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 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,
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 y2 = x3+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.
Infinite-Order Differential Operators Acting On Entire Hyperholomorphic Functions,
2021
Chapman University
Infinite-Order Differential Operators Acting On Entire Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Stefano Pinton, Irene Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
Infinite-order differential operators appear in different fields of mathematics and physics and in the past decade they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrödinger equation. Inspired by the operators arising in quantum mechanics, in this paper, we investigate the continuity of a class of infinite-order differential operators acting on spaces of entire hyperholomorphic functions. We will consider two classes of hyperholomorphic functions, both being natural extensions of holomorphic functions of one complex variable. We show that, even though these two notions of hyperholomorphic functions are quite different …
On Leibniz Cohomology,
2021
University of South Alabama
On Leibniz Cohomology, Jorg Feldvoss, Friedrich Wagemann
University Faculty and Staff Publications
In this paper we prove the Leibniz analogue of Whitehead's vanishing theorem for the Chevalley-Eilenberg cohomology of Lie algebras. As a consequence, we obtain the second Whitehead lemma for Leibniz algebras. Moreover, we compute the cohomology of several Leibniz algebras with ad joint or irreducible coefficients. Our main tool is a Leibniz analogue of the Hochschild-Serre spectral sequence, which is an extension of the dual of a spectral sequence of Pirashvili for Leibniz homology from symmetric bimodules to arbitrary bimodules.
The Adams Spectral Sequence For Topological Modular Forms,
2021
Wayne State University
The Adams Spectral Sequence For Topological Modular Forms, Robert Bruner, John Rognes
Mathematics Faculty Research Publications
The connective topological modular forms spectrum, 𝑡𝑚𝑓, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of 𝑡𝑚𝑓 and several 𝑡𝑚𝑓-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account …
On The Tropicalization Of Lines Onto Tropical Quadrics,
2021
Harvey Mudd College
On The Tropicalization Of Lines Onto Tropical Quadrics, Natasha Crepeau
HMC Senior Theses
Tropical geometry uses the minimum and addition operations to consider tropical versions of the curves, surfaces, and more generally the zero set of polynomials, called varieties, that are the objects of study in classical algebraic geometry. One known result in classical geometry is that smooth quadric surfaces in three-dimensional projective space, $\mathbb{P}^3$, are doubly ruled, and those rulings form a disjoint union of conics in $\mathbb{P}^5$. We wish to see if the same result holds for smooth tropical quadrics. We use the Fundamental Theorem of Tropical Algebraic Geometry to outline an approach to studying how lines lift onto a tropical …
Towards Tropical Psi Classes,
2021
Claremont Colleges
Towards Tropical Psi Classes, Jawahar Madan
HMC Senior Theses
To help the interested reader get their initial bearings, I present a survey of prerequisite topics for understanding the budding field of tropical Gromov-Witten theory. These include the language and methods of enumerative geometry, an introduction to tropical geometry and its relation to classical geometry, an exposition of toric varieties and their correspondence to polyhedral fans, an intuitive picture of bundles and Euler classes, and finally an introduction to the moduli spaces of n-pointed stable rational curves and their tropical counterparts.
Analysis, Constructions And Diagrams In Classical Geometry,
2021
Chapman University
Analysis, Constructions And Diagrams In Classical Geometry, Marco Panza
MPP Published Research
Greek ancient and early modern geometry necessarily uses diagrams. Among other things, these enter geometrical analysis. The paper distinguishes two sorts of geometrical analysis and shows that in one of them, dubbed “intra-confgurational” analysis, some diagrams necessarily enter as outcomes of a purely material gesture, namely not as result of a codifed constructive procedure, but as result of a free-hand drawing.
Diagrams In Intra-Configurational Analysis,
2021
Chapman University
Diagrams In Intra-Configurational Analysis, Marco Panza, Gianluca Longa
MPP Published Research
In this paper we would like to attempt to shed some light on the way in which diagrams enter into the practice of ancient Greek geometrical analysis. To this end, we will first distinguish two main forms of this practice, i.e., trans-configurational and intra-configurational. We will then argue that, while in the former diagrams enter in the proof essentially in the same way (mutatis mutandis) they enter in canonical synthetic demonstrations, in the latter, they take part in the analytic argument in a specific way, which has no correlation in other aspects of classical geometry. In intra-configurational analysis, diagrams represent …
