Schedule of Short Talks

Short Talk Abstracts

Name: Kyle Yates

School: Clemson University 

Title: Homomorphic Encryption and Secure Cloud Computing

Abstract: Cryptography is an area within mathematics that studies secure communication. The idea is that in order to keep information secure in transit, we first disguise (or encrypt) our data in some fashion before sending it. That way, an unwanted third party can not steal our information. Homomorphic encryption specifically studies systems where we can perform mathematical operations on encrypted data, while at the same time not learning any information about the data! In this talk, we’ll give a brief introduction to the ideas and applications of homomorphic encryption. 

Name: Matthew Booth 

School: University of South Carolina

Title: WLP in Low Degrees for Complete Intersections Generated by Quadratics

Abstract: Let $P=k[x_1,\dots,x_n]$ be a polynomial ring over a field $k$ of characteristic $0$, and let $I$ be a homogeneous ideal in $P$. In 1980, R. Stanley showed that if $I$ is an Artininan monomial complete intersection, then the standard graded algebra $P/I$ enjoys the Strong Lefchetz Property (SLP). Loosely speaking, this says for a generic linear form $\ell$ that the map $\times\ell^p:(P/I)_j\rightarrow (P/I)_{j+p}$ has maximal rank for all $j,p\in\mathbb{Z}^+$. If we relax this condition and only ask for maximal rank when $p=1$ (still for all $j\in\mathbb{Z}^+$), we obtain the Weak Lefschetz Property (WLP). Stanley's result initiated various threads of investigation regarding WLP, and numerous advances have followed in the intervening years. However, many of these advances assume that $I$ is a monomial ideal. One such example is in a recent paper (2022) of H. Dao \& R. Nair, which completely characterizes the failure of WLP from degree 1 to degree 2 for Artinian monomial ideals containing the squares of all the variables. They accomplished this by treating the generating ideal as the defining ideal of a Stanley-Reisner ring and studying the 1-skeleton of the associated simplicial complex. In this talk, we will use their work to prove that WLP holds from degree 1 to degree 2 for any complete intersection generated by $n$ quadratic forms and mention ongoing efforts to extend this result to almost complete interesctions, as well as the investigation of WLP from degree 2 to degree 3. This is joint work with Dr. Adela Vraciu. 

Name: Soumyadip Acharyya

School: University of South Carolina 

Title: A Generalization of U-topology on C(X)

Abstract: For a Topological Space X, let C(X) denote the Ring of all real-valued continuous functions on X. This presentation will focus on the so - called U-topology on C(X) and its generalized version obtained through a z-ideal in C(X). A chain of necessary and sufficient conditions as to when the later topological space is a topological ring will be discussed. Additionally, we will explore the connected ideals of this space.  

Name: Julia VanLandingham

School: Clemson University 

Title: Simultaneous approximation for lattice-based cryptography

Abstract: We study the worst- and average-case complexity of a reduction from the approximate shortest vector problem to the simultaneous approximation problem in order to verify the security of simultaneous approximation lattices for use in cryptography. 

Name: Madison Mabe

School: Clemson University 

Title:  A Mercurial Signature Scheme

Abstract: A cryptographic signature scheme is a set of algorithms which enables the creation and verification of digital signatures. In 2018, Elizabeth Crites and Anna Lysyanskaya introduced the idea of a “Mercurial Signature Scheme,” which allows authorized users to modify message/signature pairs, and public key/secret key pairs after their creation. In this presentation I will be discussing the basics of a signature scheme and the ideas mentioned in the above paper for the definition of a Mercurial Signature Scheme. 

Name: Kisun Lee

School: Clemson University 

Title: Introduction to numerical nonlinear algebra

Abstract: In this talk, we introduce numerical nonlinear algebra (also known as numerical algebraic geometry). The Gauss-Jordan elimination from linear algebra is used for solving a linear system. We discuss how this can be extended for solving a nonlinear system, and point out that it can be computationally expensive. As a remedy, we introduce numerical nonlinear algebra based on the homotopy continuation method. 

Name: Nisha

School: Clemson University

Title: From formulas to futures, mathematical insight into endosomal escape

Abstract: The research project focuses on developing a new method to measure the endosomal escape following the successful delivery of siRNAs (small interfering RNAs) into ovarian cancer cells using fusogenic peptides. The objective is to counteract the rapid degradation of siRNAs in the endosome, a problem that the specially designed fusogenic peptides address. These peptides, created by Dr. Alexander-Bryant, attach to siRNAs and facilitate their movement from the endosome into the cytosol. Our goal here is to develop, optimize statistical and AI models to quantify the endosomal escape of siRNAs in cancer cells. Also predict at what time and how much siRNA has been released inside cancer cells. 

