## Mathematics Department

Undergraduate Research Symposium

Spring 2014

## Monday April 28 Session - 2pm-3:40pm - LCB 121

02:00-02:40pm,

**Ryan Durr and Michael Senter**,

*Mean-square displacement and Mean first passage time in fluids with memory*

*Mentor:*Christel Hohenegger

*Abstract:*Random motion in water is well studied and understood. However, questions arise when attempting to extend Brownian models to media other than water which exhibit some viscosity. We will present research on the effects of varying parameters on particle behavior in such media. We investigated the influence of Kernel number and type on the mean-squared displacement of particles, as well as present on its influence on mean first passage time.

02:40-03:00pm,

**Wantong Du**,

*Crime Rate Analysis*

*Mentor:*Davar Khoshnevisan

*Abstract:*Crime in the United States has been declining steadily since the early to mid-1990s. Violent crime rate and property crime rate have decreased by 49% and 42%, respectively, during the last two decades. Homicide rate, in particular, is at its lowest in nearly fifty years. I attempt to model the homicide rates and forecast the rates on a national level.

03:00-03:20pm,

**Logan Calder**,

*A trading strategy for auto-correlated price processes*

*Mentor:*Jingyi Zhu

*Abstract:*Classical Brownian motion is the foundation of many stock price models. Due to the independence nature of Brownian motion, without major corrections, these models cannot account for autocorrelation observed in some price processes. Replacing classical Brownian motion with fractional Brownian motion has been one suggested remedy, but this poses serious theoretical and practical difficulties. If a price process did follow from some autocorrelated process, such as fractional Brownian motion, knowing the history of a stock could help one to make better bets. How might this be used in a trading strategy? We simulated autoregressive and fractional Brownian motion processes. Based on positively autocorrelated simulations, we will show one strategy that can lead to favorable gains. When we tested this strategy on real data, we found that quickly executing the strategy is crucial to making a profit.

03:20-03:40pm,

**Michael Primrose**,

*Imaging with waves*

*Mentors:*Fernando Guevara Vasquez and Patrick Bardsley

*Abstract:*We studied the problem of imaging a few point scatterers in 2D with waves emanating from point sources located on a linear array. The data we use for imaging are the waves recorded at a few locations that coincide with the source locations. This setup is very similar to the setup used for ultrasound imaging in medical applications. We considered both homogeneous and random media. For homogeneous media, we use the Born approximation (i.e. linearization) to model wave propagation. For random media, we first considered the problem of generating random media with known mean and correlation, and instead of using a full wave solver we used the so called travel time approximation. We then used the Kirchhoff migration method to image a few scatterers using data corresponding to media that were either homogeneous or random.

## Tuesday April 29 Session - 2pm-2:40pm - LCB 323

02:00-02:20pm,

**Kouver Bingham**,

*Reflection Groups and Coxeter Groups*

*Mentor:*Mladen Bestvina

*Abstract:*A pervasive and very beautiful type of group in group theory is one known as a Reflection Group. One of the most popular groups learned about in elementary group theory is the symmetric group on $n$ letters, and this group can be viewed as a reflection group. Reflection groups have rich historical roots and are crucial in classifying polygonal tessellations of surfaces. This talk will serve as an introduction to these groups. We'll look at several intriguing pictures of triangle tessellations of the plane, the sphere, and the hyperbolic plane, and even give the complete classification of these.

(Supported by Algebraic Geometry and Topology RTG grant)

02:20-02:40pm,

**Drew Ellingson**,

*Towards Intersection Computations in Deligne-Mumford Compactification of $M_{g,n}$*

*Mentor:*Steffen Marcus

*Abstract:*Programs to calculate intersection numbers on $M_{g,n}$ are well established, but none of these programs satisfactorily manage boundary classes. To extend the functionality of these programs, we can reduce these computations to problems in combinatorics on decorated graphs. This talk will begin with an introduction to moduli spaces and the moduli space of curves, and will conclude with a general discussion of the calculations involved in intersection computations on the Deligne-Mumford compactification.

(Supported by Algebraic Geometry and Topology RTG grant)

## Wednesday April 30 Session - 2pm-4:20pm - LCB 121

02:00-02:20pm,

**Sage Paterson**,

*Dendritic Growth: Diffusion-Limited Aggregation and other Fractal Growth Models*

*Mentor:*Elena Cherkaev

*Abstract:*Dendritic growth patterns arise in a wide variety of natural phenomena, including dielectric breakdown and particulate aggregations. Various models of this type of fractal growth have been well studied, including diffusion-limited aggregation (DLA), Laplacian growth models, and others. Diffusion-limited aggregation is a process where random walking particles cluster together to form dendritic trees. We present research on this algorithm, and the experiments and methods we developed. Specifically we investigated methods of controlling the density and fractal dimensions of the DLA clusters. We also vary the starting geometry for the growth, and model anisotropic growth. We experiment with using chaotic deterministic maps in place of pseudo-random number generators and observe nearly identical results. We also compare DLA to other models of fractal growth.

