## Mathematics Department

Undergraduate Research Symposium

Fall 2013

## Monday December 16 Session - 2pm-4pm - LCB 222

02:00-02:20pm,

**Brady Thompson**,

*The Discriminant Relation Formula*

*Mentor:*Gordan Savin

*Abstract:*One of the most significant invariants of an algebraic number field is the discriminant. The discriminant is an idea we're all familiar with since basic algebra, but idea can be generalized for number fields and becomes an ubiquitous tool in number theory. I will present general definitions and properties of the discriminant of an algebraic number field. I will also discuss the Discriminant Relation Formula, which is a tool that we can use to determine certain properties about a tower of fields. These properties include finding the ring of integers in a number field and verifying whether a field is a Hilbert class field. I will provide an example to illustrate it's usefulness.

02:20-02:40pm,

**Drew Ellingson**,

*Tropical Analogues of Classical Theorems*

*Mentor:*Steffen Marcus

*Abstract:*Tropical Geometry is a field derived from the study of the worst possible degenerations of classical Algebraic Geometry. This talk will develop the bare-bones concepts necessary to start thinking about Tropical Geometry. We then build intuition into the subject by stating and investigating a few tropical analogues of famous theorems in Algebraic Geometry. We will talk about Bézout's Theorem, and then move on to the more challenging group law on cubic curves

02:40-03:00pm,

**Drew Ellingson**,

*Computation of Top Intersections on the Moduli Space of Curves*

*Mentors:*Steffen Marcus and Drew Johnson

*Abstract:*Current algorithms for computing Top Intersections on the Moduli Space of Curves do not satisfactorily handle boundary classes. The goal of the research I have conducted with Steffen Marcus and Drew Johnson is to implement algorithms in the mathematics software SAGE to compute intersection numbers for arbitrary boundary strata. In this presentation, I will introduce the Moduli Space of Curves, its compactification, and the dual graph of a nodal curve. I will then talk about some combinatorial and graph-theoretic problems that arise in computation.

03:00-03:20pm,

**Jonathan Race**,

*Explorations in GARCH(1,1) Processes*

*Mentor:*Lajos Horváth

*Abstract:*It is often the case in financial and economic data that we need models which account for dynamic volatility, or variance. GARCH processes are a relatively recent development in such non-linear modeling. In this presentation I will review the application of GARCH processes and some necessary conditions for their existence.

03:20-03:40pm,

**Nathan Briggs**,

*Optimal three material design on the microstructure and macrostructure scale*

*Mentor:*Andrej Cherkaev

*Abstract:*The problem of optimal three material composite as formulated and solved by Cherkaev and Dzierzanowski is investigated [1]. Namely stress energy plus cost is minimized for an elastic body loaded on the boundary consisting of a strong and expensive material, a cheap but weak material, and a void. This minimization finds the optimal microstructures by solving a multivariable nonconvex minimization problem which is reduced to determination of the quasiconvex envelope of a multiwell Lagrangian, where the wells represent materials' energies plus their costs; the quasiconvex envelope represents the energy and the cost of an optimal composite [2]. After finding the microstructures the problem of optimal design of a body with these microstructures is investigated. The main focus is on a special case corresponding to a specific cost of the weak material. Finally the roll of each material in the design is investigated and applications are discussed. This work is in collaboration with Grzegorz Dzierzanowski.

[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

03:40-04:00pm,

**Sophia Hudson**,

*Exploring 2D Truss Structures Through Finite Element Simulation*

*Mentor:*Andrej Cherkaev

*Abstract:*Lattices in the Euclidean plane can be modeled as a collection of nodes and edges, forming a graph, with nodes corresponding to intersections between the trusses modeled by the edges. The problem of our particular interest is that of understanding what happens to these structures when physical properties are applied to the lattice. In this project, we apply forces to the boundary nodes of n x n truss structures, modeled as connected collections of equilateral triangles. We use a finite element model to understand the stresses and strains on the trusses and to visualize the displacement of nodes and edges from their initial conditions.

