## Mathematics Department

Undergraduate Research Symposium

Fall 2017

## December 11, 11:15am to 2pm in LCB 215

**11:15-11:30 Scott Neville**

Mentor: Arjun Krishnan

An explicit bijection from particular Kostka numbers and Permutations with specific Longest Increasing Subsequences.

**11:30-11:45 Sarah Melancon**

Mentor: Adam Boocher

Modules and Free Resolutions

**11:45-12:00 Peter Harpending**

Mentor: Elena Cherkaev

Numerical Fractional Calculus

**12:00-12:15 Bo Zhu and Chong Wang**

Mentor: Sean Lawley

Branching Process Model of Breast Cancer

**12:15-12:30 Hannah Choi**

Mentor: Sean Lawley

First Passage Time and its Ecological Applications

**12:30-12:45 Break**

**12:45-1:00 Hannah Waddel**

Mentor: Fred Adler

The Community Ecology of the Music Canon

**1:00-1:15 Katelyn Queen**

Mentor: Fred Adler

Dynamic Modelling of Subclones in Patients with Late-Stage Breast and Ovarian Cancers to Block and Reverse Chemo-resistant Tumors

**1:15 -1:30 Barrett Williams**

Mentor: Jingyi Zhu

Diffusion Equation with Cross-Derivative Term with Variable Coefficients

**1:30-1:45 Alex Henabray**

Mentor: Christel Hohenegger

Flow Around Axisymmetric Biconcave Shapes

**1:45-2:00 Max Carlson**

Mentor: Christel Hohenegger, Braxton Osting

A Numerical Solution to the Free Surface Sloshing Problem with Surface Tension

## Abstracts

**Scott Neville**

Mentor: Arjun Krishnan

Kostka Numbers and Longest Increasing Subsequences

Mentor: Arjun Krishnan

Kostka Numbers and Longest Increasing Subsequences

The Kostka Numbers appear in several natural combinatorial problems, such as symmetric polynomials or partitions. They also count the number of Young tableaux with a given shape and content. If we consider a pair of tableaux with the same shape, the RSK algorithm lets us convert them into a permutation. This bijection sends the width of our Young Tableaux to the length of the longest increasing subsequence of our permutation. If we look at permutations with a specific longest increasing subsequence, like 1,2, then we empirically see that the number of such permutations is equal to the number of non crossing permutations. This seems to generalize. We generalize an existing proof to convert pairs of Young tableaux with certain width into a single Young tableau, with a specific shape and content.

**Sarah Melancon**

Mentor: Adam Boocher

Modules and Free Resolutions

Mentor: Adam Boocher

Modules and Free Resolutions

Free resolutions are central to the field of commutative algebra. Computing a free resolution gives us lots of information about a module, and there are a variety of fascinating open questions about the properties of free modules. We will discuss what a free resolution is, what it can tell us, and what kinds of problems commutative algebraists would like to solve.

**Peter Harpending**

Mentor: Elena Cherkaev

Numerical Fractional Calculus

Mentor: Elena Cherkaev

Numerical Fractional Calculus

The fractional derivative is a generalization of the ordinary derivative, which allows differentiation to arbitrary real order. We present a numerical algorithm for computing fractional derivatives efficiently, and for solving some fractional partial differential equations. We also present a proof-of-concept program.

**Bo Zhu and Chong Wang**

Mentor: Sean Lawley

Branching Process Model of Breast Cancer

Mentor: Sean Lawley

Branching Process Model of Breast Cancer

We develop a branching process model for breast cancer growth and change accounting for three types of cell populations: Primary (cells in the breast), Lymph (live cells in nearby lymph nodes), and Metastatic (cells transplanted on other remote organs). Then we dig up the data online, use the data to find what is the best window of opportunity to recover from the breast cancer.

