. Next, we develop a number of lemmas as tools for dealing with characteristic foliations, concluding with some sightseeing with regards to the theory of so-called "movies", allowing a glimpse into the workings of a theorem due to Colin: Two smoothly isotopic embeddings of S^2 into a tight contact 3-manifold inducing the same characteristic foliation are necessarily contact isotopic. We finish with an original observation that Colin’s theorem can be used to replace a key step in Eliashberg and Fraser’s classification of topologically trivial knots, thus providing an alternate proof of that result and thereby highlighting the power of the aforementioned theorem. We provide a simplification of this proof using intermediate results we encountered along the way.

This thesis presents an algorithm for producing 4–manifold triangulations with boundary an arbitrary orientable, closed, triangulated 3–manifold. The research is an extension of Costantino and Thurston’s work on determining upper bounds on the number of 4–dimensional simplices necessary to construct such a triangulation. Our first step in this bordism construction is the geometric partitioning of an initial 3–manifold M using smooth singularity theory. This partition provides handle attachment sites on the 4–manifold Mx[0,1] and the ensuing handle attachments eliminate one of the boundary components of Mx[0,1], yielding a 4-manifold with boundary exactly M. We first present the construction in the smooth case before extending the smooth singularity theory to triangulated 3–manifolds. / Graduate

<p>The main purpose of this thesis is to study the interplay between relational structures and topology , and to portray pairings in terms of some finite poset models and order preserving maps. We show the interrelations between the categories of topological spaces, closure spaces and relational structures. We study the 4-point non-Hausdorff model S4 weakly homotopy equivalent to the circle S1. We study pairings of some objects in the category of relational structures similar to the multiplication S4 x S4- S4 S4 fails to be order preserving for posets. Nevertheless, applying a single barycentric subdivision on S4 to get S8, an 8-point model of the circle enables us to define an order preserving poset map S8 x S8- S4. Restricted to the axes, this map yields weak homotopy equivalences S8 x S8, we obtain a version of the Hopf map S8 x S8s - SS4. This model of the Hopf map is in fact a map of non-Hausdorff double map cylinders.</p>

This article is an investigation of a recently developed method of deriving a topology from a space and an elementary submodel containing it. We first define and give the basic properties of this construction, known as X/M. In the next section, we construct some examples and analyse the topological relationship between X and X/M. In the final section, we apply X/M to get novel results about Lindelof spaces, giving partial answers to a question of F.D. Tall and another question of Tall and M. Scheepers.

This article is an investigation of a recently developed method of deriving a topology from a space and an elementary submodel containing it. We first define and give the basic properties of this construction, known as X/M. In the next section, we construct some examples and analyse the topological relationship between X and X/M. In the final section, we apply X/M to get novel results about Lindelof spaces, giving partial answers to a question of F.D. Tall and another question of Tall and M. Scheepers.

Structural topology optimization methods have existing and been improving theoretically since 1980s; however, in industry, with respect to the certain conditions, proper modification is always desired. This study develops a specific method to utilize topology optimization for gearbox housing design. Gearbox housing maintains the position of the shafts to ensure the precision of gear engagement in all operational states (Naunheimer, et al., 2010). The current housing design processing used in Vicura AB, a Swedish powertrain company, is able to achieve stiff optimal housing material distribution, but difficult to fulfil gear misalignment requirement. This work overcomes the above shortages to develop a new methodology for gearbox housing topology optimization concerning the gear misalignment as well. The paper is starting with an introduction of the previous method and its defects, followed by a discussion of three possible improvements. Only one of them is feasible and two main difficulties need to be resolved to make it applicable. One of the difficulties is finding a linear assumption of the non-linear components and the other is deriving an approach for topology optimization involving both external forces and non-zero prescribed displacements. The corresponding solutions are described subsequently in detail both theoretically and practically. Then the results by implementing the new method and also the comparison with the results getting from the old method are presented. Finally, a validation of the new method is discussed and the conclusions and comments are given.

In this thesis I present a mathematical tool for understanding the spin networks that arise from the study of the loop states of quantum gravity. The spin networks that arise in quantum gravity possess more information than the original spin networks of Penrose: they are embedded within a manifold and thus possess topological information. There are limited tools available for the study of this information. To remedy this I introduce a slightly modi ed mathematical object - Braided Ribbon Networks - and demonstrate that they can be related to spin networks in a consistent manner which preserves the di eomorphism invariant character of the loop states of quantum gravity. Given a consistent de nition of Braided Ribbon Networks I then relate them back to previous trinion based versions of Braided Ribbon Networks. Next, I introduce a consistent evolution for these networks based upon the duality of these networks to simplicial complexes. From here I demonstrate that there exists an invariant of this evolution and smooth deformations of the networks, which captures some of the topological information of the networks. The principle result of this program is presented next: that the invariants of the Braided Ribbon Networks can be transferred over to the original spin network states of loop quantum gravity. From here we represent other advances in the study of braided ribbon networks, accompanied by comments of their context given the consistent framework developed earlier including: the meaning of isolatable substructures, the particular structure of the capped three braids in trivalent braided ribbon networks and their application towards emergent particle physics, and the implications of the existence of microlocal topological structures in spin networks. Lastly we describe the current state of research in braided ribbon networks, the implications of this study on quantum gravity as a whole and future directions of research in the area.