y research has been a series of projects that fall into two broad categories: studying dynamical systems that arise in complex biological problems, and using data analysis to address systemic bias in society. These varied projects have given me valuable experience in a wide array of analytical and numerical methods, and my exposure to these topics has given me the ability to consider problems from different perspectives.
My thesis focused primarily on the analytical and numerical treatments of nonlinear integro-differential equations, with applications in population dynamics of single species and food chain systems. Since graduating, I have used what I learned from these endeavors to take on projects that seek to address societal ills, which has become the main focus of my current scholarly pursuits.
Quantifying Gerrymandering
I have spent the past two years working with the Quantifying Gerrymandering group at Duke, led by Jonathan Mattingly. The long term goal of this group is to develop `the ensemble method for outlier analysis,' which is used to generate a representative sample of non-partisan maps from a distribution on redistricting. These samples are then used as a comparison against potentially partisan proposals.
The ensemble method works by first reading precinct level data of the state or county in question, and creating a planar graph where the vertices correspond to individual precincts and the edges indicate physical adjacencies. It then creates a random initial districting plan, and generates new plans by randomly sampling from possible alterations to this initial plan, ensuring that all new proposals comply to specified demographic and geographic criteria.
During my time at Duke, I worked on the implementation of a merge-split procedure for generating new district proposals. In this method, we impose a random spanning tree on each individual district, and then select two random adjacent districts to be merged. A single merged district is created with its own random spanning tree. We then scan the edges of this merged district’s tree seeking ‘valid cuts’ where the removal of the edge would result in two split districts that each comply to the given demographic and geographic criteria. We randomly select one of these ‘valid cuts,’ and this creates the two new districts. The benefit of this method is that it is reversible. The probability of our sampler accepting the new proposal depends on the total number of possible spanning trees and the number of alternative cuts we could have selected to create the same proposal, both of which can be reasonably computed. This resulted in the publication of ‘Metropolized Forest Recombination for Monte Carlo Sampling of Graph Partitions’ in the SIAM Journal on Applied Mathematics.
We also extended this merge-split method into a multi-scale framework, where we perform the merge-split procedure at each level (county, precinct, census block), but only descend levels when more resolution is required for a valid split. This framework produced very promising results, and our paper on this work, ‘Metropolized Multiscale Forest Recombination for Redistricting,’ was published in the SIAM Journal on Multiscale Modeling and Simulation in 2023.
I have several current projects in this field, centered on the applications of these methods to real world redistricting. First, one of the key issues in redistricting, from a legal standpoint, is preservation of communities of interest in the various maps generated. Broadly, communities of interest are groups of people, often geographically related, who share common characteristics or interests. The identification and preservation of such communities is critical to fair electoral representation: the Supreme Court cited community preservation when it ordered Alabama to redraw its congressional districts in 2023. This requires identification of what constitutes a community of interest in a way that can be meaningful across multiple plans. With my partners at Duke, we are preparing a paper with the working title ‘Examining Preservation of Communities of Interest in Redistricting through Markov Chain Monte Carlo Sampling.’ In this paper, we evaluate the preservation of municipalities and communities of interest in the redistricting process. To quantitatively measure the preservation of a community, we introduce the novel idea of an ‘ousted population’ that has been separated from their community. We define the ousted population as the number of residents not included in the community’s core voting districts, and explore a number of different ways to specify what a core of a community really is. We analyze some small-scale artificial models and real-world examples from North Carolina and Michigan to examine the impact of utilizing this preservation score.
