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Archive of GSS Abstracts

Spring 2017

An Interesting Fluid Dynamics Problem: Solving an Inverse Domain

Andrew Thomack

Abstract: We will look at an interesting problem in fluid dynamics which models a 2 dimensional pool of oil surrounded by water and how the boundary changes as oil is added or removed at different points. We will look at what theorems can be used to set up this problem in an accessible way, and look at a case where the problem can be explicitly solved. If time permits, we can look at pretty pictures of the validated numerical case that I am currently working on.

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d-MUL: A New Multidimensional Scalar Multiplication Algorithm

Aaron Hutchinson

Abstract: We propose new algorithms for constructing multidimensional differential addition chains and for performing multidimensional scalar point multiplication based on these chains. Our algorithms work in any dimension and offer some key efficiency and security features. In particular, our scalar point multiplication algorithm is uniform, it has high potential for constant time implementation, and it can be parallelized. We present the algorithm and some of its key features.

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Constructive Mathematics: the Real Numbers

Jean Joseph

Abstract: The real numbers can be thought of as a set with a binary relation which contains the rational numbers. We present axioms for such a structure. Any two structures satisfying those axioms are isomorphic. We define an addition and a multiplication on any such structure which turn it into a field.

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Expected Numbers of Zeros of Weyl Polynomials

Zachariah Tyree

Abstract: We'll look at the Expected Numbers of Zeros of a particular class of Harmonic Polynomials known as Weyl Polynomials. We'll develop the asymptotics and discuss the distribution of zeros in the case that the conjugation and non-conjugated contributions are of the same degree. If times permits we'll discuss the cases where the contributions have different degrees.

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Introduction to One-Time Signature Schemes

Shaun Miller

Abstract: Authenticity, integrity, and non-repudiation of data is imperative in today's society and in the post quantum era. Many signature schemes, such as RSA, DSA, and ECDSA; have security requirements that depend on the difficulty of factoring large primes or the discrete logarithm problem. Each of these schemes becomes obsolete in a post quantum era in which all of these problems may be solved in polynomial time. Hash-Based signatures (HBS) are a popular post-quantum candidate. An HBS typically relies on a one-time signature scheme which is resistant to quantum attacks. This presentation will discuss the basics of designing such a signature scheme.

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Average Arc Length and Connected Components of Level Curve of Gaussian Rational Analytic Functions

Yuxun Sun

Abstract: We will look at the the modulus of a Gaussian rational analytic function and its level curves within a disk of fixed radius. We will compute the expected arc length using Kac Rice Integral and polar coordinates. Topological aspects will also be investigated. In particular, we show the limit of the expected components over area of a disk exists using ergodicity.

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Introduction to parametrization methods with Chebyshev polynomials

Maxime Murray

Abstract: In this talk I will introduce Chebyshev polynomials and how one can use result about Fourier expansions to justify their use. Then I will introduce a parametrization method to compute (un)stable manifold for periodic orbits. The method will be applied to the Lorenz system and the circular restricted three body problem to illustrate results.

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Continuous Dependence and Differentiating Solutions of a Second Order Boundary Value Problem with Average Value Condition

Samantha Major

Abstract: Using a few conditions, continuous dependence, and a result regarding smoothness of initial conditions, we show that derivatives, with respect to each of the boundary data, of solutions to a second order boundary value problem with an average value integral condition solve the associated variational equation with interesting boundary conditions.

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Proof of Picard's Existence & Uniqueness Theorem

Olukayode Dare

Abstract: Under what conditions do ODE's have unique solutions. In our discussion we will use machinery such as Metric spaces, Lipschitz conditions, and contractions to build up to the proof of Picard's Existence & Uniqueness Theorem.

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A Tale of Two Integrals

Andrew Thomack

Abstract: We look at a problem from the 1995 Miklós Schweitzer Competition. The problem is simple to state and easy to understand, but when we try the usual measure theory techniques we find it hard to proceed. We will instead take the not so long way around through a theorem often used in topology, differential equations, and complex analysis, the Winding Number Theorem. This talk should be very accessible by the current first year graduate students and may be of particular interest to those graduate students who are currently in topology.

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Fall 2016

An Overview of Edmond's Algorithm

Ashley Valentijn

Abstract: We aim to give a brief introduction to graph theory followed by an overview of Edmond's Matching Algorithm. This algorithm uses the results from Berge's Theorem to find a maximum matching of a graph. This is done by "expanding" and "shrinking" odd circuits called blossoms.

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How to use Banach spaces to approximate solution of analytic ODE's

Maxime Murray

Abstract: In this seminar I will briefly introduce the background needed to be able to approximate a solution of a given analytic ODE. I will explain how one can rewrite the ODE into a fixed point problem for an operator defined over some Banach space, which can then be solved using the Banach fixed point theorem.

