1,720,983 research outputs found
Morphisms, Minors, and Minimal Obstructions to Convexity of Neural Codes
No full textThesis (Ph.D.)--University of Washington, 2021We study open and closed convex codes from a geometric and combinatorial point of view. We prove constructive geometric results that establish new upper bounds on the open and closed embedding dimensions of intersection complete codes. We introduce a combinatorial framework of morphisms and minors for the study of convex codes, and show that open and closed embedding dimension are monotone invariants when codes are partially ordered by minors (in particular, open or closed convex codes form a minor-closed family). We establish new discrete geometry theorems and use them to exhibit infinite families of minimally non- convex codes, including new local obstructions to convexity. We also describe families of codes with novel embedding dimension behavior: arbitrary disparity between open and closed embedding dimension, open embedding dimensions that are exponential in the number of neurons in a code, and large increases in closed embedding dimension when adding a new non-maximal codeword. We conclude with an extensive discussion of open questions
Combinatorics of CAT(0) cubical complexes, crossing complexes and co-skeletons
No full textThesis (Ph.D.)--University of Washington, 2023This thesis consists of three papers about cubical complexes: Chapter 1 is [Rowlands 22a], Chapter 2 is [Rowlands 23], and Chapter 3 is [Rowlands 22b]. Chapter 1 extends a result by Dancis to cubical complexes: Dancis proved that any d-dimensional simplicial manifold can be reconstructed from its (floor(d/2) + 1)-skeleton, and we prove an analogous result for d-dimensional cubical manifolds that can be embedded as a subcomplex into a cube I^N. Chapter 2 studies CAT(0) cubical complexes, using the framework of a poset with inconsistent pairs developed by Ardila et al. We introduce a simplicial complex called the "crossing complex" associated to each CAT(0) cubical complex, and study its properties. We deduce that this crossing complex holds much of the combinatorial information contained in the cubical complex: our main results relate their f-vectors, hyperplane/link structure, and balancedness. Finally, Chapter 3 studies the topology of complements of skeletons in polytopal complexes: we derive a long exact sequence involving homology of skeleton complements and links, and we characterise various topological properties of spaces in terms of skeleton complements. Our main application of this machinery is to CAT(0) cubical complexes: we conclude that these complexes also share several topological properties with their crossing complexes
On the g2-number of various classes of spheres and manifolds
Full text linkThesis (Ph.D.)--University of Washington, 2017-08For a -dimensional simplicial complex , we let be the number of -dimensional faces of for . One classic problem in geometric combinatorics is the following: for a given class of simplicial complexes, find tight upper and lower bounds on the face numbers and characterize the complexes that attain these bounds. This dissertation studies these questions in various classes of simplicial complexes including balanced manifolds, flag manifolds and simplicial spheres. A -dimensional simplicial complex is called balanced if its graph is -colorable. In Chapter 2, we determine the minimum number of vertices needed to provide balanced triangulations of \Sp^{d-2}-bundles over \Sp^1. Similar results apply to all balanced triangulated manifolds with and . In Chapter 3, we turn to the Upper Bound Conjecture for balanced simplicial spheres. We find the first two examples of non-octahedral balanced 2-neighborly spheres. Each construction is of dimension 3 and with 16 vertices. Along the way, we show that the rank-selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition. A simplicial complex is flag if it is the clique complex of its graph. In Chapter 4, we settle the Upper Bound Conjecture for flag 3-manifolds, establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. In Chapter 5, we characterize homology manifolds with . We prove that every prime homology manifold with is obtained by centrally retriangulating a polytopal sphere with along a certain subcomplex. This implies that all homology spheres with are polytopal spheres. In Chapter 6, we prove that for any prime homology -sphere () with and any edge , the graph is generically -rigid. This confirms a conjecture of Nevo and Novinsky
On f-vectors of polytopes and matroids
No full textThesis (Ph.D.)--University of Washington, 2016-08The f-vector of a simplicial complex is a fundamental invariant that counts the number of faces in each dimension. A natural question in the theory of simplicial complexes is to understand the relationship between the f-vector of the simplicial complex and the properties of the topological, algebraic and combinatorial structures associated to the simplicial complex. This dissertation studies this question for two different classes of simplicial complexes: simplicial polytopes and matroid independence complexes. For the class of simplicial polytopes we study what are the possible f-vectors of polytopes that are good approximations of a convex body with smooth enough boundary. In particular, in Chapter 2, we settle a longstanding conjecture of Kalai asserting that good approximations of smooth convex body K are far from extremal in the sense of the lower bound theorem in a precise way. We make the result quantitative in the case the convex body is of type C^2. Little is known about f-vectors of matroid independence complexes. A full characterization is believed to be out of reach and several conjectures about properties of such vectors are wide open. A famous one is a conjecture of Stanley that predicts certain behavior based on the properties of the Stanley-Reisner ring of the matroid independence complex. The main goal of the second part of this document is to study this conjecture. In Chapter 3, we prove that the conjecture holds for rank 4 matroids by means of a new combinatorial method. In Chapter 4 we study the external activity complex of a matroid which is topologically simpler than the independence complex of the matroid and contains the information of -vector. Chapter 5 introduces the notion of a quasi-matroidal class of ordered simplicial complexes, the notion is used to provide extensions of various properties of matroids, including a refinement of Stanley's conjecture that is proved to hold in a variety of cases, including Schubert matroids