Next, I have worked with research students at Grinnell to identify communities of interest in the state of Iowa. We first collect over a thousand individual demographic, social, economic, and housing statistics at the census tract level in Iowa. We then condense these data into summary statistics commonly used to define communities, such as racial/ethnic makeup, educational attainment, household income, occupation, monthly rent, and home value. We perform hierarchical linkage clustering at a census tract level to create geographical regions with similar summary statistics. Additionally, we collect geographical data such as electrical service areas or school district boundaries, which identify regions with shared resources or infrastructure. All of these geographical regions, both those resulting from the clustering of summary statistics and those defined by shared resources and infrastructure, represent potential definitions of communities in Iowa: they are all regions where the population share common characteristics. But any single region alone cannot properly capture the idea of community. Instead, we look at their overlap to define the underlying communities. We cluster these geographic regions using a Modified Hausdorff Distance, creating heat maps of overlapping regions. Each resulting cluster’s heat map identifies a community in Iowa. This heat map definition of a community aligns exactly with the definition of a community described above, meaning that we will be able to use that preservation score along with these identified communities to analyze and inform future redistricting efforts in Iowa and other states.
Finally, I am working with several Grinnell students on a more stand-alone gerrymandering project that analyzes a specific algorithm used for sampling random spanning trees from a graph, an important step in the merge-split procedure I helped develop at Duke. The algorithm in question, Russo’s Algorithm, is a relatively new method for sampling random spanning trees. In particular, we are interested in its application in weighted graphs. Russo’s algorithm works by starting with an initial spanning tree and performing a chain of edge additions and removals that eventually converges to a desired distribution. Our goal in this project is to characterize how the convergence rate of this process changes with respect to variations in the size of the graph, the magnitude of the edge weights, the disparity in the edge weights, and how the weights are arranged within the graph. We examined the algorithm across a wide variety of weighting schemes, and demonstrated how Russo’s algorithm responds in these scenarios. We have also explored some ideas around how we select the initial tree to start the chain, and whether this choice can lead to faster convergence. We have obtained promising results so far, and this project is also in preparation for publication this year.
Des Moines Traffic Citations (continuing)
Since coming to Grinnell, I have worked closely with partners in Iowa advocacy groups who have collected data on traffic citations, arrests, and dismissed cases in the capital city, my hometown, Des Moines. Using this comprehensive traffic citation, warning, and arrest data from the Iowa Department of Transportation (Iowa DOT) and the Des Moines Police Department (DMPD), we highlight the effects of race on traffic stops and policing behavior. Previous work has shown large disparities in the rates of citations and arrests of minority groups in the City of Des Moines, and our work aims to describe these disparities: when, where, and why they occur. We use statistical and quasi-experimental methods to assess racial profiling and further analyze citation locations to examine geo-level patterns, including disparities in the distances between citation locations and home addresses, as well as census block-level differences in traffic citations. Meanwhile, with our holistic overview of all traffic interactions of the past year, we aimed to quantify pretextual stops by examining the charge categories assigned to different racial groups. The Grinnell students who worked with me on this project for the past two summers presented their results, both years, both at the Iowa Summit on Justice and Disparities, and at the international Joint Mathematics Meeting, the largest mathematics meeting in the world. We plan to compile a paper on our results by the end of summer 2025.
Understanding Voting Patterns and Interactions with Gerrymandering (2021):
This past summer, I worked with a team of undergraduates from the Data+ program at Duke in an exploratory analysis of elections in North Carolina. Our goal was to examine a set of statewide elections from the past 20 years to look for potential spatial patterns, and see how these patterns interacted with the geographic complexities of districting plans.
The first task of the summer was to try clustering the elections. The students first used PCA to get a measure of the distance between elections. They also developed several of their own distance metrics based on the well known ‘uniform swing’ hypothesis. For each set of pairwise distances obtained, the students applied an agglomerative clustering algorithm. The resulting clusters were quite stable.
Once the students had identified the clusters of interest, they started exploring various generative methods for creating new synthetic elections. The goal was to examine an ensemble of potential districting maps for a set of synthetic elections. My students presented a poster of our results at the end of the summer.
American Predatory Lending
In the summer of 2020, I mentored a group of students from the Data+ program at Duke. This project was a continuation of a much larger American Predatory Lending group at Duke which is attempting to compile a history of the 2007/2008 financial crisis. In past years, this group has used a variety of proprietary data sets to help describe the mortgage market of the 2000s decade at a national level. This summer, my team's task was to focus this analysis on North Carolina.