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Polynomial Lemniscate Configurations and Lemniscate Trees

Michael Epstein

Abstract: We consider the combinatorial class of labeled, nonplanar, rooted trees in which vertices have outdegree at most two, such that the labels increase along any directed path in the tree. We refer to these as lemniscate trees since this class arises as the solution to an enumeration problem involving topological equivalence classes of generic polynomial lemniscate congurations. With the goal of establishing a baseline for studies of nonlocal statistics in random polynomials, we investigate the typical shape of a lemniscate tree sampled uniformly at random from the combinatorial class. We apply complex analytic methods to a bivariate generating function in order to determine (asymptotically in the size of the tree) the mean and variance of the number of vertices of outdegree two.

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Injectivity and Some of Its Generalizations

Akeel Omairi

Abstract: An R-module M is called injective if for all R-module homomorphisms ?: E ? N and ?: E ? M where ? is injective, there exists an R-module homomorphism ?: N ? M such that ? ? ? = ?. Various generalizations of injective modules are extensively studied since several years. For instance, Quasi-injective, Finitely generated quasi- injective (FQ-injective), and Principally quasi-injective( PQ-injective). Injective ? Quasi-injective ? FQ-injective ? PQ-injective. In 2005, Z. Zhu, J. Chen, and X Zhang introduced the concept of (m, n)-quasi-injetive modules which is a generalization of injective modules and combines concepts FQ-injective and PQ-injective.

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Transport Layer Security

Hai Pham

Abstract: Transport Layer Security (TLS) provides secure communications such as email, e-faxing, data transfer, web browsing on the internet. TLS connections predominantly use one of the two families of cipher: RC4 based and AES-CBC based. However, both have encountered major problems (include some references?). ChaCha20 is a high speed stream cipher, developed by D.J. Bernstein with the hope to improve the performance of secure communication and provide a better experience. The core of ChaCha20 revolved around the add-rotate-XOR operations. Google has selected ChaCha20 along with its partner Poly1305 as a replacement for RC4. In addition, CloudFlare supported it since 2005. In this talk, we will take a look into the construction of ChaCha20 and perform an implementation example of it.

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Graph Reconstruction

Brandon Langenberg

Abstract: The Reconstruction Conjecture (1941) asserts that every finite simple undirected graph on three or more vertices is determined, up to isomorphism, by its collection of vertex-deleted subgraphs. The Reconstruction Conjecture is generally regarded as one of the foremost unsolved problems in graph theory. Further, he will be presenting some of the ideas from the paper by Bondy and Hemminger "Graph Reconstruction - A Survey."

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Constructive Mathematics: The Intermediate Value Theorem

Jean Joseph

Abstract: We will give a brief overview of constructive mathematics and explain why the intermediate value theorem cannot be proved, constructively.

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Spring 2016

Title: A Gentle Introduction to Dynamical Systems

Presenter: Shane Kepley

Abstract: We give an overview of the field of Dynamical Systems. Specifically, we will explore examples of discrete and continuous dynamical systems from biology and continuum mechanics. This talk should be accessible to anyone with a basic calculus background.

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Title: Groups, Designs, and Kramer-Mesner Matrices

Presenter: Michael Epstein

Abstract: The aim of this talk is to give a brief introduction to block designs and show how designs with prescribed automorphism groups can be constructed by solving certain associated matrix equations.

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Title: Uniform Contraction Principle and Application to Dynamical Systems

Presenter: Maxime Murray

Abstract: The presentation will introduce how we can use the uniform contraction principle in order to show the existence of particular objects in dynamical systems. We will show how to expand equilibrias, (un)stable manifolds and connecting orbits in the right choice of series whose coefficients relay in a Banach space. Some results for the Lorenz equation will also be presented as an example of application.

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Title: Abstract Nonsense: An Introduction to Category Theory

Presenter: Aaron Hutchinson

Abstract: We aim to give a brief introduction to the basic principals of category theory. Topics covered will include categories and functors, and we will look at many examples of where these ideas appear in areas of mathematics you have likely already studied.

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Title: Solutions to Dynamic Equations on Different Time Scales

Presenter: Sher Chetri

Abstract: Time-scale calculus is first introduced by Stefan Hilger in his PhD thesis in 1988. It has been created in order to unify the study of differential and difference equations, offering formal study of hybrid discrete–continuous dynamical system. Using the properties of delta derivative, delta anti-derivative and the concept of Hilger's Complex Plane, we discuss the analytical and graphical behavior of particularly chosen first and second order dynamic equations on different time scales.

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