Connectedness- Its Evolution And Applications, 2019 Ursinus College

#### Connectedness- Its Evolution And Applications, Nicholas A. Scoville

*Topology*

No abstract provided.

Dense Geometry Of Music And Visual Arts: Vanishing Points, Continuous Tonnetz, And Theremin Performance, 2019 Independent researcher, Palermo, Italy

#### Dense Geometry Of Music And Visual Arts: Vanishing Points, Continuous Tonnetz, And Theremin Performance, Maria Mannone, Irene Iaccarino, Rosanna Iembo

*The STEAM Journal*

The dualism between continuous and discrete is relevant in music theory as well as in performance practice of musical instruments. Geometry has been used since longtime to represent relationships between notes and chords in tonal system. Moreover, in the field of mathematics itself, it has been shown that the continuity of real numbers can arise from geometrical observations and reasoning. Here, we consider a geometrical approach to generalize representations used in music theory introducing continuous pitch. Such a theoretical framework can be applied to instrument playing where continuous pitch can be naturally performed. Geometry and visual representations of concepts of ...

Parametric Natura Morta, 2019 Independent researcher, Palermo, Italy

#### Parametric Natura Morta, Maria C. Mannone

*The STEAM Journal*

Parametric equations can also be used to draw fruits, shells, and a cornucopia of a mathematical still life. Simple mathematics allows the creation of a variety of shapes and visual artworks, and it can also constitute a pedagogical tool for students.

Topological Properties Of A 3-Rung Möbius Ladder, 2018 Stephen F. Austin State University

#### Topological Properties Of A 3-Rung Möbius Ladder, Rebecca Woods

*Electronic Theses and Dissertations*

In this work, we discuss the properties of the 3-rung Möbius ladder on the torus. We also prove ℤ_{2} is an orientation preserving topological symmetry group of the 3-rung Möbius ladder with sides and rungs distinct, embedded in the torus.

Finite Simple Graphs And Their Associated Graph Lattices, 2018 Middle Tennessee State University

#### Finite Simple Graphs And Their Associated Graph Lattices, James B. Hart, Brian Frazier

*Theory and Applications of Graphs*

In his 2005 dissertation, Antoine Vella explored combinatorical aspects of finite graphs utilizing a topological space whose open sets are intimately tied to the structure of the graph. In this paper, we go a step further and examine some aspects of the open set lattices induced by these topological spaces. In particular, we will characterize all lattices that constitute the opens for finite simple graphs endowed with this topology, explore the structure of these lattices, and show that these lattices contain information necessary to reconstruct the graph and its complement in several ways.

Topological Detection Of The Dimension Of The Stimuli Space, 2018 Illinois State University

#### Topological Detection Of The Dimension Of The Stimuli Space, Aliaksandra Yarosh

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

Equivalent Constructions Of Cartan Pairs, 2018 University of New Mexico

#### Equivalent Constructions Of Cartan Pairs, Phung Thanh Tran

*Math Theses*

Feldman and Moore [4] introduce Cartan subalgebra of the von Neumann algebra M on a separable Hilbert space H from the natural subalgebra of M(R, sigma), the twisted algebra of matrices over the relation R on a Borel space (X, B, muy). They show that if M has a Cartan subalgebra A, then M is isomorphic to M(R, sigma) where A is the twisted algebra onto the diagonal subalgebra L^inf (X, muy). The relation R is unique to isomorphism and the orbit of the two-cohomology class on R in the torus T, which is the automorphism group ...

Studying The Space Of Almost Complex Structures On A Manifold Using De Rham Homotopy Theory, 2018 The Graduate Center, City University of New York

#### Studying The Space Of Almost Complex Structures On A Manifold Using De Rham Homotopy Theory, Bora Ferlengez

*All Dissertations, Theses, and Capstone Projects*

In his seminal paper *Infinitesimal Computations in Topology*, Sullivan constructs algebraic models for spaces and then computes various invariants using them. In this thesis, we use those ideas to obtain a finiteness result for such an invariant (the de Rham homotopy type) for each connected component of the space of cross-sections of certain fibrations. We then apply this result to differential geometry and prove a finiteness theorem of the de Rham homotopy type for each connected component of the space of almost complex structures on a manifold. As a special case, we discuss the space of almost complex structures on ...

Topological Recursion And Random Finite Noncommutative Geometries, 2018 The University of Western Ontario

#### Topological Recursion And Random Finite Noncommutative Geometries, Shahab Azarfar

*Electronic Thesis and Dissertation Repository*

In this thesis, we investigate a model for quantum gravity on finite noncommutative spaces using the topological recursion method originated from random matrix theory. More precisely, we consider a particular type of finite noncommutative geometries, in the sense of Connes, called spectral triples of type ${(1,0)} \,$, introduced by Barrett. A random spectral triple of type ${(1,0)}$ has a fixed fermion space, and the moduli space of its Dirac operator ${D=\{ H , \cdot \} \, ,}$ ${H \in {\mathcal{H}_N}}$, encoding all the possible geometries over the fermion space, is the space of Hermitian matrices ${\mathcal{H}_N}$. A distribution of ...