We first had to find ways to use the data present in the Home Mortgage Disclosure Act (HMDA) data set to help identify predatory practices because we did not have access to the proprietary data used previously. Usually predatory practices can be identified by looking at delinquent loans or repetitive refinancing loans for the same property. But the HMDA data set was lacking in many ways, and did not allow for either of those more typical approaches. Instead, we used indicators like the rate spread (difference between its APR and a survey-based estimate of current APRs on comparable prime mortgage loans) and denial rates to help identify racial disparity. Our analysis showed that black applicants were denied loans at almost twice the rate as white applicants across the state, and received significantly higher rate spreads when they were able to obtain loans, even when accounting for applicant income or the value to income ratio for the loan. My students presented a poster of our results at the end of the summer.
Women in Professional Hierarchies
After graduation, I worked on a collaboration attempting to model how the distribution of women at each level of a professional hierarchy changes over time, which led to a paper `Mathematical Model of Gender Bias and Homophily in Professional Hierarchies.’ In our model, we examined the role that cultural and psychological factors, such as promotion bias and homophily (self-seeking), play in the ascension of women through these hierarchies. Incorporating these factors into a model that accounts for the proportion of women at each level in a hierarchy resulted in a system of nonlinear differential equations. This system exhibits a rich range of dynamics, and, importantly, indicates that gender parity is not inevitable. We collected data on the gender fractionation over time of a number of different professional hierarchies, mainly examining the progression of women through different fields in academia. We used this data to verify our model, and quantify the degree of homophily and bias in each hierarchy. For this project, I helped in the derivation of the original model, and contributed many of the numerical results.
Nonlocal Population Dynamics (2014-2018)
My graduate work focused on migratory traveling wave behaviors in models of population dynamics governed by nonlocal species interactions. Nonlocality here refers to interactions between individuals that occur over a distance. To model this, we considered the weighted spatial average of the population, which is represented by a convolution of the population against a kernel function. The addition of these convolutions leads to nonlinear integro-partial differential equations that can be analytically intractable. My research focused on the formation of patterned states in these models and how they interact with the traveling wave, species migration problem.
My first project led to the paper `Traveling Waves in a Nonlocal, Piecewise Linear Reaction-Diffusion Population Model.' The model considered small populations that are governed by local natural growth and decay. When the population increases beyond a threshold, growth was instead controlled by nonlocal competition. This piecewise linear assumption, along with a specific choice of kernel function, allowed us to reduce the integro-partial differential equation to a system of algebraic equations, which enabled an analytic characterization of traveling waves in the presence of nonlocality.
My second project involved a three-species food chain system in the context of biological control, and led to the paper `Biological Control with Nonlocal Interactions.' The ecological system in question consisted of a crop, a pest infesting the crop, and an artificially introduced superpredator designed to devour the pest. The goal of the model was to examine the possibility of biological control, where the pest was fully eliminated. In my research, I extended this model to a system of integro-partial differential equations to allow for species mobility and superpredator nonlocality. I found that resurgences of the pest species can occur in certain parameter regimes, such as when the pest is highly mobile relative to the superpredator. I was able to identify parameter regimes where robust control is attainable.
Future Interest in Social Justice:
Both my Des Moines traffic project and the gerrymandering projects have the potential to generate research into the foreseeable future. In the traffic project we continue to receive updates, and more complete data, and are making connections with other possible sources of data. Further, some of my collaborators are of national prominence and are willing to help me make the necessary connections outside of Iowa so that we could study patterns in other cities. In gerrymandering, there remain promising avenues of exploration with the question of identifying communities of interest. I also recently led an exploratory project looking at electoral fairness in Thailand’s multi-party system which can help to clarify the use of these computational techniques for a fundamentally different electoral system.
