Algebraic Geometry Commons^™

Studying Extended Sets From Young Tableaux, Eric Nofziger

Rose-Hulman Undergraduate Mathematics Journal

Young tableaux are combinatorial objects related to the partitions of an integer and have various applications in representation theory. They are particularly useful in the study of the fibers arising from the Springer resolution. In recent work of Graham-Precup-Russell, an association has been made between a given row-strict tableau and three disjoint subsets of {1,2,...,n}. These subsets are then used in the study of extended Springer fibers, so we call them extended sets. In this project, we use combinatorial techniques to classify which of these extended sets correlate to a valid row-strict or standard tableau and give bounds on the …

A Cluster Structure On The Coordinate Ring Of Partial Flag Varieties, Fayadh Kadhem

LSU Doctoral Dissertations

The main goal of this dissertation is to show that the (multi-homogeneous) coordinate ring of a partial flag variety C[G/P_K^−] contains a cluster algebra for every semisimple complex algebraic group G. We use derivation properties and a canonical lifting map to prove that the cluster algebra structure A of the coordinate ring C[N_K] of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure \hat{A} living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra \hat{A} is equal …

Generalization Of Bi-Canonical Degrees, Joseph Brennan, Laura Ghezzi, Jooyoun Hong, Wolmer Vasconcelos

Publications and Research

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module ω. Attached to each such ring R, when ω is an ideal, there are integers–the type of R, the reduction number of ω–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In (Ghezzi et al. in JMS 589:506–528, 2017) and (Ghezzi et al. in JMS 571:55–74, 2021) we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where ω is not necessarily an ideal. We also discuss generalizations to rings …

The Local Lifting Problem For Curves With Quaternion Actions, George Mitchell

Dissertations, Theses, and Capstone Projects

The lifting problem asks whether one can lift Galois covers of curves defined over positive characteristic to Galois covers of curves over characteristic zero. The lifting problem has an equivalent local variant, which asks if a Galois extension of complete discrete valuation rings over positive characteristic, with algebraically closed residue field, can be lifted to characteristic zero. In this dissertation, we content ourselves with the study of the local lifting problem when the prime is 2, and the Galois group of the extension is the group of quaternions. In this case, it is known that certain quaternion extensions cannot be …

Reduction Of L-Functions Of Elliptic Curves Modulo Integers, Félix Baril Boudreau

Electronic Thesis and Dissertation Repository

Let $\mathbb{F}_q$ be a finite field of size $q$, where $q$ is a power of a prime $p \geq 5$. Let $C$ be a smooth, proper, and geometrically connected curve over $\mathbb{F}_q$. Consider an elliptic curve $E$ over the function field $K$ of $C$ with nonconstant $j$-invariant. One can attach to $E$ its $L$-function $L(T,E/K)$, which is a generating function that contains information about the reduction types of $E$ at the different places of $K$. The $L$-function of $E/K$ was proven to be a polynomial in $\mathbb{Z}[T]$.

In 1985, Schoof devised an algorithm to compute the zeta function of an …

The Design And Implementation Of A High-Performance Polynomial System Solver, Alexander Brandt

Electronic Thesis and Dissertation Repository

This thesis examines the algorithmic and practical challenges of solving systems of polynomial equations. We discuss the design and implementation of triangular decomposition to solve polynomials systems exactly by means of symbolic computation.

Incremental triangular decomposition solves one equation from the input list of polynomials at a time. Each step may produce several different components (points, curves, surfaces, etc.) of the solution set. Independent components imply that the solving process may proceed on each component concurrently. This so-called component-level parallelism is a theoretical and practical challenge characterized by irregular parallelism. Parallelism is not an algorithmic property but rather a geometrical …

Recognizing And Reducing Ambiguity In Mathematics Curriculum And Relations Of Θ-Functions In Genus One And Two: A Geometric Perspective, Shantel Spatig

All Graduate Plan B and other Reports

Anxiety and mathematics come hand in hand for many individuals. This is due, in
part, to the fact that the only experience they have with mathematics is what some
mathematics educators refer to as "schoolmath," which uses a somewhat different
language than real mathematics. The language of schoolmath can cause individu-
als to have confusion and develop misconceptions related to several mathematical
concepts. One such concept is a fraction. In chapter one of this report, one possible
reason for this is discussed and a possible solution is purposed.
In chapter three of this report, genus-two curves admitting an elliptic involution …

Identifying Trace Affine Linear Sets Using Homotopy Continuation, Julianne Mckay

All Theses

We investigate how the coefficients of a sparse polynomial system influence the sum, or the trace, of its solutions. We discuss an extension of the classical trace test in numerical algebraic geometry to sparse polynomial systems. Two known methods for identifying a trace affine linear subset of the support of a sparse polynomial system use sparse resultants and polyhedral geometry, respectively. We introduce a new approach which provides more precise classifications of trace affine linear sets than was previously known. For this new approach, we developed software in Macaulay2.

Academic Hats And Ice Cream: Two Optimization Problems, Valery F. Ochkov, Yulia V. Chudova

Journal of Humanistic Mathematics

This article describes the use of computer software to optimize the design of an academic hat and an ice cream cone!

