Talks

Student Research Presentations

Grace Burton, Applications of double counting

Abstract: Do you ever just look at a graph and wonder in how many different ways you could count its edges and vertices? If so, this is the talk for you! Following ideas presented in Proofs from THE BOOK, we will apply double counting to obtain a couple of interesting relations between the number of edges and vertices in a graph.

Erick Gonzalez, Topological Symmetry Group of C_13

Abstract: Spatial graph theory is the study of graphs embedded in $\mathbb{R}^3$. The $K_7$ family is the set of all graphs obtained by a finite sequence of $\Delta-Y$ moves performed on $K_7$. This family is of special interest to knot theorists since its elements are the only intrinsically knotted graphs with $21$ edges. One of the graphs in this family is $C_{13}$, which is obtained by performing a $Y-\Delta$ move on the Heawood graph. We study the embeddings of $C_{13}$ and determine which automorphism groups of $C_{13}$ can be induced by groups of homeomorphism of $\mathbb{R}^3$

Amanda Liles, Braids and the Temperley-Lieb Algebra

Abstract: Braid theory is a subarea of topology that was first introduced by scientists studying the structure of DNA strands. Although the theory did not turn out to apply as expected by biologists, it still has lots of applications outside of mathematics. Mathematicians became very interested in mathematical braids due to their natural beauty and strong mathematical principles. In this presentation, we will introduce the braid groups and an invariant for braids. Along the way, we will also look at another algebraic structure, called the Temperley - Lieb Algebra (named after its founders). We will show how to construct a homomorphism that takes a braid and gives out an element in this algebra. Then we compose this homomorphism with another function such that the composition of the two functions yields a polynomial associated to a given braid.

Maria Diaz, The characteristic polynomial of a spread in PG(3,q)

Abstract: A spread of Fq^4 is a collection of 2-dimensional subspaces of Fq^4 that partition all the non-zero vectors of Fq^4. Spreads are important since they can be considered as ‘geometric bases’ for affine planes of a certain kind (translation planes). In this talk, we will introduce the concept of the characteristic polynomial of a spread in Fq^4 and then see what the degree of this polynomial is for a variety of spreads of Fq^4 for small values of q. We will focus specially on the so-called regular spread and spreads obtained by doing certain line-replacements in a regular spread.

Olivia Soghomonian, Fractals and the Golden Ratio

Abstract: This presentation will explore the fractals, specifically, the Koch Snowflake, and its properties, along with some of the mathematics behind its construction. A connection between the golden ratio and its appearance in fractals will be addressed and explored in an example of fractal trees.

Jagdeep Basi, Quandles!

Abstract: Mathematical knots are typically studied using projections into 2-space, that we refer to as knot diagrams. A quandle is a set used to label the strands of a knot diagram, along with an operation defined by axioms analogous to the Reidemeister moves; moves that determine when two knot diagrams represent the same knot. We will introduce the concept of quandles with visual examples and focus on a familiar algebraic structure. We will also discuss the quandle coloring invariant and some more recent invariants found in the quandle coloring quiver.

Sam Cleofas, Hyperplane Arrangements over Finite Fields

Abstract: Hyperplane arrangements are commonly studied in vector spaces over R and C. As expected, a generalization of hyperplane arrangements to vector spaces over finite fields will have a very different flavor, as many concepts, such as the number of bounded and unbounded regions created by the hyperplane arrangement, cannot be generalized to spaces of the finite type. However, studying hyperplane arrangements in a finite world opens the doors to combinatorial methods usually used in the study of finite projective spaces. This presentation will be on work done about blocking sets of the complement of a hyperplane arrangement in a vector space over a finite field Fq that generalize the work of Settepanella (2013).

Emily Varney, Be There or Be (Winged) Square

Abstract: Which numbers can be written as the sum of two squares? In this presentation we will go over a simple, yet unexpected geometric interpretation that shows how certain primes can be written as the sum of two squares. We utilize winged squares and winged shapes to accomplish this. The proof is written in its full glory in Proofs from THE BOOK by Aigner and Ziegler. You won’t want to miss it!

