## Mathematics Department

Undergraduate Research Symposium

Fall 2015

## December 16 - 2:30pm-5:00pm - Room LCB 323

2:30-2:45,

**Stephen McKean**,

*Fluid Dynamics and Traffic Flow*

*Mentor:*Don Tucker

*Abstract:*The flow of liquids and gasses is well explained through physical laws and theories. However, in terms of mathematical theories and predictions, traffic flow is not understood to the same extent as the physics of matter flow. Is there a correlation between matter and traffic, and can the physical theories of fluid dynamics be generalized to the abstracted case of traffic flow?

2:45-3:00 ,

**Siben Li**,

*Statistical Analysis of Lightcurves*

*Mentor:*Lajos Horvath

*Abstract:*A Type Ia supernova (SN Ia) is the explosion of a carbon–oxygen white dwarf. The observed brightness can be used to measure the distance to the exploded star. These measurements can be used to deduce the expansion of the Universe. The analysis of the measurements led to the discovery that the Universe is dominated by an unknown form of energy that acts in the opposite sense of energy (“dark matter”). Our analysis is based on lightcurves, and we will assume a model to perform our analysis. In this research we would like to study the following question: Is the model supported by the data? We try 3 different methods to answer this question: likelihood method, least squares, and weighted least squares. If not, what is the better model?

3:00-3:15,

**Michael Zhao**,

*Maximal Orders of Quaternion Algebras*

*Mentor:*Gordan Savin

*Abstract:*Suppose we have a base field $k$, a quadratic extension $L$ and a quaternion algebra $H$. Let $\mathcal{O}_L$ and $\mathcal{O}_H$ be two maximal orders of $L$ and $H$. I'll discuss an order is, and the differences between orders in the split and non-split case (i.e. when $H$ is a matrix algebra or a field). Additionally, I'll discuss \emph{optimal} embeddings, which are embeddings $f: L -> H$ with $f(L) \cap \mathcal{O}_H = f(\mathcal{O}_L)$. Finally, I'll present some preliminary results concerning the relation between integral binary hermitian forms and maximal orders.

3:15-3:30,

**Jake Madrid**,

*A Study of Knot and Link Invariants*

*Mentor:*Sayonita Ghosh Hajra

*Abstract:*A knot is the embedding of a closed curve in space. One of the fundamental problems in knot theory is to distinguish knots. Knot invariants are objects assigned to different knot projections and are invaluable in solving this problem. In this talk, we will explore several of these knot invariants and will also discuss Vassiliev invariants. Finally, we will discuss and analyze an algorithm used to compute an invariant called Stabilized Hat Heegaard Floer homology.

3:30-3:45,

**Hunter Simper**,

*Is the maximum number of singular points for an irreducible tropical curve the same as a complex curve?*

*Mentor:*Aaron Bertram

*Abstract:*A brief overview of definitions and concepts in the tropical setting. Followed by an introduction to the above problem by examining how key differences between the tropical version of Bezout and the usual one preclude the direct translation of the common proof for this fact.

4:00-4:15 ,

**Mackenzie Simper**,

*Bak-Sneppen Backwards*

*Mentor:*Tom Alberts

*Abstract:*The Bak-Sneppen model is a Markov chain which serves as a simplified model of evolution in a population of spatially interacting species. We study the backwards Markov chain for the Bak-Sneppen model and derive its corresponding reversibility equations. We show that, in contrast to the forwards Markov chain, the dynamics of the backwards chain explicitly involve the stationary distribution of the model, and from this we derive a functional equation that the stationary distribution must satisfy. We use this functional equation to derive differential equations for the stationary distribution of Bak-Sneppen models in which all but one or all but two of the fitnesses are replaced at each step.

4:15-4:30,

**Jammin Gieber**,

*Eigenvalues in Chaos*

*Mentor:*Elena Cherkaev

*Abstract:*I will be talking about how the eigenvalues change in the Lorenz strange attractor and about some of the bifurcations and where they occur along the flow.

4:30-4:45,

**Chong Wang**,

*Surprising Mathematics of Longest Increasing Subsequences*

*Mentor:*Tom Alberts

*Abstract:*I will talk about patience sorting and the Robinson-Schensted algorithm.

4:45-5:00,

**Tauni Du**,

*The Role of mRNA Decay in a Genetic Switch*

*Mentor:*Katrina Johnson and Fred Adler

*Abstract:*Genes can be switched on or off by regulatory proteins. For example, two genes may each synthesize a protein that downregulates the other gene, creating a repressor-repressor switch that has two stable steady states: one being when the first gene is "on" and the second gene is repressed, and the other in which the situation is reversed. Previous study of a model by Cherry and Adler showed that the existence of such a switch depends on factors including the shapes of the two repression functions. Adding to the model an additional point of control - mRNA dynamics - resulted in further restrictions upon the parameters that can lead to a functional genetic switch. Specifically, compared to the original model, it is twice as difficult for a system that takes into account second-order mRNA decay to have a working switch.