Name: Jakini Kauba

School: Clemson University 

Title: Quantifying Hierarchy and Documenting Inequity in PhD-granting Mathematical Sciences Departments in the United States

Abstract: In this paper we provide an example of the application of quantitative techniques, tools, and topics from mathematics and data science to analyze the mathematics community itself in order to quantify inequity and document elitism. 

This work is a contribution to the new and growing field recently termed "mathematics of Mathematics,'' or "MetaMath.'' Our goal is to rebut, rebuke, and refute the idea that the mathematical sciences in the United States is a meritocracy by using data science and quantitative analysis. Using research and data about PhD-granting institutions in the United States, we quantify, document, and highlight inequities in departments at U.S. institutions of higher education that produce PhDs in the mathematical sciences. Specifically, we determine that a small fraction of mathematical sciences departments receive a large majority of federal funding awarded to support mathematics in the United States and that women are dramatically underrepresented in these departments. Additionally, we quantify the extent to which women are underrepresented in almost all mathematical sciences PhD-granting institutions in the United States.

Name: Victoria Chebotaeva

School: University of South Carolina 

Title: SEIR epidemic models with cross-diffusion coefficients

Abstract: We examine the effects of cross-diffusion dynamics in epidemiological models. Using reaction-diffusion models to model the spread of infectious diseases, we focus on situations in which the movement of individuals is affected by the concentration of individuals of other categories. In particular, we present a model where susceptible individuals move away from infected and infectious individuals.

Our results show that accounting for this cross-diffusion dynamics leads to a noticeable effect on epidemic dynamics. It is noteworthy that this leads to a delay in the onset of epidemics and an increase in the total number of people infected. This new representation improves the spatiotemporal accuracy of the SEIR Erlang model, allowing us to explore how spatial mobility driven by social behavior influences the disease trajectory.

One of the key findings of our study is the effectiveness of adapted control measures. By implementing strategies such as targeted testing, contact tracing, and isolation of infected people, we demonstrate that we can effectively contain the spread of infectious diseases. Moreover, these measures allow achieving such a result, while minimizing the negative impact on society and the economy.

Name: Megan Powell 

School: University Of North Carolina At Asheville

Title: Saving the Devil is in the Details

Abstract: Tasmanian Devil populations have been decimated by one of few known transmissible cancers called facial tumor disease (DFTD).  This talk will show the outcome of a computational model of the potential recovery of devil populations with the intervention strategies including a vaccination program that is currently in development.  We find the surprising result that getting the vaccine to the largest number of devils is more impactful that the efficacy of the vaccine itself. 

Name: Carter Hinsley 

School: Georgia State University 

Title: Heteroclinic $n$-cycle-opedia: Extending the Sharkovsky stratification

Abstract: Oleksandr Sharkovsky showed that cycles in a continuous iterated map obey a universal ordering. Later, he extended this ordering to a stratification by interleaved cycles and homoclinic trajectories, which are temporary excursions from cycles. We discuss how this stratification extends to the more complicated case of heteroclinic connections, where a trajectory departs from nearby one cycle to travel to another. This extension permits new methods for the detection of chaos in dynamical systems. 

Name: Yunan Wang 

School: Clemson University 

Title: Generalized Correction Factors in Hardy-Littlewood Conjecture

Abstract: The Goldbach Conjecture posits that every even natural number greater than 2 can be written as the sum of two primes. There are various conjectures regarding \(r(n)\), the number of ways to represent \(n\) as a sum of two primes. In 1923, Hardy and Littlewood conjectured that \[r(n) \sim 2 C_2\left(\prod_{p \mid n; p \geq 3} \frac{p-1}{p-2}\right) \frac{n}{(\log (n))^2},\] where \(C_2\) is the Hardy-Littlewood constant. This conjecture includes a "correction factor," stemming from Sylvester's 1871 conjecture, which was later proved incorrect by Hardy and Littlewood. This talk will focus on the rationale behind the correction factor in the conjecture and will extend the discussion to generalize correction factors for \(r_2(n)\) related to twin primes and \(r_3(n)\) for cousin primes. 

Name: Sarah Otterbeck

School: Clemson University

Title:  The Inclusive Equation: Ways to Add Engagement to Your Class for All Students

Abstract: The traditional math classroom, with its rows of desks and emphasis on paper-and-pencil work, may often leave students disengaged and feeling like math “isn’t for them”. But what if we could rewrite the equation? This talk explores practical strategies to create an inclusive math classroom where every student feels engaged, empowered, and capable of succeeding. Forget the one-size fits all instruction! We will explore a toolkit of multi-modal engagement strategies that cater to diverse learners and hopefully increases engagement in your classroom. Leave the session with at least 1 strategy you could implement next class! 

