### July 30, 2020 REU Conference

- Schedule
Here is the schedule. The schools listed are the school of the summer project, not the students' school. Talks in the first session are staggered, but not in the second session. 1:00: Opening Remarks, Kevin Pilgrim, Indiana University Bloomington. Track A, first session 1:15: Dynamics on Dilation Tori, Mason Haberle, IUB 1:40: Geometric properties of 3-ellipses and their relationship to numerical range, Drew Anderson and Jordan Crawford, Taylor 2:05: Extensions of the uniform random polygon model, Xingyu Cheng and Pedro Morales, Rose Hulman 2:30: Calculating the Probability of Linking of Random Polygons, Spencer Eddins and Yasmin Aguillon, Rose Hulman Track B, first session 1:25: The Adoption of M-Pesa: A Percolation Approach to Network Goods, Lisa Reed, Janet Stefanov, and Zerrin Vural, Rose Hulman 1:50: Markov models for DNA sequencing on trees, Alex Xue, IUB

2:15: Dynamic transitions of the Swift-Hohenberg equation with third-order dissipation, Kevin Li, IUB

2:50 to 3:15: Coffee Break

Track A, second session

3:15: Geometric Finite Subdivision Rules for Polygons, Rebecca E. Barry and Jenavie Lorman, IUB

3:40: Normality properties of composition matrices, Hallie Kaiser, Katy O'Malley, and Grace Weeks, Taylor

4:05: A Coalition Game on Finite Groups, Ebtihal Mostafa Abdelaziz, Goshen College

4:30: Knots and Links in Modular Tensor Categories, Sung Kim, IUB

Track B, second session:

3:15: Using Mathematical Modeling to Predict Flow and Oxygenation of the Retina Amanda Albright, and Grace Mattingly, IUPUI

3:40: Modeling the Impact of Adoptive Transfer in Murine Heart Transplant Rejection, Mei Knudson and Lauren Mossman, IUPUI

4:05: Three-Dimensional Steady and Near Steady-State Cancer Cell Model, Allison Cruikshank and Jacob Woodrome, IUPUI

4:30: Simple Python Implementation of the Cellular Potts Model, Trevor Gordley and Benjamin Thomas, IUPUI

4:50: Closing remarks- Titles, speakers
## Indiana REU conference 2020 schedule

- Zoom Links
Track A (and joint events): https://iu.zoom.us/j/97012233245?pwd=LzhyU1ZhS1ZMdWQxQVhSdXJDejdpdz09 Meeting ID: 970 1223 3245, Passcode: 923470

Track B: https://iu.zoom.us/j/94801376273?pwd=TE5IRnRRTmtKSUFDSmpRWEJHd293Zz09 Meeting ID: 948 0137 6273, Passcode: 361793

Track C:https://zoom.us/j/9245245146?pwd=NVA4dFdTZnJQZWMxRUx2TjR1UTdudz09 Meeting ID: 924 524 5146 Passcode: 9x?Xp#

**TRACK A (and joint events)**

dpthurst@iu.edu is inviting you to a scheduled Zoom@IU meeting. Topic: 2020 REU Conference (Track A) Time: Jul 30, 2020 01:00 PM Indiana (East) Join from computer or mobile: https://iu.zoom.us/j/97012233245?pwd=LzhyU1ZhS1ZMdWQxQVhSdXJDejdpdz09 Meeting ID: 970 1223 3245 Password: 923470 One tap mobile +13126266799,,97012233245# US (Chicago) +16465588656,,97012233245# US (New York) Dial by your location +1 312 626 6799 US (Chicago) +1 646 558 8656 US (New York) +1 301 715 8592 US (Germantown) +1 346 248 7799 US (Houston) +1 669 900 6833 US (San Jose) +1 253 215 8782 US (Tacoma) Meeting ID: 970 1223 3245 IU videoconferencing equipment: 26 970 1223 3245 Videoconferencing equipment outside of IU: SIP: 97012233245@zoomcrc.com H.323: 162.255.37.11 (US West) 162.255.36.11 (US East) 221.122.88.195 (China) 115.114.131.7 (India Mumbai) 115.114.115.7 (India Hyderabad) 213.19.144.110 (EMEA) 103.122.166.55 (Australia) 209.9.211.110 (Hong Kong SAR) 64.211.144.160 (Brazil) 69.174.57.160 (Canada) 207.226.132.110 (Japan) Meeting ID: 970 1223 3245 Password: 923470