**Hannah Choi**

Mentor: Sean Lawley

First Passage Time and its Ecological Applications

Mentor: Sean Lawley

First Passage Time and its Ecological Applications

TBA

**Hannah Waddel**

Mentor: Fred Adler

The Community Ecology of the Music Canon

Mentor: Fred Adler

The Community Ecology of the Music Canon

Symphony orchestras today have access to a musical canon stretching back at least four hundred years and play a critical role in its establishment. Full symphony orchestras are expensive to operate and have limited performance time, which only allows a finite number of songs and composers to be considered canonical. Most works by new composers, if performed at all, never get performance time and enter the canon. Community ecology describes the structure and interactions of species that are competing for limited resources. Treating composers or songs as species, we are using quantitative ecological methods to characterize the ways that music interacts in the canon, with a particular focus on the ways that composers and pieces enter or remain in the canon. The canon already appears to follow some common species composition patterns, where a small number of species constitute the bulk of the biomass in an ecosystem and most species are rare. In our data from 9 symphony orchestras, over performances of 207,000 pieces by 296 composers, 91 composers had less than 120 performances of their work while Mozart alone accounted for nearly 12,000 performances. This result lends us confidence that some ecological methods may be useful to describe the canon. The efficacy of ecological models in describing the behavior of the canon lends insights into what processes occur in the field of music composition that cause a composer to either fade into obscurity or become enshrined through centuries of performance.

**Katelyn Queen**

Mentor: Fred Adler

Dynamic Modelling of Subclones in Patients with Late-Stage Breast and Ovarian Cancers to Block and Reverse Chemo-resistant Tumors

Mentor: Fred Adler

Dynamic Modelling of Subclones in Patients with Late-Stage Breast and Ovarian Cancers to Block and Reverse Chemo-resistant Tumors

(By Katelyn J. Queen, Dr. Fred Adler, Dr. Jason Griffiths) During the course of chemotherapy treatment in cancer patients, genetically related groups of cells in the tumors called subclones can become resistant to treatments. This study uses mathematical and statistical models to understand this process, and eventually, we hope, to find ways to delay or even reverse the development of the chemo-resistant tumors associated with breast and ovarian cancers. Our data come from metastatic tumor cells from the pleura of patients with breast and ovarian cancer collected before, during, and after different types of treatments. These cells are then genetically sequenced with a method called single-cell RNA seq that gives the genetic composition in detail. Our main mathematical tool will be integral projection models, which can track continuous change in the stage of disease progression over time, and will be used for the first time to map phenotypic changes in metastatic cells. These methods will allow us to derive, simulate and analyze a dynamic model of subclone changes during the transformation of a chemo-sensitive tumor to a chemo-resistant tumor, which in turn would create the opportunity to then block or reverse this transformation in patients with late- stage breast and ovarian cancers. As an added complexity, the progression of cancers can be facilitated by non-cancerous cells like white blood cells. We will extend our data analysis techniques to determine how cancer subclones interact with other cell types, and how these interactions might be used to delay the transition to a chemo-resistant state. Currently, the project is focused on a simple, dynamic, ordinary differential equations model of cancer, that entertains two different cancer cell states as well as therapy and the immune system.

**Barrett Williams**

Mentor: Jingyi Zhu

Diffusion Equation with Cross-Derivative Term with Variable Coefficients

Mentor: Jingyi Zhu

Diffusion Equation with Cross-Derivative Term with Variable Coefficients

TBA

**Alex Henabray**

Mentor: Christel Hohenegger

Fluid Flow Around Biconcave Surfaces

Mentor: Christel Hohenegger

Fluid Flow Around Biconcave Surfaces

Fluid mechanics is defined as the mathematical study of fluids and their behavior at rest and in motion. Scientists use fluid mechanics to explore many natural phenomena, including the red spot on Jupiter and the behavior of tornados. Mathematicians who specialize in this field are often concerned with developing mathematical models and deriving equations that accurately describe fluid behavior, especially around obstacles. English mathematician Sir George Stokes derived the equation that describes viscous fluid flow around a sphere, and other mathematicians have expanded on Stokesâ€™ work with ellipsoidal objects . Modeling fluid flow around obstacles with irregular shapes is very challenging, and these problems typically do not have analytical solutions. The purpose of this research project is to study fluid flow around biconcave shapes, using infinite series and Gegenbauer polynomials. These models will then be examined using MATLAB to determine how accurate they model the fluidâ€™s behavior.

**Max Carlson**

Mentor: Christel Hohenegger, Braxton Osting

A Numerical Solution to the Free Surface Sloshing Problem with Surface Tension

Mentor: Christel Hohenegger, Braxton Osting

A Numerical Solution to the Free Surface Sloshing Problem with Surface Tension

I will be presenting a scalable, parallel numerical method for computing approximate solutions to the free surface sloshing problem with surface tension and discussing the performance of this method.