Tropical Geometry Of T-Varieties With Applications To Algebraic Statistics, 2022 University of Kentucky

#### Tropical Geometry Of T-Varieties With Applications To Algebraic Statistics, Joseph Cummings

*Theses and Dissertations--Mathematics*

Varieties with group action have been of interest to algebraic geometers for centuries. In particular, toric varieties have proven useful both theoretically and in practical applications. A rich theory blending algebraic geometry and polyhedral geometry has been developed for T-varieties which are natural generalizations of toric varieties. The first results discussed in this dissertation study the relationship between torus actions and the well-poised property. In particular, I show that the well-poised property is preserved under a geometric invariant theory quotient by a (quasi-)torus. Conversely, I argue that T-varieties built on a well-poised base preserve the well-poised property when the base …

Topics In Moufang Loops, 2022 University of Denver

#### Topics In Moufang Loops, Riley Britten

*Electronic Theses and Dissertations*

We will begin by discussing power graphs of Moufang loops. We are able to show that as in groups the directed power graph of a Moufang loop is uniquely determined by the undirected power graph. In the process of proving this result we define the generalized octonion loops, a variety of Moufang loops which behave analogously to the generalized quaternion groups. We proceed to investigate para-F quasigroups, a variety of quasigroups which we show are antilinear over Moufang loops. We briefly depart from the context of Moufang loops to discuss solvability in general loops. We then prove some results on …

Equisingular Approximation Of Analytic Germs, 2021 The University of Western Ontario

#### Equisingular Approximation Of Analytic Germs, Aftab Yusuf Patel

*Electronic Thesis and Dissertation Repository*

This thesis deals with the problem of approximating germs of real or complex analytic spaces by Nash or algebraic germs. In particular, we investigate the problem of approximating analytic germs in various ways while preserving the Hilbert-Samuel function, which is of importance in the resolution of singularities. We first show that analytic germs that are complete intersections can be arbitrarily closely approximated by algebraic germs which are complete intersections with the same Hilbert-Samuel function. We then show that analytic germs whose local rings are Cohen-Macaulay can be arbitrarily closely approximated by Nash germs whose local rings are Cohen- Macaulay and …

Acceleration Skinning: Kinematics-Driven Cartoon Effects For Articulated Characters, 2021 Clemson University

#### Acceleration Skinning: Kinematics-Driven Cartoon Effects For Articulated Characters, Niranjan Kalyanasundaram

*All Theses*

Secondary effects are key to adding fluidity and style to animation. This thesis introduces the idea of “Acceleration Skinning” following a recent well-received technique, Velocity Skinning, to automatically create secondary motion in character animation by modifying the standard pipeline for skeletal rig skinning. These effects, which animators may refer to as squash and stretch or drag, attempt to create an illusion of inertia. In this thesis, I extend the Velocity Skinning technique to include acceleration for creating a wider gamut of cartoon effects. I explore three new deformers that make use of this Acceleration Skinning framework: followthrough, centripetal stretch, and …

ℂ-Motivic Modular Forms, 2021 Max-Planck-Institut für Mathematik

#### ℂ-Motivic Modular Forms, Bogdan Gheorghe, Daniel C. Isaksen, Achim Krause, Nicolas Ricka

*Mathematics Faculty Research Publications*

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations.We compute the Steenrod algebra in this context, and we construct a “motivic modular forms” spectrum over ℂ.

Cache-Friendly, Modular And Parallel Schemes For Computing Subresultant Chains, 2021 The University of Western Ontario

#### Cache-Friendly, Modular And Parallel Schemes For Computing Subresultant Chains, Mohammadali Asadi

*Electronic Thesis and Dissertation Repository*

The RegularChains library in Maple offers a collection of commands for solving polynomial systems symbolically with taking advantage of the theory of regular chains. The primary goal of this thesis is algorithmic contributions, in particular, to high-performance computational schemes for subresultant chains and underlying routines to extend that of RegularChains in a C/C++ open-source library.