A new direction for my work in social justice research is examining the fairness of jury composition. A fundamental aspect of the United State’s judicial system is the right to a fair and impartial trial by a jury of one’s peers. This right encompasses a ‘fair cross-section requirement,’ which is effectively the idea that the pool of possible jurors needs to be sufficiently diverse so as to represent the community involved in the case. The requirement itself derives principally from the Sixth Amendment right to an impartial jury and the Fourteenth Amendment’s Equal Protection Clause. Under the seminal case of the Duren v. Missouri a defendant making a fair cross-section argument based on race must establish, among other things, that representation in the jury pool is not fair and reasonable in relation to the number of such persons of that race in the population, and that the under-representation of the group results from a systematic exclusion of the racial group in the jury selection process. As with many things the devil is in the details, and practical roadblocks have all but ended such challenges in many states.
For a defendant to argue a fair cross-section claim, they must present evidence that a distinctive group is being excluded from the jury selection process. So knowledge of what groups are being included in the jury selection process is critical to any such evidence. And unfortunately, a lack of data is not considered evidence of exclusion. The biggest obstacle to fair cross-section challenges currently is a lack of available data on the demographic makeup of juries. A recent report from a group at Berkeley found that 31 states simply do not collect demographic jury data on race and ethnicity and that list includes 8 of the 11 states with the largest Black populations in the country. Although a large group of jurisdictions do not collect race and ethnicity data on their juries, they do still collect some sort of data: usually in the form of names, addresses, or zip codes of the potential jurors for a given day/week. This is not the racial data needed for a fair cross-section claim, but this type of data can be used as a proxy for race in some cases, and can be used to impute the racial data - inferring the race of the individual based on their name or home address.
The use of imputed data, while helping overcome the lack of proper demographic jury data, does come with its own challenges. The most significant is the need for the collection and cleaning of the data, the application of sophisticated statistical methods, and the communication of those statistical results to an audience of primarily laypeople (judges, lawyers, policymakers, advocates). My ultimate goal for this future project is to create a ‘road-map’ – a document to help guide advocates through this process of data collection, analysis, and reporting – and prepare it for wide dissemination through my connections in the Iowa/Nebraska NAACP. However, there are many remaining research challenges in collecting, imputing, and analyzing the necessary data. In order to address these research questions, I have a series of sub-projects that together would form the basis of my road-map, and help lead to its completion. Notably, Iowa is actually on the cutting edge of this field because of nearly unique legal victories by my collaborators at the NAACP. Furthermore, since Iowa is one of the states that collects demographic jury data, it can act very effectively as a test-case for the tools and analysis I hope to develop. This series of sub-projects performing a complete analysis in Iowa, exploring the use of imputed data in Iowa, examining the presence and quality of data in other states, and preparing a data toolkit with supporting case studies, will be a major focus of my research over the next several years.
Future Interest In Population Dynamics
While my research interests in recent years have branched out into a wide variety of different projects, my interest in populations dynamics abides.
One promising avenue of research is related to comparing the nonlocal population systems with real world data. The inclusion of nonlocality in models of population dynamics is often justified with arguments that a scarcity of resources will force competition between individuals over larger distances, as they all search for the shared, scarce resource. This means that the species these models are typically applied to tend to be few in numbers and live in remote environments, making them difficult to study. Nonlocal behaviors, however, can also arise when the species involved are highly mobile so that two individuals can interact over large distances. This means that some migratory bird species might be good candidates for comparison, and it will be interesting to examine the available data on various migrations.
Another area of interest is the study of these migratory problems in higher dimensions. A problem of interest would be when a large, planar wave hits some obstacle (such as a migratory herd maneuvering around a large lake or a city). Boundary interactions are known to give rise to patterned states in these ecological models, so the interactions with the obstacle could provide some interesting dynamics. In numerical simulations, it would be necessary to ensure that the artificially imposed boundary around the simulated domain does not cause spurious patterns to form. One approach might be to introduce artificial damping in an extended layer around the obstacle, with the artificial domain boundaries determined by the unobstructed planar wave solution.