**TRACK B**dpthurst@iu.edu is inviting you to a scheduled Zoom@IU meeting. Topic: 2020 REU Conference (Track B) Time: Jul 30, 2020 01:00 PM Indiana (East) Join from computer or mobile: https://iu.zoom.us/j/94801376273?pwd=TE5IRnRRTmtKSUFDSmpRWEJHd293Zz09 Meeting ID: 948 0137 6273 Password: 361793 One tap mobile +13126266799,,94801376273# US (Chicago) +16465588656,,94801376273# US (New York) Dial by your location +1 312 626 6799 US (Chicago) +1 646 558 8656 US (New York) +1 301 715 8592 US (Germantown) +1 346 248 7799 US (Houston) +1 669 900 6833 US (San Jose) +1 253 215 8782 US (Tacoma) Meeting ID: 948 0137 6273 IU videoconferencing equipment: 26 948 0137 6273 Videoconferencing equipment outside of IU: SIP: 94801376273@zoomcrc.com H.323: 162.255.37.11 (US West) 162.255.36.11 (US East) 221.122.88.195 (China) 115.114.131.7 (India Mumbai) 115.114.115.7 (India Hyderabad) 213.19.144.110 (EMEA) 103.122.166.55 (Australia) 209.9.211.110 (Hong Kong SAR) 64.211.144.160 (Brazil) 69.174.57.160 (Canada) 207.226.132.110 (Japan) Meeting ID: 948 0137 6273 Password: 361793

- Abstracts
**IUB****Markov models for DNA sequencing on trees****Alex Xue**

DNA sequences can be modeled by character sequences that evolve through insertion, deletion, and substitution events. The sequences are associated to vertices of a rooted tree, and they evolve along the edges of the tree through an evolutionary process. Without assumptions on this process, it can be difficult to infer information about the sequences or how the sequences relate to each other. The TKF91 process is one model that is particularly amenable to computations, while still being somewhat biologically realistic. In studying DNA sequence evolution, one important question is: given information about the sequences at the leaves of the tree, can we reconstruct the structure of the tree and the parameters behind the evolutionary process? Such results are called invertibility results, and in this talk, we will discuss some invertibility results relating to the TKF91 process.

**Dynamic transitions of the Swift-Hohenberg equation with third-order dissipation****Kevin Li**

The standard Swift-Hohenberg equation was introduced to describe the onset of Rayleigh-Benard convection, which considers a horizontal layer of viscous fluid heated from below. It is crucial to the study of non-equilibrium physic due to the natural formation of convection patterns as the fluid is heated. As a control parameter crosses a critical value, the equation exhibits distinct pattern forming behavior. Such dynamic transitions can be understood through analysis of the center manifold, which is a finite dimensional invariant manifold on which the loss of stability occurs. In particular, we focus on the Swift-Hohenberg equation with a third-order dissipation term in one spacial dimension with a periodic boundary condition. The dynamics on the center manifold can be reduced to a single or double Hopf bifurcation, allowing us to fully characterize the transitions.

**Dynamics on Dilation Tori****Mason Haberle**

Dilation surfaces are a recent generalization of the well-studied class of translation surfaces. They are flat surfaces produced by identifying parallel sides of polygons, possibly of different lengths. Even the most elementary dilation surfaces exhibit mysterious new behaviors not found in their translation surface progenitors. The long-term dynamics of straight-line flows are especially rich. Instead of filling in subsurfaces, generic flows attract to closed orbits. Certain flows even have attractors which resemble Cantor sets. The collection of tori with a single boundary component, a simple class of dilation surfaces, is host to all these dynamical behaviors and more. In this talk, we introduce translation and dilation surfaces and explore the dynamics of straight-line flows on this class of dilation tori.

