Summer 2020 Student Talks
Speaker: Vanessa Vega
Title: Excluded minors for nearly-paving matroids
Date: Thursday, July 30, 2020
Time: 10:00 - 11:00 am
Abstract: Matroids capture an abstract notion of independence that generalizes linear independence in linear algebra, edge independence in graph theory, as well as algebraic independence. Given a particular property of matroids, all the matroids possessing that property form a matroid class. A common research theme in matroid theory is to characterize matroid classes so that, given a matroid M, it is possible to determine whether or not M belongs to a given class. An excluded minor of a minor-closed class is a matroid N that is, in a sense, minimal with respect to not being in the minor-closed class. An attractive way to characterize a minor-closed class of matroids is to determine the complete list of excluded minors for the minor-closed class. For example, the class of planar graphs is characterized (Kuratowski’s Theorem) by graphs that do not have any minor that is isomorphic to K_5 or K_3,3. In this presentation, we introduce several fundamental minor-closed classes of matroids; namely, uniform matroids and paving matroids. We define closely related minor-closed classes, which we call nearly-uniform and nearly-paving matroids. Finally, we provide an excluded minor characterization for nearly-uniform and nearly-paving matroids.
Spring 2020 Student Talks
We are excited to announce many student talks during the spring 2020 quarter. They are all on Zoom. The schedule is here and the abstracts are below. Some talks have been recorded the videos are linked below.
|12-1pm||Stacey Cox and Rama Alshreteh||Uniform Convergence of Polynomials|
|June 3||2:45-3:45pm||Tyler Ray||Studying Cubic Equations Through Paper Folding|
|June 4||12-1pm||Danielle Gaynair||
Elliptic Curves and Cryptography
|June 4||12-1pm||Elijah Howells||A Discussion of Short Term Actuarial Modeling|
|June 5||12-1pm||Madison Manjarrez and Shantanu Tavkar||Applications of the Discrete Log Problem in Concepts of Cryptography|
|June 5||2-3pm||Sergio Lopez Palomares||RSA and Lattice-Based Cryptosystems|
|June 9||1-2pm||Vanessa Castro, Stacey Cox & Matthew Nuyten||Knots Invariants and the Jones Polynomial|
|June 10||2:40-3:40pm||Crystal Diaz||Permutation and Monomial Progenitors|
To access the Zoom links ( sign in with your CoyoteID).
Speakers: Stacey Cox and Rama Alshreteh
Title: Uniform Convergence of Polynomials
Date: June 3, 2020
Time: 12 - 1 pm PDT
Uniform convergence is known to be a stronger and more useful form of convergence when compared to that of point-wise convergence. Join us as we compare the two forms and use that comparison to show that any continuous function is the uniform limit of polynomials on a compact interval. This is, in fact, the focus of Stone-Weierstrass’ Theorem in which Bernstein polynomials are used to prove this result. We will also demonstrate the generalization of this theorem through providing our own example of continuous functions that do not resemble the other polynomials provided.
Speaker: Tyler Ray
Title: Studying Cubic Equations Through Paper Folding
Date: June 3, 2020
Time: 2:45-3:45 pm PDT
In this presentation, we will explore cubic equations through paper folding. In particular, we will discuss methods for folding paper to construct cubic curves and to find solutions to general cubic equations. We will also analyze the mathematics behind the construction techniques and connect the algebra to the geometry.
Speaker: Danielle Gaynair
Date: June 4 2020
Time: 12 - 1 pm PDT
Title: Elliptic Curves and Cryptography
For this presentation I will be talking about specific ideas in Cryptography. Cryptography is the art of concealing the content of messages. I will specifically be going into detail on how elliptic curves are used to secure online communication, in particular, how to create shared secret keys and authenticating them. I will explain the Algebra and Geometry of Elliptic Curves over finite fields. From there I will show in detail how they are used to secure widely used algorithms, such as, Elliptic Curve Discrete Log Problem and Elliptic Curve Digital Signature Algorithm. Throughout this quarter I have coded these algorithms up from scratch using Python and SageMath. I will show clear examples on how these concepts of math and concepts of encryption work together.
