*12*^{th}UTK Undergraduate Math Conference 2018**Talks**(alphabetically by speaker)

PDF for printing

**Noah Caldwell**and

**Jordan Brown**, UTK (#10)

*Image Classification with Neural Networks and Python*We will discuss the process of image classification using a Convolutional Neural Network with the library Tensorflow.

*Adviser: Vasilios Alexiades*

**Alana Cooper**, UTK (#9)

*Spectrum for Multigraph Designs on 4 Vertices and 7 Edges*Let G be a multigraph obtained by doubling 3 distinct edges of the paw graph (K3 with a pendant edge). By λKn we mean the λ-fold complete multigraph of order n. We investigate necessary and sufficient conditions on n and λ for the existence of a G-decomposition of λKn for each of the three possibilities for G.

*Adviser: Saad El-Zanati*

**Alan Gan**, UTK (#7)

*Python and Parallel Computing*Parallel computing is one of the most important tools of modern computing. Traditionally, parallel computing is done through Message Passaging Interface (MPI) Library, in conjunction with Fortran or C/C++. Now however, newer programming languages such as Python and java, also offer bindings to MPI. In this talk we detail the use of parallel computing with Python, to massively speed up the run time of a numerical partial differential equation code. Furthermore, we compare the performance of Python with that of Fortran and C/C++, demonstrating why most heavy computations are still done in Fortran and C/C++, despite the general trend of programming drifting away from these languages.

*Adviser: Vasilios Alexiades*

**Emmanuel Hartman**, UNC Asheville NC (#8)

*Using Topology to Study Group Theory*We will explore a method of generating a topology from a group, G. Our main focus concerns how this construction translates properties from group theory to topological structures and vice versa. For instance, we will show that the normal closure of a subset, S, of G is equal to the closure of S with respect to our topological space on G. Additionally, we will use this construction to form new proofs in group theory that depend on theorems from topology.

*Adviser: Gregory Boudreaux*

**Kyla Linn**, UTK (#2)

*Impact of Oseltamivir*Influenza virus infects millions of humans every year and the current vaccines provides only limited protection against the flu. While most infected individuals can control virus replication well, some individuals, generally very young and elderly may succumb to the disease. The commonly used treatment for flu is oseltamivir (Tamiflu). While the mechanism of action of oseltamivir as neuraminidase inhibitor is well understood, impact of oseltamivir on influenza virus dynamics in patients has been controversial. Many major clinical trials with oseltamivir have been done by pharmaceutical companies such as Roche and only recently primary data from several of such trials have been released publically. We digitized and re-analyzed experimental data on influenza virus shedding in human volunteers infected with influenza A (1 trial) or B viruses (2 trials). We found that impact on oseltamivir on the virus shedding dynamics in these trials was strongly dependent on i) selection of patients were infected with the virus, and ii) the detection limit in the measurement assay. Given these uncertainties it was not always possible to match the published median virus shedding curves with curves we generated from raw data. Additional analyses did confirm that oseltamivir had an impact on the duration of shedding but this result was varied by the drug dose and viral species. Oseltamivir did not have an impact on the virus growth or clearance rates. Interestingly, a subset of patients showed extended shedding for the whole duration of experiments (8-9 days) even in the presence of the drug suggesting that some individuals could disproportionally contribute to the flu spread in the population.

*Adviser: Vitaly Ganusov*

**Connor Malin**, U Alabama, Tuscaloosa AL (#1)

*On Subsums of Series with Positive Terms*We give a necessary and complete sufficient for subsums of series with positive terms to completely cover the interval [0,s], where s is the sum (possibly infinity) of the series.

*Adviser: David Cruz-Uribe*

**Jessica Matthews**, North Carolina Institute for Climate Studies

Opening Lecture:

*Mathematics in Climate Science*Mathematics shows up in all aspects of climate science, from dataset production through extrapolation to future scenarios. The world’s largest active archive of weather and climate data, NOAA’s National Centers for Environmental Information (NCEI), is located in Asheville, NC. In this talk we will detail several research applications leveraged from data holdings in this massive archive. In particular we will highlight: the mathematics of deriving climate measurements from satellite observations; a project using vegetation data to analyze annual start-of-springtime dates; and using mathematical models to estimate when sea ice will disappear from the Arctic.

**William Reese**, NCSU, Raleigh NC (#6)

*Low-rank Spectral Representations for Solutions of Elliptic PDEs with Random Coefficients Functions*We focus on the Poisson equation with a random coefficient function, in one space dimension. We represent the log of the random coefficient, modeled as a Gaussian field, with a Karhunen Loeve (KL) expansion. The KL modes are computed numerically, by discretizing the corresponding generalized eigenvalue problem using quadrature. We establish accuracy of the computed KL modes numerically, by conducting convergence studies with respect to resolution of the quadrature formula. Our goal is to understand the properties of the solution of the boundary value problem, which is also a random field, by studying its spectral properties. In particular, we compute the KL representation of the solution of the boundary value problem, and study the decay of the eigenvalues, and also convergence of the pointwise variance, as well as numerical illustrations of impact of increasing the number of KL modes on realizations of the process. We find that the solution of the boundary value problem can be represented reliably with a truncated KL expansion with a small number of terms.

*Adviser: Alen Alexanderian*

**Jeremiah Smith**, Harding U, Searcy AR (#4)

*The Thousand Year Problem*This presentation focuses on the congruent number problem. First posed by the Arabs in the 10th century, this problem has been the subject of much mathematical research. Though there are many ways to represent the problem including rational right triangles with integer area, the focus will be on the elliptic curve representation and will end with an elegant theorem partially proved by Tunnell in 1983.

*Adviser: Ronald Smith*

**Andrew Wintenberg**, UTK (#5)

*Butson Type Hadamard Matrices*Complex Hadamard matrices are orthogonal matrices with the additional property that all entries have modulus one. They are found throughout mathematics in areas like combinatorics, operator algebras and harmonic analysis, with interesting applications in quantum information theory. A fundamental subclass known as Butson type have entries that are roots of unity. They can approximate general complex Hadamard matrices; however even for lower dimensions, they remain mysterious. Collecting examples of Butson type matrices is the first step in further understanding them. In this talk we discuss how abstract algebra can be used to develop an algorithm to generate these interesting matrices.

*Adviser: Remus Nicoara*

**Viktor Zenkov**and

**Aileen Barry**, UTK (#3)

*Cell Moving in a Constrained Environment*In the deadly disease of malaria, infected mosquitoes inject malaria parasites during a blood meal, and the parasites make their way to the human liver, where they enter liver cells, multiply, and burst the cells. In response, we have vaccines consisting of T cells specific to the malaria parasites. Constrained to moving within the tube-like structure (sinusoids) of the liver, the T cells can kill a parasite when they make contact. It is unclear if the T cells move purposely with "attraction" to the parasites, or if they move randomly and only kill parasites when they stumble upon them. Current analysis of position data over time of the movements of T cells near malaria parasites treats the data as if all the cells can move freely in open space; it does not take into account the fact that the cells are constrained to move in sinusoids. This investigation models simulated cells constrained to a tube structure. We construct simple tube structures composed of three line segments attached end-to-end in 2D space. We then place a stationary parasite and a moving T cell on opposite ends of each structure, modeling both cells as points. We let the T cell move with a constant probability p of moving toward (Euclidean distance) the parasite. We explore the optimal probability p such that the T cell reaches the parasite on most structures. This investigation helps us understand if T cells are more successful in reaching parasites when the T cells utilize some random movement. The investigation also provides the first step towards analyzing movement data with sinusoid structural constraints in place.

*Adviser: Vasilios Alexiades*