Name: Ben Gobler

School: Clemson University 

Title: Taffy, Trees, and Tangles

Abstract: In this playful talk, we will explore the mathematics of taffy pulling machines. Counting the number of taffy layers as the machine runs reveals familiar combinatorial patterns. In fact, the taffy machine exhibits the same behavior as the Calkin-Wilf tree, a binary tree which enumerates the rational numbers without duplicates. The Calkin-Wilf tree is closely related to another famous class of objects known as rational tangles, which appear to share some characteristics with taffy pulls. For the big finale, we reveal the incredibly beautiful relationship between tangles and taffy, bringing all of the pieces together in a natural and satisfying way. 

Name: Evan Butterworth

School: Clemson University 

Title: On the Horizontal Method of Lines for Nonlinear Adsorption Problems

Abstract: We consider a nonlinear transport problem to model the chromatography process of high-capacity multimodal membranes. The absorption model for this process is nonlinear and can be represented by either explicitly or implicitly defined isotherms. Through the utilization of the horizontal method of lines, a variety of time-integration schemes are developed. The main advantage of this method is its flexibility in the manipulation of the nonlinear term when compared to the traditional method of lines. We use the FEniCSx environment to generate numerical solutions, employing its parallel capabilities to evaluate different temporal integration schemes coupled with nonlinear solution algorithms. Additional tools in FEniCSx are utilized to develop animations and plots of these solutions, highlighting its compatibility with a variety of data visualization tools. 

Name: Christopher Serkan

School: University of North Georgia 

Title: Overcoming the challenge of transitioning from high school mathematics to college mathematics

Abstract: Mathematical departments have been seeking ways to reduce high student failure rates in college-level math courses for three decades. Developing and implementing effective strategies for increasing student success in mathematics courses requires an understanding of students' mathematical backgrounds, their conceptions of teaching and learning mathematics, and their expectations of college faculty. As part of this study, we examined students' perceptions of the differences between mathematics instruction in college and high school. This study examined freshman students' beliefs regarding why they struggle in college mathematics courses. By gaining a deeper understanding of what a student is looking for in a mathematics classroom, it is possible to gain a better understanding of what he or she needs. A total of one hundred twenty first semester college students were asked to write anonymously about their perceived differences and similarities between mathematics instruction in high school and college. In Wheeler and Montgomery (2009), it was found that "students' previous experiences in instructional environments were closely related to their beliefs" (p. 289) since students will operate within this new intellectual college endeavor based on their previous educational experiences. Through qualitative research methodology known as grounded theory, the students' responses were analyzed to extract the essence of the students’ perception of differences between college and high school mathematics classes so that the essence could be used to communicate and explore the meaning of those differences. Students struggled with a smooth transition to postsecondary education, according to the results. 

Name: MUHAMMAD MUBASHAR SHAHZAD

School: Government College University Faisalabad

Title: Application of New Quintic Hyperbolic B-Spline functions for Solving Time Fractional Boussinesq Equation

Abstract: This paper introduces an innovative collocation method that manipulating quintic Hyperbolic B-spline functions for the numerical solutions of the time fractional fourth-order Boussinesq equation. To capture the time derivative of non-integral order, the Caputo definition has been adopted. Space discretization is achieved through the utilization of quintic hyperbolic basis B-spline functions, while time discretization is implemented using a forward finite difference scheme. The proposed method has undergone rigorous testing on a range of problems, each characterized by distinct end conditions relevant to various physical applications. Our numerical results are thoughtfully compared against well-established exact solutions to assess the method's accuracy and reliability. This research contributes to the advancement of numerical techniques in tackling challenging fourth-order partial differential equations encountered in diverse scientific and engineering contexts. 

Name: Antsa Tantely Fandresena Rakotondrafara

School: Clemson University 

Title: An friendly introduction to code-based cryptography

Abstract: Cryptography is used to protect our sensitive data. In this talk, I will give a gentle introduction to code-based cryptography. We will use small toy examples to illustrate the concept of linear codes, encryption and decryption. 