Subresultants are one of the most fundamental tools in computer algebra. They are at the core of numerous algorithms including, but not limited to, polynomial GCD computations, polynomial system solving, and symbolic integration. When the subresultant chain of two polynomials is involved in a client …

Windows In Algebraic Geometry And Applications To Moduli, 2021 University of Massachusetts Amherst

#### Windows In Algebraic Geometry And Applications To Moduli, Sebastian Torres

*Doctoral Dissertations*

We apply the theory of windows, as developed by Halpern-Leistner and by Ballard, Favero and Katzarkov, to study certain moduli spaces and their derived categories. Using quantization and other techniques we show that stable GIT quotients of $(\mathbb{P}^1)^n$ by $PGL_2$ over an algebraically closed field of characteristic zero satisfy a rare property called Bott vanishing, which states that $\Omega^j_Y \otimes L$ has no higher cohomology for every j and every ample line bundle L. Similar techniques are used to reprove the well known fact that toric varieties satisfy Bott vanishing. We also use windows to explore derived categories of moduli …

Equivariant Smoothings Of Cusp Singularities, 2021 University of Massachusetts Amherst

#### Equivariant Smoothings Of Cusp Singularities, Angelica Simonetti

*Doctoral Dissertations*

Let $p \in X$ be the germ of a cusp singularity and let $\iota$ be an antisymplectic involution, that is an involution free on $X\setminus \{p\}$ and such that there exists a nowhere vanishing holomorphic 2-form $\Omega$ on $X\setminus \{p\}$ for which $\iota^*(\Omega)=-\Omega$. We prove that a sufficient condiition for such a singularity equipped with an antisymplectic involution to be equivariantly smoothable is the existence of a Looijenga (or anticanonical) pair $(Y,D)$ that admits an involution free on $Y\setminus D$ and that reverses the orientation of $D$.

Using Lie Sphere Geometry To Study Dupin Hypersurfaces In R^N, 2021 College of the Holy Cross

#### Using Lie Sphere Geometry To Study Dupin Hypersurfaces In R^N, Thomas E. Cecil

*Mathematics Department Faculty Scholarship*

A hypersurface *M* in **R**^{n} or S^{n} is said to be *Dupin* if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be *proper Dupin* if each principal curvature has constant multiplicity on *M*, i.e., the number of distinct principal curvatures is constant on *M*. The notions of Dupin and proper Dupin hypersurfaces in **R**^{n }or *S ^{n}* can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective …

Distribution Of The P-Torsion Of Jacobian Groups Of Regular Matroids, 2021 The University of Western Ontairo

#### Distribution Of The P-Torsion Of Jacobian Groups Of Regular Matroids, Sergio R. Zapata Ceballos

*Electronic Thesis and Dissertation Repository*

Given a regular matroid $M$ and a map $\lambda\colon E(M)\to \N$, we construct a regular matroid $M_\lambda$. Then we study the distribution of the $p$-torsion of the Jacobian groups of the family $\{M_\lambda\}_{\lambda\in\N^{E(M)}}$. We approach the problem by parameterizing the Jacobian groups of this family with non-trivial $p$-torsion by the $\F_p$-rational points of the configuration hypersurface associated to $M$. In this way, we reduce the problem to counting points over finite fields. As a result, we obtain a closed formula for the proportion of these groups with non-trivial $p$-torsion as well as some estimates. In addition, we show that the …

Searching For New Relations Among The Eilenberg-Zilber Maps, 2021 Western University

#### Searching For New Relations Among The Eilenberg-Zilber Maps, Owen T. Abma

*Undergraduate Student Research Internships Conference*

The goal of this project was to write a computer program that would aid in the search for relations among the Eilenberg-Zilber maps, which relate to simplicial objects in algebraic topology. This presentation outlines the process of writing this program, the challenges faced along the way, and the final results of the project.

Studies Of Subvarieties Of Classical Complex Algebraic Geometry, 2021 Western University

#### Studies Of Subvarieties Of Classical Complex Algebraic Geometry, Wenzhe Wang

*Undergraduate Student Research Internships Conference*

My project in this USRI program is to study subvariety of classical complex algebraic geometry. I observed the orbit of elements in the unit sphere in space ℂ² ⊗ ℂ², the structure of unit sphere of ℂ² ⊗ ℂ². After this, I tried to generalize the result to ℂ^n ⊗ ℂ^n.