Speaker: Elijah Howells
Title: A Discussion of Short Term Actuarial Modeling
Date: June 4, 2020
Time: 12 - 1 pm PDT
This project covers mathematical methods used to conduct actuarial analysis in the short term, such as policy deductible analysis, maximum covered loss analysis, and mixtures of distributions. Assessment of a loss variable's distribution under the effect of a policy deductible, as well as one with an implemented maximum covered loss, and under both a policy deductible and maximum covered loss will also be covered. The derivation, meaning, and use of cost per loss and cost per payment will be discussed, as will those of an aggregate sum distribution, stop loss policy, and maximum likelihood estimation. For each topic, special cases based on distribution will be described and discussed. These methods and subjects are used to assess and manage risk, typically for insurance providers, but can also be adapted to a number of other fields.
Speakers: Madison Manjarrez and Shantanu Tavkar
Title: Applications of the Discrete Log Problem in Concepts of Cryptography
Date: June 5, 2020
Time: 12 - 1 pm PDT
During this talk we will be focusing on the Discrete Log Problem (DLP) and its application to key and information exchange as well as signature authentication. We will discuss the makings of the DLP and how it can be difficult to solve. This difficulty is used to the advantage of security during information exchanges. We will be presenting our efficient algorithms that streamline the math processes involved in key creation, message encryption and decryption, and signature verification. Specifically, we will be sharing our key and message exchange algorithm, El Gamal, which is built in Sage. We will highlight some of the important mathematical functions such as the Extended Euclidean Algorithm, on which El Gamal depends for efficiency. Building off this algorithm, we will demonstrate the added security through our Digital Signature Algorithm (DSA) to combat third-party tampering. Combining these two algorithms, we will present an efficient message and authentication signature exchange that relies on the difficulty of the DLP.
Speaker: Sergio Lopez Palomares
Title: RSA and Lattice-Based Cryptosystems
Date: June 5, 2020
Time: 2 - 3 pm PDT
RSA is an asymmetric cryptography algorithm used by computers to encrypt and decrypt messages which provides a peace of mind that data retains its integrity. Despite RSA’s current protection, it is not a question of “if” but “when” technological advances enable potentially malicious third parties to efficiently break its security. Shor’s algorithm needs only the invention of a quantum computer to break RSA. How do we ensure that we safeguard data against technology that has yet to be invented? Understanding this dilemma makes it ever so pressing to look at innovative and quantum resistant cryptosystems, such as lattice-based encryption to be prepared when that day arrives.
In this talk, I will begin by telling you about RSA. I will present how RSA relies on the difficulty of factoring large numbers. I will show how it can be used to both send encrypted messages and sign messages. I will show code that I wrote from scratch in Sage which implements RSA. This code uses such important mathematical algorithms as the extended Euclidean algorithm and the fast powering algorithm. Next, I will tell you about lattice-based cryptosystems. I will explain how lattice-based encryption relies on the difficulty of the closest vector problem (CVP). I will outline GGH cryptosystem and provide examples using both Sage and Geogebra.
Speakers: Vanessa Castro, Stacey Cox & Matthew Nuyten
Title: Knots Invariants and the Jones Polynomial
Date: June 9, 2020
Time: 1-2 pm PDT
In this talk we explore different mathematical ways to tell knots apart. Once we choose a projection on the plane for a 3D knot, we study its similarities with other projections and discuss different ways to conclude all such projections are projections of the same knot. One way of proving this is to transform one projection into another using Reidemeister’s moves. Reidemeister proved in 1927 that if two projections belong to the same knot, one can be transformed into the second by a series of applications of three moves. Finally, using Reidemeister’s moves to prove that the Jones polynomial is an invariant of knots, we construct this polynomial for knots and prove they belong to different 3D objects.
Speaker: Crystal Diaz
Title: Permutation and Monomial Progenitors
Date: June 10, 2020
Time: 2:40-3:40 pm PDT
We searched several monomial and permutation progenitors for symmetric presentations of important images, nonabelian simple groups, their automorphism groups, or groups that have these as their factor groups. Our target non-abelian simple groups included sporadic groups, linear groups, and alternating groups.
In this presentation, We will describe our search for the homomorphic images through the permutation progenitor 2^(*15):(D_5×3) and construction of a monomial representation through the group 2^3:3.
We will construct PGL(2,7) over 2^3:3 on 6 letters and L_2(11) over 2^2:3 on 8 letters.
We will also give our construction of S_5×2 and L_2(25) as homomorphic images of the monomial progenitors 3^(*3) :_m D4 and S^(*6):S_5.
In addition, we will describe as to how to solve the extension problem for finite groups through the example of the group (4×2^2):A_4.
We note that the symmetric presentations and constructions given in this presentation are original, to the best of our knowledge.