Poster Abstracts

Name: Miller Christen 

School: Clemson University 

Title: Images of BoHeMIa: Optimizing Sample Size Using Visualization

Abstract: Bohemian matrices, characterized by random entries sampled from a discrete set or “population” of integers with bounded height, have been found to create fascinating visualizations when plotting the density of their eigenvalues. However, the computation time of all possible eigenvalues increases exponentially as the matrix dimension increases. This project focuses on determining optimal sample sizes for precise representation of eigenvalues in Upper Unit Hessenberg matrix structures. The application of singular value decomposition for density plot images was used to compare the true population datasets of chosen matrices and varying sample sizes. In examining the trending behavior of the angle between singular values of our plot images, we seek to ensure consistency in the optimal sample size. Initial findings suggest that a sample size of 10^7 matrices suffices to approximate the true density plot across populations of dimensions 5 or 6. An ongoing investigation aims to determine whether this sample size is efficient for much larger dimensions by analyzing sample to sample without computing the complete dataset and across different matrix structures. This study intends to expand the accessibility and optimize computational time for the research of Bohemian matrices.

Name: Tania Hazra

School: University of South Carolina

Title: Overcoming a Math-Physics Conflict in a Dielectric Function Distribution

Abstract: Electrostatic analysis is essential for studying various important biological processes at the atomistic level, which involve charged objects such as proteins, DNAs and RNAs, immersed in an aquatic environment with mobile ions. The Poisson Boltzmann Equation (PBE), an implicit solvent model, has been widely used to simulate electrostatic interactions between the solute macromolecular and the surrounding solvent molecules. The Dielectric constant distribution from solute to solvent is a vital part to consider. A super-Gaussian (sG) Dielectric Poisson Boltzmann Equation solver was introduced in 2018. This dielectric function is smooth, and this feature made the Alternating Direction Implicit Scheme (numerical algorithm to solve the PBE) unconditionally stable in the water and vacuum states both. By nature, the smoothness of the dielectric function across the solute-solvent interface offers a concavity in the vacuum state. In some sense, such inflation contradicts our physical intuition, i.e., the dielectric function in the vacuum state should decay monotonically outside the protein. To avoid such a non-monotonicity issue, a new reference state with a large enough dielectric value is employed in the super-Gaussian PB model. Based on the electrostatic free energy calculated using this new reference state, a multiple regression model is developed in this project to estimate the original free energy.

Name: Preethika Yetukuri

School: Clemson University 

Title: An Exploration of Statistical Quality Control Processes

Abstract: Statistical Quality Control processes are often used in a variety of contexts in different industries for replication of procedures to take place. This research aims to explore the statistics and probability calculations associated with SQC methodologies and potential opportunities to advance established quality control techniques as well as exploring what acceptance sampling means. Additionally, this research aims to take on an analysis of the similarities and differences between multiple SQC processes, hoping to take on a computational mathematics and statistics approach in engineering decision making, as well as how techniques can be implemented for future use in the realm of education.

Name: Sahil Chindal

School: Virginia Commonwealth University

Title: Predicting Dengue Incidence In Central Argentina Using Google Trends Data

Abstract: Dengue is a mosquito-borne disease prominent in tropical and subtropical regions of the world but has been emerging in temperate areas. In C\'{o}rdoba, a city in temperate central Argentina, there have been several dengue outbreaks in the last decade following the city's first outbreak in 2009. Internet data, such as social media posts and search engine trends, have proven to be useful for predicting the spread of infectious diseases. As the first step in developing a predictive model of dengue incidence in C\'{o}rdoba using Google Trends data, we have conducted a study of relationships between Google search terms and dengue incidence during recent outbreaks in the city. Specifically, using relevant search terms as predictors and dengue case data as the response variable, our trained model can identify which search terms are significant for predicting dengue cases. We study relationships between predictor and response variables in real-time and with lags in our predictive model. We employ several methods to identify the significance of search terms, and we find that terms such as "mosquito", "dengue", "aedes", "aegypti", "dengue virus", and "virus del dengue", are often strongly correlated with dengue incidence. We observe that the lag data, as compared to real-time data, has a better fit and predictive performance for dengue cases. Our predictive model that utilizes Google Trends data can be integrated with climate, sociodemographic, and other types of information as part of a comprehensive early warning system that predicts outbreaks and informs public health and mosquito control policies.

Name: Evan Hall 

School: Clemson University

Title: A Subset Metric and Sequence Subset Codes

Abstract:  For a bounded metric space $X$, we define a metric on the set of all finite subsets of $X$.  This generalizes the  sequence-subset distance introduced by Wentu Song, Kui  Cai and  Kees A. Schouhamer Immink to study error correcting codes for DNA based data storage. Furthermore, we present constructions of families of Sequence-Subset codes. We exhibit specific examples of the constructions  using BCH Codes, Reed Solomon Codes and other cyclic codes.