From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, 2021 Wayne State University

#### From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, Andrew Salch, Adam Regalski, Hassan Abdallah, Raviteja Suryadevara, Michael J. Catanzaro, Vaibhav A. Diwadkar

*Mathematics Faculty Research Publications*

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting *structure from the* (non-summarized) *fMRI data itself* are heretofore nonexistent. …

Elliptic Curves And Their Practical Applications, 2021 Missouri State University

#### Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv

*MSU Graduate Theses*

Finding rational points that satisfy functions known as elliptic curves induces a finitely-generated abelian group. Such functions are powerful tools that were used to solve Fermat's Last Theorem and are used in cryptography to send private keys over public systems. Elliptic curves are also useful in factoring and determining primality.

Probability Distributions For Elliptic Curves In The Cgl Hash Function, 2021 Brown University

#### Probability Distributions For Elliptic Curves In The Cgl Hash Function, Dhruv Bhatia, Kara Fagerstrom, Max Watson

*Mathematical Sciences Technical Reports (MSTR)*

Hash functions map data of arbitrary length to data of predetermined length. Good hash functions are hard to predict, making them useful in cryptography. We are interested in the elliptic curve CGL hash function, which maps a bitstring to an elliptic curve by traversing an inputdetermined path through an isogeny graph. The nodes of an isogeny graph are elliptic curves, and the edges are special maps betwixt elliptic curves called isogenies. Knowing which hash values are most likely informs us of potential security weaknesses in the hash function. We use stochastic matrices to compute the expected probability distributions of the …

A Cone Conjecture For Log Calabi-Yau Surfaces, 2021 University of Massachusetts Amherst

#### A Cone Conjecture For Log Calabi-Yau Surfaces, Jennifer Li

*Doctoral Dissertations*

In 1993, Morrison conjectured that the automorphism group of a Calabi-Yau 3-fold acts on its nef cone with a rational polyhedral fundamental domain. In this thesis, we prove a version of this conjecture for log Calabi-Yau surfaces. In particular, for a generic log Calabi-Yau surface with singular boundary, the monodromy group acts on the nef effective cone with a rational polyhedral fundamental domain. In addition, the automorphism group of the unique surface with a split mixed Hodge structure in each deformation type acts on the nef effective cone with a rational polyhedral fundamental domain. We also prove that, given a …

Evaluating The Historical Accuracy Of Blackwork Embroidery With Fractal Analysis, 2021 University of Lynchburg

#### Evaluating The Historical Accuracy Of Blackwork Embroidery With Fractal Analysis, Rhiannon Cire

*Undergraduate Theses and Capstone Projects*

The intricate monochromatic embroidery that graced the collars and cuffs of Renaissance nobility and domestic materials from that era has been little studied beyond the historical costuming and crafting communities. This style, known as blackwork, for it was traditionally done in black silk on white linen, exemplifies how complex and visually-appealing designs can arise from repetition of simple forms, often demonstrating the fractal property of self-similarity. Though most blackwork patterns are not true fractals, fractal analysis offers a means of objectively quantifying their complexity and new lens through which to examine this embroidery technique. The purpose of this study was …

On Elliptic Curves, 2021 Missouri State University

#### On Elliptic Curves, Montana S. Miller

*MSU Graduate Theses*

An elliptic curve over the rational numbers is given by the equation *y ^{2} = x^{3}+Ax+B*. In our thesis, we study elliptic curves. It is known that the set of rational points on the elliptic curve form a finitely generated abelian group induced by the secant-tangent addition law. We present an elementary proof of associativity using Maple. We also present a relatively concise proof of the Mordell-Weil Theorem.

Lecture 03: Hierarchically Low Rank Methods And Applications, 2021 King Abdullah University of Science and Technology

#### Lecture 03: Hierarchically Low Rank Methods And Applications, David Keyes

*Mathematical Sciences Spring Lecture Series*

As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the …

Lecture 00: Opening Remarks: 46th Spring Lecture Series, 2021 University of Arkansas, Fayetteville

#### Lecture 00: Opening Remarks: 46th Spring Lecture Series, Tulin Kaman

*Mathematical Sciences Spring Lecture Series*

Opening remarks for the 46th Annual Mathematical Sciences Spring Lecture Series at the University of Arkansas, Fayetteville.