Marissa Hernandez, SVD and Cosine Similarity Approaches for Detecting Epistasis

Abstract: There are a variety of methods used to understand and interpret one’s phenotype, the physical expression of one or more genes; one method, in particular, would be epistasis. Epistasis is used to assess gene variation in an attempt to find a combination of Single Nucleotide Polymorphisms (SNPs) that contribute to a certain phenotype. The difficulty with implementing epistasis is that it could be challenging to find a correlation between SNPs because rarely does one SNP affect a phenotype. So, in an effort to ease the complexity of this issue, we used the technique of Singular Value Decomposition (SVD) to compare individuals’ SNPs. When using this method on simulated data, we were able to find the top N cosine similar individuals and analyze which SNPs were most prominent between them. We aim to continue this success with the use of real data gathered on rats to test what SNPs contribute to their weight and lean mass.

Kenneth Ferrell II, Darboux-Stieltjes Integral

Abstract: Did you know that the Riemann Integral is not the only one? In this talk, we will show the constructions of the Darboux Integral and the Darboux-Stieltjes Integral. We will then give examples and discuss the properties of them.

Katelyn Clark and Vathana To, Exploring Time Series Analysis and Volatility Models on Financial Exchange Rates

Abstract: Time series analysis is central in understanding how variables of interest change over time. Fields such as economics, mathematical finance, and astronomy rely on analyzing sequences of values recorded over time. In this presentation, we introduce an exposition on time series analysis by providing examples on the ARIMA model and by connecting time series analysis to three volatility models (GARCH, EGARCH, and GJR-GARCH). We fit and investigate the models’ performances on various financial data sets.

Summer Al-Hamdani, On Binomial Combinations of Chebyshev Polynomials

Abstract: The sequence of Chebyshev polynomials of the second kind {Um(z)}∞ m=0 is a well-known sequence of orthogonal polynomials whose zeros lie on the interval (−1, 1) and are dense there as m → ∞. For 0 < α < 1, we discuss the zeros of the sequence of polynomials {Pm(z)}∞ m=0 generated by the reciprocal of (1 − t) α(1 − 2zt + t2) expanded as a power series in t. This sequence can equivalently be obtained from a linear combination of Chebyshev polynomials whose coefficients have a binomial form. We prove that the number of zeros of Pm(z) outside the interval (−1, 1) is bounded by a constant independent of m.

Trevor Ferguson, Counting sums of squares

Abstract: In the Fundamenta nova theoriae functonum ellipticarum, Carl Gustav Jacobi proved expressions for counting the number of ways a natural number can be expressed as a sum of 2,4,6, and 8 squared integers. Srinivasa Ramanujan's work in q-series provide arguably easier proofs for Jacobi's expressions. In this talk, I will introduce I-series and Jacobi's Triple Product Identity, a fundamental theorem in q-theory. I will then provide an outline of how to prove Jacobi's expression for the number of ways a natural number can be expressed as the sum of 2 squared integers using q-series.

Vathana To, ZimulatE: An R package for simulating and modeling zero-inflated data

Abstract: In analyzing count data with an excess occurrence of zero values, one may refer to using the standard Poisson and negative binomial models. However, much of recent literature points to the models’ inability to accommodate the number of zeros. In this presentation, we provide an introduction to the development of zero-inflated and hurdle models and discuss their use in analyzing count data with excess zeros. We demonstrate our R package, ZimulatE, to simulate zero-inflated data and use counts of disease spread to illustrate model performance.

Jack Luong, Zeros of Polynomials Generated by a Bivariate Contiguous Relation

Abstract: Bivariate recurrence relations are of interest due to their application in various combinatorial problems. For example, they can be used to count the number of rook paths from one corner of an infinite chessboard to another corner. In this talk, we study a table of polynomials obtained from contiguous bivariate recurrence relations. We determine how the initial conditions and coefficients of the recurrence relation affect the reality of zeros of the polynomials.

Audrey Baumheckel, Very Triangular Numbers

Abstract: We investigate the binary representations of triangular numbers, and look for a characterization of those that have a triangular number of 1s in their binary representation (these are called very triangular numbers). It is known that there are infinitely many such numbers, but there is no known formula for the $nth$ very triangular number. We present some preliminary results regarding very triangular numbers, and discuss some ideas on how to understand them and their distribution among all positive integers.

Monico Maniquiz, Gamma Function and Asymptotics

Abstract: Special functions are particular mathematical functions that have established names and notations due to their importance in various fields of science and mathematics. This talk will cover some of these functions, especially the Gamma function, and how we can approximate them.