Homological algebra of Stanley-Reisner rings and modules
Full text linkThesis (Ph.D.)--University of Washington, 2018Associated to each simplicial complex and each field \field is the Stanley--Reisner ring \field[\Delta]. The answers to a multitude of questions related to simplicial complexes have historically been found through a thorough examination of the algebraic structure of \field[\Delta]. There is a rich pre-existing body of literature equating combinatorial and topological statements about the structure of a simplicial complex with statements about \field[\Delta]; this dissertation expands upon the dictionary translating such statements by examining algebraic structures derived from \field[\Delta]. In particular, we mainly focus on the local cohomology modules H_\mideal^i(\field[\Delta]) and the Ext modules \Ext^i(\field, \field[\Delta]). Roughly speaking, a simplicial complex is called Buchsbaum if its geometric realization is similar to a manifold. In Chapter 2, we study the homological structure of \field[\Delta] and some of its quotients by linear forms when fails to be Buchsbaum in a way that may be considered ``minimal.'' We obtain a large family of rings with interesting combinations of the (ring-theoretic) properties of Buchsbaumness and quasi-Buchsbaumness, while developing a geometric interpretation of their presence. In Chapter 3, we turn our attention to complexes that exhibit some degree of symmetry via group actions. Here it is shown that the induced action on H_\mideal^i(\field[\Delta]) can be described in a similar manner to the one induced on the simplicial cohomology modules of and some of its subcomplexes. Some applications to the study of face numbers are provided. If the definition of a simplicial complex is slightly relaxed, then one arrives at the notion of a simplicial poset. Chapter 4 is devoted to the study of these objects and their associated face rings. We provide extensions of well-known results describing the structure of the Ext and local cohomology modules of simplicial complexes to this larger class and further examine the Buchsbaum property. In Chapter 5, we study the class of balanced triangulations of manifolds and obtain lower bounds on entries in the -vector phrased in terms of topological invariants. This proves a conjecture of Klee and Novik
A Survey of Tverberg Type Problems
Full text linkThesis (Master's)--University of Washington, 2016-12Tverberg's theorem, which celebrates its fiftieth anniversary this year, is a central result in the fields of discrete geometry and topological combinatorics. Proved in 1966, it was a major step in solving questions whether, given a complex, all affine (or more generally, continuous) maps into some Euclidean space have some specified intersection property. Many other extensions and variations stem from this first result, such as the topological Tverberg conjecture and the colorful Tverberg theorem. Much work is still being done to generalize and extend Tverberg's theorem, which has resulted in several recent and major breakthroughs. Most notably, Frick's surprising counterexample to the topological Tverberg conjecture was only discovered in 2015, and the "constraint method" used to construct the counterexample lends itself to many other applications in proving different extensions of the topological Tverberg conjecture. In this thesis, we will look at some classical theorems that inspired Tverberg's theorem, the current state of affairs vis-a-vis the Tverberg conjecture, and other closely related problems
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Face Numbers of Polytopes, Posets, and Complexes
No full textThesis (Ph.D.)--University of Washington, 2022A key tool that combinatorialists use to study simplicial complexes and polytopes is the {\bf -vector} (or face vector), which records the number of faces of each dimension. In order to better understand the face numbers, relations involving both equalities and inequalities on -vectors have been extensively studied. In this dissertation we discuss the author's contributions to these topics.
The classical Dehn--Sommerville relations assert that the -vector of an Eulerian simplicial complex is symmetric. In Chapter 2, we establish three generalizations of the Dehn--Sommerville relations: one for the -vectors of pure simplicial complexes, another one for the flag -vectors of balanced simplicial complexes and graded posets, and yet another one for the toric -vectors of graded posets with restricted singularities. In all of these cases, we express any failure of symmetry in terms of ``errors coming from the links." For simplicial complexes, this further extends Klee's semi-Eulerian relations. In Chapters 3 and 4, we change our focus from equalities to inequalities on -vectors. In 1967, Gr\"unbaum conjectured that any -dimensional polytope with vertices has at least
-faces. In Chapter 3, we prove this conjecture and also characterize the cases in which equality holds. In Chapter 4, several extensions of Gr\"unbaum's conjecture are established. Specifically, it is proved that every lattice with diamond property and atoms has at least elements of rank . Furthermore, in the case of lattices that are face lattices of strongly regular CW complexes representing normal pseudomanifolds with up to vertices, a characterization of equality cases is given. Finally, sharp lower bounds on the number of -faces of strongly regular CW complexes representing normal pseudomanifolds with vertices are obtained. These bounds are given by the face numbers of certain polytopes with vertices
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