Mathematics Department
Undergraduate Research Symposium
Fall 2018
Monday, December 10, 12:15 to 1:45 pm in LCB 222
12:15-12:30 Justin Baker
Mentor: Elena Cherkaev
Designed Swarming behavior Using Optimal Transportation Networks
12:30-12:45 Charlotte Blake
Mentor: Yekaterina Epshteyn
Efficient Numerical Algorithms for Automatically Processing Data with Application to Materials Science
12:45-1 Audrey Brown
Mentor: Alla Borisyuk
Analysis of Mice Olfactory Response Data
1-1:15 Dylan Johnson
Mentor: Karl Schwede, Daniel Smolkin, Marcus Robinson
Searching for Rings with USTP
1:15-1:30 Dylan Soller
Mentor: Anna Romanov
Finite Gelfand Pairs
1:30-1:45 Eric Allen
Mentor: Lajos Horvath
To be Stationary or to be Non-Stationary, GARCH simulations in RStudio
Abstracts
Eric Allen
Mentor: Lajos Horvath
To be Stationary or to be Non-Stationary, GARCH simulations in RStudio
The GARCH (Generalized Autoregressive Conditional Heteroskedastic) models are used to model stock volatility. Stock volatility is defined as how much and how frequently a stock price changes in the stock market. We run GARCH simulations in order to analyze a times series of data which represent these incremental changes. A time series is a series of values taken at successive, equally- spaced times. The time series represents a sequence of discrete-time data. I used packages, “rugarch” and “fgarch” in RStudio to run stationary and non-stationary GARCH simulations. Multiple simulations were conducted with fixed alpha, beta, and omega to produce plots of the time series and density function. Simulations to estimate values for alpha, beta, and omega in order to predict the accuracy of the “ugarchspec” function were also conducted.
Justin Baker
Mentor: Elena Cherkaev
Designed Swarming Behavior Using Optimal Transportation Networks
Different models of swarming behavior can be used to study, analyze, and optimize the behavior of large populations, building evacuation by emergency response teams, and model groups of robots, people and animals. The current project considers the problem of designing the swarming behavior, and formulates this problem as an optimal transportation problem. We formulate the optimal transportation problem as a discretized linear programming problem. We use the dual problem to maximize efficiency of the designed transportation network. Finally, we numerically compute the solution using Python and develop a visualization of the network and solution for several hypothetical models
Charlotte Blake
Mentor: Ekaterina Epshteyn
Efficient Numerical Algorithms for Automatically Processing Data with Application to Materials Science
Our research focused on developing robust numerical algorithms that take images of crystal grains as the input and automatically output relevant data, including information about grain area, perimeter, and number of neighbors. In this presentation, we will discuss the process of obtaining each type of data, including the difficulties along the way. We will also present the obstacles that appeared as a part of the design of such algorithms and how they were resolved. Special focus will be given to the aspects of the algorithms related to the computational geometry questions of corner identification and polygon approximation of boundaries.
Audrey Brown
Mentor: Alla Borisyuk
Analysis of Mice Olfactory Response Data
1-1:15 Dylan Johnson
Mentor: Karl Schwede, Daniel Smolkin, Marcus Robinson
Searching for Rings with USTP
1:15-1:30 Dylan Soller
Mentor: Anna Romanov
Finite Gelfand Pairs
1:30-1:45 Eric Allen
Mentor: Lajos Horvath
To be Stationary or to be Non-Stationary, GARCH simulations in RStudio
Abstracts
Eric Allen
Mentor: Lajos Horvath
To be Stationary or to be Non-Stationary, GARCH simulations in RStudio
The GARCH (Generalized Autoregressive Conditional Heteroskedastic) models are used to model stock volatility. Stock volatility is defined as how much and how frequently a stock price changes in the stock market. We run GARCH simulations in order to analyze a times series of data which represent these incremental changes. A time series is a series of values taken at successive, equally- spaced times. The time series represents a sequence of discrete-time data. I used packages, “rugarch” and “fgarch” in RStudio to run stationary and non-stationary GARCH simulations. Multiple simulations were conducted with fixed alpha, beta, and omega to produce plots of the time series and density function. Simulations to estimate values for alpha, beta, and omega in order to predict the accuracy of the “ugarchspec” function were also conducted.
Justin Baker
Mentor: Elena Cherkaev
Designed Swarming Behavior Using Optimal Transportation Networks
Different models of swarming behavior can be used to study, analyze, and optimize the behavior of large populations, building evacuation by emergency response teams, and model groups of robots, people and animals. The current project considers the problem of designing the swarming behavior, and formulates this problem as an optimal transportation problem. We formulate the optimal transportation problem as a discretized linear programming problem. We use the dual problem to maximize efficiency of the designed transportation network. Finally, we numerically compute the solution using Python and develop a visualization of the network and solution for several hypothetical models
Charlotte Blake
Mentor: Ekaterina Epshteyn
Efficient Numerical Algorithms for Automatically Processing Data with Application to Materials Science
Our research focused on developing robust numerical algorithms that take images of crystal grains as the input and automatically output relevant data, including information about grain area, perimeter, and number of neighbors. In this presentation, we will discuss the process of obtaining each type of data, including the difficulties along the way. We will also present the obstacles that appeared as a part of the design of such algorithms and how they were resolved. Special focus will be given to the aspects of the algorithms related to the computational geometry questions of corner identification and polygon approximation of boundaries.
Audrey Brown
Mentor: Alla Borisyuk
Analysis of Mice Olfactory Response Data
Relationships within mice olfactory response data was investigated in existing data from eight mice. Previously identified patterns in odor response similarity was further analyzed, and patterns in glomerular response similarity was analyzed using clustering methods. The first method used was hierarchal clustering, for which several different linkage and distance measurements were experimented with. The second method used was K-means clustering Finally, Gaussian distribution clustering was used. For odor response clustering, it was determined that, though with some variability, clustering patterns tend to follow previously observed patterns in odor response similarity. For clustering by glomerular response, different clustering methods were experimented with. Future plans include using glomerular clustering to identify glomeruli from mouse to mouse, and testing correlation between glomerular response and glomerular anatomical position.
Dylan Johnson
Mentors: Karl Schwede, Daniel Smolkin, Marcus Robinson
Searching for Rings with USTP
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. For k a field, we show that the toric rings k[x,y], k[x,y,z], and k[w,x,y,z]/(wx-yz) have USTP. All 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. Indeed, we also classify the 2 and 3 dimensional toric rings for which Smolkin's and Carvajal-Rojas's method may show that he rings have USTP. Finally, we consider a specific type of toric ring, called Hibi rings, and work on applying this method to them in larger dimensions.
Dylan Soller
Mentor: Anna Romanov
Finite Gelfand Pairs
Finite Gelfand paris are algebro-combinatorial objects that arise naturally in various areas of mathematics including statistics, coding theory, and combinatorics. In this presentation, we will discuss ways in which finite Gelfand pairs are related to Markov chains and association schemes. We will also define the “cracking point” of a finite group, and discuss new results including the cracking points of the symmetric groups.