02:20-02:40pm,

**Camille Humphries**,

*Numerical Methods for Conservation Laws and Shallow Water Models*

*Mentors:*Yekaterina Epshteyn and Jason Albright

*Abstract:*Conservation laws are widely used in many areas of science and engineering. In this talk, I will describe a modern class of numerical methods called central schemes [1] that are designed to accurately approximate solutions to conservation laws. First, to illustrate the capabilities of these schemes, we will look at two models: the linear advection equation and Burger's equation. In the case of non-linear conservation laws, central schemes are implemented in conjunction with slope-limiters and high-order time discretizations (such as Strong-Stability Preserving Runge-Kutta Methods [5]). We will conclude this talk with an introduction to a current area of research, in which numerical methods based on central-upwind schemes [2,3,4] are being used to solve shallow water systems that model ocean waves, hurricane flood surges, and tsunamis.

[1] A. Kurganov, E. Tadmor, "New High Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations", Journal of Computational Physics 160 (2000), 241-282.

[2] A. Kurganov, S. Noelle, G. Petrova, "Semi-Discrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations", SIAM Journal on Scientific Computing 23 (2001), 707-740.

[3] A. Kurganov, D. Levy, "Central-Upwind Schemes for the Saint-Venant System", Mathematical Modelling and Numerical Analysis, 36 (2002), 397-425.

[4] S. Bryson, Y. Epshteyn, A. Kurganov, G. Petrova, "Well-Balanced Positivity Preserving Central-Upwind Scheme on Triangular Grids for the Saint-Venant System", Mathematical Modelling and Numerical Analysis, 45 (2011), 423-446.

[5] S. Gottlieb, C.-W. Shu, E. Tadmor, "High order time discretization methods with the strong stability property", SIAM Review 43 (2001) 89-112.

02:40-03:00pm,

**Nathan Briggs**,

*Multimaterial optimal composites for 3D elastic structures*

*Mentor:*Andrej Cherkaev

*Abstract:*The problem of multimaterial optimal elastic structures has already been investigated. Namely given a strong and expense, a weak and inexpensive, and void an optimal structure can be computed. This structure consists of various composite microstructures. We expand this problem to 3D by finding the optimal microstructures. By laminating pure materials together, computing effective fields and properties, and enforcing the Hashin Shtrikman bounds an optimal composite microstructure can be found for any admissible eigenvalues of the stress tensor. The problem is considered for the specific case when the cost functional of the weak material touches the quasiconvex envelope formed by the strong material and void.

[1] A. Cherkaev and G. Dzierzanowski. Three-phase plane composites of minimal elastic stress energy: High-porosity structures. International Journal of Solids and Structures, 50, 25-26, pp. 4145-4160, 2013.

[2] A. Cherkaev. Variational Methods for Structural Optimization. Springer

[3] Z. Hashin, S. Shtrikman. A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids, 1963, Vol. 11

03:00-03:20pm,

**Nathan Briggs**,

*Immersed Interface Method as a numerical solution to the Stefan Problem*

*Mentor:*Yekaterina Epshteyn

*Abstract:*The IIM algorithm in 1D [1] is based on well-known Crank-Nicolson algorithm on a fixed grid with adjustments made to account for discontinuities of the derivative at the interface, as well as adjustments to account for the moving boundary. In particular it is shown what happens when the interface crosses grid points. The method is extended to 2D as well, and results and conclusions are presented.

Finally the Stefan Problem is relaxed so the solution at the interface is no longer known, but the jump discontinuity (if any) at the boundary is known. The problem of shape optimization is formulated as the moving boundary problem similar to Stefan Problem, and the possibility of using the IIM to solve it is explored.

[1] Zhilin Li and Kazufumi Ito. The immersed interface method, volume 33 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. Numerical solutions of PDEs involving interfaces and irregular domains.

(Math 4800: Selected Numerical Algorithms and Their Analysis)

03:20-03:40pm,

**Stephen Durtschi**,

*Evolutionary Algorithms Applied to a Pathing Game*

*Mentor:*Yekaterina Epshteyn

*Abstract:*An approximate solution to the Traveling Salesman problem has been successfully developed using both Genetic Algorithms and Swarm Algorithms. This work applies similar techniques to attempt to find optimal paths for the board game Ticket to Ride, a game which requires players to choose a best path between cities. Strengths and weaknesses of both methods are discussed.

(Math 4800: Selected Numerical Algorithms and Their Analysis)

03:40-04:00pm,

**Wenyi Wang**,

*Imaging defects in a plate with waves*

*Mentors:*Fernando Guevara Vasquez and Dongbin Xiu

*Abstract:*This project is about detecting and imaging damage (such as cracks) in a plate by using ultrasonic waves. The wave source is an ultrasonic transducer which is carried by a robot that can move on the plate. The data used for imaging are the waves recorded at a receiver, which is another ultrasonic transducer carried by the robot. We simulated wave propagation in the plate by keeping only the first two Lamb modes and using the Born or linearization approximation. The imaging was done using Kirchhoff migration and assumed an a priori known path for the robot on the plate. The application of this research is to aircraft structural health monitoring and is done in collaboration with Thomas Henderson (School of Computing, University of Utah).

04:00-04:20pm,

**Wyatt Mackey**,

*Determinant of genuine representations of the spin cover of $S_n$*

*Mentor:*Dan Ciubotaru

*Abstract:*Deriving the character table for the symmetric group can be done in many ways. One interesting way is by writing it as a product of matrices. One result of this, if we are careful about the construction of the matrices, is a simple formula for the determinant of the character table. This paper presents such a construction. We then investigate the determinant of the character table of a spin cover of the symmetric group, and present preliminary computations, and an interesting conjecture for further computations. We finally investigate possible methods of proof for our conjecture by relating it back to the character table of the symmetric group.