Syzygy Order Of Big Polygon Spaces, 2018 The University of Western Ontario

#### Syzygy Order Of Big Polygon Spaces, Jianing Huang

*Electronic Thesis and Dissertation Repository*

For a compact smooth manifold with a torus action, its equivariant cohomology is a finitely generated module over a polynomial ring encoding information about the space and the action. For such a module, we can associate a purely algebraic notion called syzygy order. Syzygy order of equivariant cohomology is closely related to the exactness of Atiyah-Bredon sequence in equivariant cohomology. In this thesis we study a family of compact orientable manifolds with torus actions called big polygon spaces. We compute the syzygy orders of their equivariant cohomologies. The main tool used is a quotient criterion for syzygies in equivariant cohomology ...

Persistence Equivalence Of Discrete Morse Functions On Trees, 2018 Ursinus College

#### Persistence Equivalence Of Discrete Morse Functions On Trees, Yuqing Liu

*Mathematics Summer Fellows*

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this notion of equivalence to other notions of equivalent discrete Morse functions. We then compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree. We conclude with an example illustrating our construction.

Local Higher Category Theory, 2018 The University of Western Ontario

#### Local Higher Category Theory, Nicholas Meadows

*Electronic Thesis and Dissertation Repository*

The purpose of this thesis is to give presheaf-theoretic versions of three of the main extant models of higher category theory: the Joyal, Rezk and Bergner model structures. The construction of these model structures takes up Chapters 2, 3 and 4 of the thesis, respectively. In each of the model structures, the weak equivalences are local or ‘stalkwise’ weak equivalences. In addition, it is shown that certain Quillen equivalences between the aforementioned models of higher category theory extend to Quillen equivalences between the various models of local higher category theory.

Throughout, a number of features of local higher category theory ...

From Sets To Metric Spaces To Topological Spaces, 2018 Ursinus College

#### From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville

*Topology*

No abstract provided.

Nearness Without Distance, 2018 Ursinus College

Toroidal Embeddings And Desingularization, 2018 California State University, San Bernardino

#### Toroidal Embeddings And Desingularization, Leon Nguyen

*Electronic Theses, Projects, and Dissertations*

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.

Interdisciplinary Fun With Knapp Chairs, Mit's Erik And Martin Demaine, 2018 University of San Diego

#### Interdisciplinary Fun With Knapp Chairs, Mit's Erik And Martin Demaine, Ryan T. Blystone

*Research Week*

No abstract provided.

A Study Of Topological Invariants In The Braid Group B2, 2018 East Tennessee State University

#### A Study Of Topological Invariants In The Braid Group B2, Andrew Sweeney

*Electronic Theses and Dissertations*

The Jones polynomial is a special topological invariant in the field of Knot Theory. Created by Vaughn Jones, in the year 1984, it is used to study when links in space are topologically different and when they are topologically equivalent. This thesis discusses the Jones polynomial in depth as well as determines a general form for the closure of any braid in the braid group B2 where the closure is a knot. This derivation is facilitated by the help of the Temperley-Lieb algebra as well as with tools from the field of Abstract Algebra. In general, the Artin braid group ...

Affine And Projective Planes, 2018 Missouri State University

#### Affine And Projective Planes, Abraham Pascoe

*MSU Graduate Theses*

In this thesis, we investigate affine and projective geometries. An affine geometry is an incidence geometry where for every line and every point not incident to it, there is a unique line parallel to the given line. Affine geometry is a generalization of the Euclidean geometry studied in high school. A projective geometry is an incidence geometry where every pair of lines meet. We study basic properties of affine and projective planes and a number of methods of constructing them. We end by prov- ing the Bruck-Ryser Theorem on the non-existence of projective planes of certain orders.

The Average Measure Of A K-Dimensional Simplex In An N-Cube, 2018 Missouri State University

#### The Average Measure Of A K-Dimensional Simplex In An N-Cube, John A. Carter

*MSU Graduate Theses*

Within an n-dimensional unit cube, a number of k-dimensional simplices can be formed whose vertices are the vertices of the n-cube. In this thesis, we analyze the average measure of a k-simplex in the n-cube. We develop exact equations for the average measure when k = 1, 2, and 3. Then we generate data for these cases and conjecture that their averages appear to approach n^{k/2} times some constant. Using the convergence of Bernstein polynomials and a k-simplex Bernstein generalization, we prove the conjecture is true for the 1-simplex and 2-simplex cases. We then develop a generalized formula for ...

Divergence Of Cat(0) Cube Complexes And Coxeter Groups, 2018 The Graduate Center, City University of New York

#### Divergence Of Cat(0) Cube Complexes And Coxeter Groups, Ivan Levcovitz

*All Dissertations, Theses, and Capstone Projects*

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we characterize right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani-Thomas that addressed the class of 2-dimensional right-angled Coxeter groups. This characterization also has a direct application to the theory of random right-angled Coxeter groups. As another application of the divergence bounds obtained for cube complexes, we provide an inductive graph theoretic criterion on a right-angled Coxeter ...