## Tuesday December 17 Session - 2pm-4pm - LCB 222

02:00-02:20pm,

**Logan Calder**,

*Fractal Models of Finance*

*Mentor:*Jingyi Zhu

*Abstract:*Fractals have already been used to solve technological problems in communication, and the fractal power of modeling natural features is widely known. Fractals have been used to create realistic animated landscapes and special effects in movies. The complex features found in living organisms can be recreated by repeating a simple pattern. Even in aspects of more modern systems, such as the risk in financial markets, fractals provide more understanding of reality than standard models.

Typically, models of finance have been based on the normal probability distribution. The data though, doesn't fit the model. There are two many big changes in market prices to allow for an easy model to come from the normal distribution. The normal distribution also doesn't allow for dependent events. Instead, the fractal dimension of market graphs may better help us understand the dependence of price changes on each other and therefore allow us to predict more accurately the variance of price changes over time.

The short term and long term dependence of price changes can be determined in one of two ways: Finding the fractal dimension of the graph of market prices over time, or plotting the log of variance of price changes versus the log of different time intervals.

We found that price changes in gold have short term dependence over a period of about 100 days. This allows for easier determination of variance of changes over different time intervals.

02:20-02:40pm,

**Michael Senter**,

*Random Motion in Media with Memory*

*Mentor:*Christel Hohenegger

*Abstract:*Robert Brown discovered random particle motion in the 19th century. We will discuss the model developed by Langevin to describe this motion, as well as the results of Ornstein and Uhlenbeck. We will then proceed to look at random motion in media with memory.

02:40-03:00pm,

**Camille Humphries**,

*Numerical Methods for the Advection Equation: Comparison of Lax-Friedrichs and Central Schemes*

*Mentor:*Yekaterina Epshteyn

*Abstract:*This presentation will include an introduction to the advection equation and will focus on two numerical approximation schemes. The structure of the Lax-Friedrichs and Central Schemes will be presented and explained. A comparison of the accuracy and error ratios in both schemes will be shown for a test function.

03:20-03:40pm,

**Ryan Durr**,

*The Quantification of Exit Times for Varying Fluid Models*

*Mentor:*Christel Hohenegger

*Abstract:*This project begins with a classic theoretical approach to modeling using the Langevin equation. This model assumes that there is no lasting effect from the fluid on the kinematics of a particle. The advantages of this approach is that there is a known analytic solution that can verify the mathematical simulation. The concluding portion of this project is to model the kinematics of a particle that is traversing a fluid with lasting effects and to quantify its exit times. This model does not have analytic solution. The goal of this research is to quantify the exit time for the different fluid models.

03:40-04:00pm,

**Wyatt Mackey**,

*On the McKay Graphs of the Projective Representations of $S_n$*

*Mentor:*Dan Ciubotaru

*Abstract:*The McKay correspondence details a specific connection between the finite automorphisms of $\mathbb{R}^3$ and the Dynkin diagrams of certain Lie algebras. In particular, one creates the ``Mckay Graphs" by observing multiplicities of representations in the tensor products of the representations of a group with its spin representation, then using this to create a weighted, directed graph. Applying this process on the projective covers of the finite automorphisms of $\mathbb{R}^3$, we achieve equivalent graphs to certain interesting Dynkin diagrams.

The finite groups of automorphisms of $\mathbb{R}^3$ correspond to the cyclic groups, the dihedral groups, and $A_3, \ A_4, \ S_3,$ and $S_4$. This paper is interested in examining possible patterns in the McKay graphs of the projective covers of the symmetric group $S_n$ for larger $n$; in particular, we examine the graphs of $n=5$ and $n=6$. To this end, we wrote a program capable of doing all necessary computations to draw the McKay graphs, given the character table of the group. We did not find similar results to McKay's correspondence, however. Interestingly, we find that the McKay graph of the projective cover of $S_5$ is non-planar, quite different from the cases of $n \le 4$, wherein all of the graphs were trees. Surprisingly, this does not hold for the projective cover of $S_6$, which is again planar.