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.