**Knots and Links in Modular Tensor Categories****Sung Kim**

Modular Tensor Categories (MTCs) are algebraic structures that are equivalent to (2+1)-topological quantum field theories. Knot theory has become a powerful practical tool to help us understand and distinguish MTCs. Since the advent of MTCs, it has been conjectured that these categories are classified just by their modular data (S,T) where S and T are invariants derived from the Hopf link and once-twisted unknot respectively. However, Mignard and Schauenburg recently disproved this conjecture by studying a specific class of MTCs known as the twisted quantum double of finite groups. As a result, the study of other knot and link invariants beyond the modular data is important to advance in the classification of MTCs. In this talk, I will provide a basic introduction to MTCs and a new construction technique known as zesting. I will also discuss a particular link invariant, the W-matrix, that is derived from the Whitehead link and how zesting affect knot and link invariants.

**Geometric Finite Subdivision Rules for Polygons****Rebecca E. Barry and Jenavie Lorman**

Finite subdivision rules (FSR) are a way to organize tilings of the plane. A FSR is purely topological and does not necessarily have a geometry. Geometric FSRs let us repeatedly subdivide a shape into smaller copies of itself and then enlarge the smaller tiles to be the size of the original shape in order to cover the plane. We develop processes for finding FSRs and categorizing them to help eliminate those that do not have geometries. Geometric FSRs must subdivide all edges infinitely, but must not subdivide corners infinitely. We find all geometric FSRs where a triangle or quadrilateral is divided into two copies of itself (i.e., 2-reptiles).

**IUPUI****Using Mathematical Modeling to Predict Flow and Oxygenation of the Retina****Amanda Albright, IUPUI****Grace Mattingly, IUPUI**

Glaucoma is the second leading cause of blindness worldwide and is characterized by retinal ganglion cell death. Previous studies have linked impaired blood flow and increased venous oxygen saturation to glaucoma, but there is an ongoing controversy regarding whether these factors are primary or secondary to the disease. Mathematical modeling has emerged as a useful tool to help decipher the role of hemodynamics in glaucoma. This work introduces a theoretical model of the human retinal microvasculature that accounts for spatial heterogeneity, flow regulation, and oxygen transport within the system. A spatially heterogeneous model of arterioles in C++ is linked to a compartmental model of capillaries and venules in MATLAB to generate a complete hybrid model that can yield a more accurate prediction of flow regulation and oxygen saturation throughout the system as metabolic demand, pressure, and flow regulation mechanisms are varied. The flow and partial pressure of oxygen at each terminal arteriole is used to calculate a metabolic wall signal that is generated in the capillaries and venules and conducted back upstream to the arterioles. This metabolic signal along with the myogenic and shear-dependent responses lead to changes in arteriolar diameter and vascular smooth muscle activation and ultimately affect the distribution of oxygen in the retinal tissue and blood vessels. The results of this ongoing work will provide clarity on the cause-and-effect relationship between the retinal ganglion cell death and impaired blood flow as well as have the potential to impact diagnosis and treatment strategies for glaucoma patients.

**Modeling the Impact of Adoptive Transfer in Murine Heart Transplant Rejection****Mei Knudson: Carleton College, Northfield MN****Lauren Mossman: St.Olaf College, Northfield MN**

Due to the body's innate immune response, organ transplant patients must receive lifelong immunosuppression treatment to prevent graft rejection. However, immunosuppression compromises the quality of life of patients by putting them at risk for life threatening conditions such as opportunistic infections, heart disease, and cancer. This study examines the effect of a promising treatment alternative known as adoptive transfer (AT). The adoptive transfer of regulatory T cells (Tregs) delivers activated Tregs to the organ recipient, thereby reducing the destructive immune response and promoting graft survival. A system of ordinary differential equations was used to analyze the impact of dosing magnitude, frequency, and timing of adaptive transfer treatment on immune cell populations and graft survival. The model suggests that administering 13 evenly-spaced doses of Tregs (starting on the day of transplantation) delays rejection more effectively than administering a single large dose on the day of transplantation. Furthermore, the model predicts that it is more advantageous to administer a smaller dose of Tregs two days after transplantation than to administer a larger dose of Tregs on the day of transplantation. However, the model results show that treatment with AT alone is not sufficient to prevent transplant rejection. Therefore, the model is adapted to investigate combined AT and immunosuppression therapy strategies that aim to minimize the amount of immunosuppression delivered to the patient.