A third area of interest would be to examine a nonlocal model where the extent of the nonlocality itself varied in space, i.e., the kernel function would be dependent on the spatial variable(s). Since nonlocality is often used as a means of representing a scarcity of a shared resource, this type of model could represent a situation where the relative abundance/scarcity of a resource was not uniformly distributed in the environment.
My thesis focused primarily on the analytical and numerical treatments of nonlinear integro-differential equations, with applications in population dynamics of single species and food chain systems. Since graduating, I have used what I learned from these endeavors to take on projects that seek to address societal ills, which has become the main focus of my current scholarly pursuits.
Quantifying Gerrymandering
I have spent the past two years working with the Quantifying Gerrymandering group at Duke, led by Jonathan Mattingly. The long term goal of this group is to develop `the ensemble method for outlier analysis,' which is used to generate a representative sample of non-partisan maps from a distribution on redistricting. These samples are then used as a comparison against potentially partisan proposals.
The ensemble method works by first reading precinct level data of the state or county in question, and creating a planar graph where the vertices correspond to individual precincts and the edges indicate physical adjacencies. It then creates a random initial districting plan, and generates new plans by randomly sampling from possible alterations to this initial plan, ensuring that all new proposals comply to specified demographic and geographic criteria.
During my time at Duke, I worked on the implementation of a merge-split procedure for generating new district proposals. In this method, we impose a random spanning tree on each individual district, and then select two random adjacent districts to be merged. A single merged district is created with its own random spanning tree. We then scan the edges of this merged district’s tree seeking ‘valid cuts’ where the removal of the edge would result in two split districts that each comply to the given demographic and geographic criteria. We randomly select one of these ‘valid cuts,’ and this creates the two new districts. The benefit of this method is that it is reversible. The probability of our sampler accepting the new proposal depends on the total number of possible spanning trees and the number of alternative cuts we could have selected to create the same proposal, both of which can be reasonably computed. This resulted in the publication of ‘Metropolized Forest Recombination for Monte Carlo Sampling of Graph Partitions’ in the SIAM Journal on Applied Mathematics.
We also extended this merge-split method into a multi-scale framework, where we perform the merge-split procedure at each level (county, precinct, census block), but only descend levels when more resolution is required for a valid split. This framework produced very promising results, and our paper on this work, ‘Metropolized Multiscale Forest Recombination for Redistricting,’ was published in the SIAM Journal on Multiscale Modeling and Simulation in 2023.
I have several current projects in this field, centered on the applications of these methods to real world redistricting. First, one of the key issues in redistricting, from a legal standpoint, is preservation of communities of interest in the various maps generated. Broadly, communities of interest are groups of people, often geographically related, who share common characteristics or interests. The identification and preservation of such communities is critical to fair electoral representation: the Supreme Court cited community preservation when it ordered Alabama to redraw its congressional districts in 2023. This requires identification of what constitutes a community of interest in a way that can be meaningful across multiple plans. With my partners at Duke, we are preparing a paper with the working title ‘Examining Preservation of Communities of Interest in Redistricting through Markov Chain Monte Carlo Sampling.’ In this paper, we evaluate the preservation of municipalities and communities of interest in the redistricting process. To quantitatively measure the preservation of a community, we introduce the novel idea of an ‘ousted population’ that has been separated from their community. We define the ousted population as the number of residents not included in the community’s core voting districts, and explore a number of different ways to specify what a core of a community really is. We analyze some small-scale artificial models and real-world examples from North Carolina and Michigan to examine the impact of utilizing this preservation score.
