## Mathematics Department

Undergraduate Research Symposium

Summer 2018

## Thursday, August 23, 2:15 to 3:45 pm, room JTB 120

**2:15-2:30 Jack Garzella**

Mentor: Fernando Guevara-Vasquez

Understanding Spring Netowrk Approximations for Continuous Elastic Bodies

**2:30- 2:45 Adam Lee**

Mentor: Daniel Zavitz, Alla Borisyuk

Relationship between the connectivity of directed networks with discrete dynamics and their attractors.

**2:45-3:00 Dylan Johnson**

Mentor: Karl Schwede, Daniel Smolkin, Marcus Robinson

Searching for Rings with Uniform Symbolic Topology Property

**3:00-3:30 Faith Pearson and Dylan Soller**

Mentor: Anna Romanova, Peter Trapa

Cracking Points of Finite Gelfand Pairs

**3:30-3:45 Audrey Brown**

Mentor: Alla Borisyuk

Intro to research: Analysis of Mice Olfactory Response Data

## Abstracts

Audrey Brown

Mentor: Alla Borisyuk

Intro to research: Analysis of Mice Olfactory Response Data

In the olfactory system, the original neural response is produced when odors bond to an olfactory receptor neuron (ORN) in the nose. The response then travels through glomeruli, and mitral cells (in mammals). The original input to ORN’s is dependent on the type of odor--some ORN types will respond more to one odor than others. The similarity of ORN response was investigated in recent data from eight mice. A response to each odor across ORN's was considered as a high-dimensional vector. Similarity between each pair of odors was calculated using cosine similarity. The odors were then grouped by their chemical class. By comparing the distributions and averages of the odor similarity using histograms, box plots, and matrices, it was determined that the general trend across mice is that similarity between odor responses is higher within a chemically similar group than between chemical groups. It was also determined by comparing odor response to overall activity that there is not a clear correlation between activity level and odor similarity.

Jack Garzella

Mentor: Fernando Guevara-Vasquez

Understanding Spring Netowrk Approximations for Continuous Elastic Bodies

What is the best way to approximate a continuous elastic body? The differential equations governing such systems are not well-known, or necessarily solved, even in the 2D case. We study ways of trying to approximate an elastic body, and determine the Lamé parameters of a substance based on only data gathered from the boundary of the object. We first use discrete Spiring Networks, and conclude that this method doesn't work well. We then use a Finite Element discretization, and conclude that this is a better approach.

Adam Lee

Mentor: Daniel Zavitz, Alla Borisyuk

Relationship between the connectivity of directed networks with discrete dynamics and their attractors.

Inspired by neural activity, we examine simplified directed networks with discrete dynamics. These simplified dynamics allow nodes to have an active, neutral, and refractory state. After each discrete time step, the signal moves from active nodes to nodes in the neutral state. Eventually, the activity forms a stable, cyclic pattern called an attractor or dies. We examine the relationship between the connectivity of these networks and the number and size of their attractors.

Dylan Johnson

Mentor: Karl Schwede, Daniel Smolkin, Marcus Robinson

Searching for Rings with Uniform Symbolic Topology Property

Using recent work from D. Smolkin and J. Carvajal-Rojas, we seek to identify new commutative, Noetherian rings with Uniform Symbolic Topology Property, abbreviated USTP. First, we show that, for k a field, the toric ring k[x,y] has USTP. Then, we consider k[w,x,y,z]/(wx-yz), and make progress toward showing that this ring also has USTP. Both are toric rings already known to have the property, but we provide an alternative proof using a different method, one which we hope to extend to rings unknown to have USTP.

Faith Pearson and Dylan Soller

Mentor: Anna Romanova, Peter Trapa

Cracking Points of Finite Gelfand Pairs

This presentation examines the representation theory of finite Gelfand pairs, which are algebro-combinatorial objects of interest in finite group theory and harmonic analysis. In this presentation, we will construct our Gelfand pairs of interest, and define the cracking point of a finite group. We will present a new smaller upper bound for cracking points of certain groups, and also present our significant progress in finding the cracking points of all symmetric groups.