**Three-Dimensional Steady and Near Steady-State Cancer Cell Model****Allison Cruikshank, University of Nebraska-Lincoln****Jacob Woodrome, Rose-Hulman Institute of Technology**

70-90% of cancer-related deaths involve metastasis, where cancer cells travel from a primary tumor in one location in order to form a secondary tumor in another location. In breast cancer, which affects roughly 3.5 million Americans per year, such metastasis commonly occurs in bone. To better understand metastatic cancer cells, Barber and Zhu created a rudimentary two-dimensional cancer cell model and illustrated its capabilities by considering the cell???s behavior in a diagnostic microfluidic device. Because many related experiments have taken place in relatively slow or quiescent flow, past student researchers developed a ???fluidless??? version of the two-dimensional model for considering near steady-state environments where external fluidic forces were negligible. In this talk, we will share our preliminary efforts and future plans towards generalizing this fluidless model to three-dimensions. Such cell models are composed of interconnected networks of linear viscoelastic elements (damped springs). Balancing the forces acting on these elements result in a system of differential equations, when the cell is in motion, and a system of linear equations, when the cell is at steady state. Work in progress includes using Matlab to solve these two systems of equations for the dynamics and equilibrium states of cells. Similar to the two-dimensional model, ignoring fluid forces allows a great increase in model efficiency relative to the fluid-inclusive versions of the model. In the future, this model can be expanded to include more physiologically realistic cells including more realistic spring-network geometry, spring properties, and cell interactions with the surrounding environment. The model can also be used to speed up the calibration process for the full fluid-inclusive models by using the fluidless models for slow flow experiments available from literature. More importantly, such models can be used to study cancer cell dynamics in clinical (e.g. treatment development) and in vivo situations.

**A Simple Python Implementation of the Cellular Potts Model****Trevor Gordley, University of Illinois at Urbana-Champaign****Benjamin Thomas, Louisiana State University**

Background: The Cellular Potts Model (CPM) is a powerfully simple biological model. By using Monte Carlo steps and an energy function called the Hamiltonian, the CPM can create complex biological models. Yet there are few resources to clearly explain the details of CPM implementation to beginners.

Purpose: We seek to outline the fundamental concepts of the Cellular Potts Model and provide an accessible example of implementation with a graphic user interface. This example would function both as a research tool and a learning opportunity that can provide complete understanding to people new to the somewhat insular community of biological research.

Methods: The CPM was implemented in python script. Eight files, averaging less than two hundred lines each, provide a complete implementation, along with straightforward accommodation for additional experimentation by users. The Module class allows new modules that can alter the Hamiltonian energy function to incorporate new principles and phenomena. The Potts class performs the main loop, and the PVideo class runs simulations while creating visualizations. While the code is intended to allow easy access and manipulation, the graphic user interface allows users to learn from the model even with little to no coding experience. Results: The researchers sought to represent a few keystone behaviors of the CPM. Forces upon cell objects were simulated, as well as perimeter minimization and contact energies between different cell types. These behaviors can be simulated with a short python file, without the steep wall of jargon and complexity that prevents sophisticated software from gaining popularity in the mainstream. This implementation is not intended to replace sophisticated software, but rather to provide a less steep learning curve to beginners.