Bbt Acoustic Alternative Top Bracing Cadd Data Set-Norev-2022jun28, Bill Hemphill

STEM Guitar Project’s BBT Acoustic Kit

This electronic document file set consists of an overview presentation (PDF-formatted) file and companion video (MP4) and CADD files (DWG & DXF) for laser cutting the ETSU-developed alternate top bracing designs and marking templates for the STEM Guitar Project’s BBT (OM-sized) standard acoustic guitar kit. The three (3) alternative BBT top bracing designs in this release are
(a) a one-piece base for the standard kit's (Martin-style) bracing,
(b) 277 Ladder-style bracing, and
(c) an X-braced fan-style bracing similar to traditional European or so-called 'classical' acoustic guitars.

The CADD data set for each of the three (3) top bracing designs includes …

Bbt Side Mold Assy, Bill Hemphill

STEM Guitar Project’s BBT Acoustic Kit

This electronic document file set covers the design and fabrication information of the ETSU Guitar Building Project’s BBT (OM-sized) Side Mold Assy for use with the STEM Guitar Project’s standard acoustic guitar kit. The extended 'as built' data set contains an overview file and companion video, the 'parent' CADD drawing, CADD data for laser etching and cutting a drill &/or layout template, CADD drawings in AutoCAD .DWG and .DXF R12 formats of the centerline tool paths for creating the mold assembly pieces on an AXYZ CNC router, and support documentation for CAM applications including router bit specifications, feeds, speed, multi-pass …

Unomaha Problem Of The Week (2021-2022 Edition), Brad Horner, Jordan M. Sahs

UNO Student Research and Creative Activity Fair

The University of Omaha math department's Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester's end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes.

Now there are three difficulty tiers to POW problems, roughly corresponding to …

The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza

Dissertations, Theses, and Capstone Projects

Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf …

On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi

Rose-Hulman Undergraduate Mathematics Journal

It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$-invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of either 0 …

The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

Electronic Theses, Projects, and Dissertations

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …

Lattice Reduction Algorithms, Juan Ortega

Electronic Theses, Projects, and Dissertations

The purpose of this thesis is to propose and analyze an algorithm that follows
similar steps of Guassian Lattice Reduction Algorithm in two-dimensions and applying
them to three-dimensions. We start off by discussing the importance of cryptography in
our day to day lives. Then we dive into some linear algebra and discuss specific topics that
will later help us in understanding lattice reduction algorithms. We discuss two lattice
problems: the shortest vector problem and the closest vector problem. Then we introduce
two types of lattice reduction algorithms: Guassian Lattice Reduction in two-dimensions
and the LLL Algortihm. We illustrate how both …

Towards A Generalization Of Fulton's Intersection Multiplicity Algorithm, Ryan Sandford

Electronic Thesis and Dissertation Repository

In this manuscript we generalize Fulton's bivariate intersection multiplicity algorithm to a partial intersection multiplicity algorithm in the n-variate setting. We extend this generalization of Fulton's algorithm to work at any point, rational or not, using the theory of regular chains. We implement these algorithms in Maple and provide experimental testing. The results indicate the proposed algorithm often outperforms the existing standard basis-free intersection multiplicity algorithm in Maple, typically by one to two orders of magnitude. Moreover, we also provide some examples where the proposed algorithm outperforms intersection multiplicity algorithms which rely on standard bases, indicating the proposed algorithm is …

On The Geometry Of Multi-Affine Polynomials, Junquan Xiao

Electronic Thesis and Dissertation Repository

This work investigates several geometric properties of the solutions of the multi-affine polynomials. Chapters 1, 2 discuss two different notions of invariant circles. Chapter 3 gives several loci of polynomials of degree three. A locus of a complex polynomial p(z) is a minimal, with respect to inclusion, set that contains at least one point of every solution of the polarization of the polynomial. The study of such objects allows one to improve upon know results about the location of zeros and critical points of complex polynomials, see for example [22] and [24]. A complex polynomial has many loci. It is …

Kissing The Archimedeans, Anthony Webb

All NMU Master's Theses

In this paper the three dimensional kissing problem will be related to the Platonic and Archimedean solids. On each polyhedra presented their vertices will have spheres expanding such that the center of each of these outer spheres are the vertices of the polyhedron, and these outer spheres will continue to expand until they become tangent to each other. The ratio will be found between the radius of each outer sphere, and the radius of an inner sphere such that each inner sphere's center is the circumcenter of the polyhedron, and the inner sphere is tangent to each outer sphere. Every …

Anticanonical Models Of Smoothings Of Cyclic Quotient Singularities, Arie A. Stern Gonzalez

Doctoral Dissertations

In this thesis we study anticanonical models of smoothings of cyclic quotient singularities. Given a surface cyclic quotient singularity $Q\in Y$, it is an open problem to determine all smoothings of $Y$ that admit an anticanonical model and to compute it. In \cite{HTU}, Hacking, Tevelev and Urz\'ua studied certain irreducible components of the versal deformation space of $Y$, and within these components, they found one parameter smoothings $\Y \to \A^1$ that admit an anticanonical model and proved that they have canonical singularities. Moreover, they compute explicitly the anticanonical models that have terminal singularities using Mori's division algorithm \cite{M02}. We study …

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, 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, 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, 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, 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, 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, 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, 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, Thomas E. Cecil

Mathematics Department Faculty Scholarship

A hypersurface M in Rⁿ or Sⁿ 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 Rⁿ or Sⁿ 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, 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 …