Research Advisor: Dr. Andres Tovar

**Rose-Hulman****Calculating the Probability of Linking of Random Polygons****Spencer Eddins, Yasmin Aguillon**

The Uniform Random Polygon model, defined by Arsuaga et al., mimics other models of DNA entanglement but differs in that it generates two polygons via independently generating the vertices in some convex space and probability distribution, such as the cube with uniform probability. The expected squared linking number of the two polygons, for any number of vertices, is completely determined by three constants, p, u, and v, which are defined as the product of the signed crossings in certain configurations of edges. We present an exact value of p in terms of Sylvester???s Four Point Problem, and we found partial results for an exact calculation of u using linear spatial embeddings of K5.

**Extensions of the uniform random polygon model****Xingyu Cheng, Pedro Morales**

Understanding long polymer chains such as DNA in confined spaces is of interest in biology. Since long polymer chains in confined spaces behave like closed loops, Arsuaga et al. introduced the uniform random polygon model in order to better understand such loops in confined spaces using probabilistic and knot theoretical techniques. In this talk, we extend their results to loops with random number of edges as well as describe applications of these extensions and results to the complete graph K6.

**The Adoption of M-Pesa: A Percolation Approach to Network Goods****Lisa Reed, Janet Stefanov, and Zerrin Vural**

In 2007, Kenya's mobile network operator Safaricom launched M-Pesa, a mobile phone-based money transfer service. Today, over 95% of Kenyan households use M-Pesa, making Kenya one of the first developing countries to fully embrace mobile payment systems. M-Pesa merits further academic investigation due to the resulting economic growth and reduction of poverty since its inauguration.

Here we reference percolation theory from statistical physics to develop a theoretical model of the spread of M-Pesa from 2007 to 2014. We consider M-Pesa a network good that spreads primarily via word of mouth and assume its chance of adoption is determined by the utility a person can derive from it. This utility increases with the number of M-Pesa users one has in their social network. We simulate the spread of M-Pesa throughout Kenya by using social network models and measure the deviation between experimental and empirical data of the increase of users as a percentage of the Kenyan population over time. Our model may be useful in analyzing the potential for the propagation of mobile money in other developing countries. We hope our findings will highlight the positive impact to be made by mobile money systems, and motivate others to realize similar impacts in developing countries.

**Goshen College****A Coalition Game on Finite Groups****Ebtihal Mostafa Abdelaziz, Goshen College**

This is an initial investigation into possible connections between the mathematical theories of groups and coalition games. An example of a group is the set of symmetries of a square: (0-, 90-, 180-, and 270-degree rotations and flips across the four lines of symmetries) with composition of motions as the binary operation. A coalition game is a set of players and a numerical worth for each coalition (a nonempty subset of players), and an allocation divides the worth of the all-player coalition among the players. Given an allocation, the excess of a coalition is the sum of their payoffs minus the worth of the coalition, which is one way to quantify how happy the coalition is with the allocation. The prenucleolus is an allocation that maximizes the minimum coalition excess. One coalition game on a group uses the group elements as the players, and the worth of a coalition is the number of elements in the subgroup generated by the coalition. I show that for any such coalition game the prenucleolus payoff is nonnegative for each player and zero for the identity element player. In addition, the prenucleolus for two infinite classes of groups were determined: the symmetries of a regular n-gon and the integers 0, 1, ..., n ??? 1 using addition modulo n as the binary operation.

**Taylor****Normality properties of composition matrices****Hallie Kaiser, Katy O'Malley, and Grace Weeks**

We explore two main concepts in relation to truncated composition operators: the conditions required for the binormal and commutative properties. Both of these topics are important in linear algebra due to their connection with diagonalization. We begin with a simple normal solution before moving onto the more complex binormal solutions. Then we cover conditions for a composition operator to commute with a general matrix. Finally, we end with ongoing questions for future work.

**Geometric properties of 3-ellipses and their relationship to numerical range**

An ellipse, or a "2-ellipse", is the nonempty set of all points for which the sum of the distances to two given points (called the foci) is constant. A 3-ellipse is similar: the set of all points for which the sum of the distances to three foci is constant. Several pictures of 3-ellipses are included to give the reader an idea of their general appearance.**Drew Anderson, Jordan Crawford**