Next, I have worked with research students at Grinnell to identify communities of interest in the state of Iowa. We first collect over a thousand individual demographic, social, economic, and housing statistics at the census tract level in Iowa. We then condense these data into summary statistics commonly used to define communities, such as racial/ethnic makeup, educational attainment, household income, occupation, monthly rent, and home value. We perform hierarchical linkage clustering at a census tract level to create geographical regions with similar summary statistics. Additionally, we collect geographical data such as electrical service areas or school district boundaries, which identify regions with shared resources or infrastructure. All of these geographical regions, both those resulting from the clustering of summary statistics and those defined by shared resources and infrastructure, represent potential definitions of communities in Iowa: they are all regions where the population share common characteristics. But any single region alone cannot properly capture the idea of community. Instead, we look at their overlap to define the underlying communities. We cluster these geographic regions using a Modified Hausdorff Distance, creating heat maps of overlapping regions. Each resulting cluster’s heat map identifies a community in Iowa. This heat map definition of a community aligns exactly with the definition of a community described above, meaning that we will be able to use that preservation score along with these identified communities to analyze and inform future redistricting efforts in Iowa and other states.
Finally, I am working with several Grinnell students on a more stand-alone gerrymandering project that analyzes a specific algorithm used for sampling random spanning trees from a graph, an important step in the merge-split procedure I helped develop at Duke. The algorithm in question, Russo’s Algorithm, is a relatively new method for sampling random spanning trees. In particular, we are interested in its application in weighted graphs. Russo’s algorithm works by starting with an initial spanning tree and performing a chain of edge additions and removals that eventually converges to a desired distribution. Our goal in this project is to characterize how the convergence rate of this process changes with respect to variations in the size of the graph, the magnitude of the edge weights, the disparity in the edge weights, and how the weights are arranged within the graph. We examined the algorithm across a wide variety of weighting schemes, and demonstrated how Russo’s algorithm responds in these scenarios. We have also explored some ideas around how we select the initial tree to start the chain, and whether this choice can lead to faster convergence. We have obtained promising results so far, and this project is also in preparation for publication this year.
Des Moines Traffic Citations (continuing)
Since coming to Grinnell, I have worked closely with partners in Iowa advocacy groups who have collected data on traffic citations, arrests, and dismissed cases in the capital city, my hometown, Des Moines. Using this comprehensive traffic citation, warning, and arrest data from the Iowa Department of Transportation (Iowa DOT) and the Des Moines Police Department (DMPD), we highlight the effects of race on traffic stops and policing behavior. Previous work has shown large disparities in the rates of citations and arrests of minority groups in the City of Des Moines, and our work aims to describe these disparities: when, where, and why they occur. We use statistical and quasi-experimental methods to assess racial profiling and further analyze citation locations to examine geo-level patterns, including disparities in the distances between citation locations and home addresses, as well as census block-level differences in traffic citations. Meanwhile, with our holistic overview of all traffic interactions of the past year, we aimed to quantify pretextual stops by examining the charge categories assigned to different racial groups. The Grinnell students who worked with me on this project for the past two summers presented their results, both years, both at the Iowa Summit on Justice and Disparities, and at the international Joint Mathematics Meeting, the largest mathematics meeting in the world. We plan to compile a paper on our results by the end of summer 2025.
Understanding Voting Patterns and Interactions with Gerrymandering (2021):
This past summer, I worked with a team of undergraduates from the Data+ program at Duke in an exploratory analysis of elections in North Carolina. Our goal was to examine a set of statewide elections from the past 20 years to look for potential spatial patterns, and see how these patterns interacted with the geographic complexities of districting plans.
The first task of the summer was to try clustering the elections. The students first used PCA to get a measure of the distance between elections. They also developed several of their own distance metrics based on the well known ‘uniform swing’ hypothesis. For each set of pairwise distances obtained, the students applied an agglomerative clustering algorithm. The resulting clusters were quite stable.
Once the students had identified the clusters of interest, they started exploring various generative methods for creating new synthetic elections. The goal was to examine an ensemble of potential districting maps for a set of synthetic elections. My students presented a poster of our results at the end of the summer.
American Predatory Lending
In the summer of 2020, I mentored a group of students from the Data+ program at Duke. This project was a continuation of a much larger American Predatory Lending group at Duke which is attempting to compile a history of the 2007/2008 financial crisis. In past years, this group has used a variety of proprietary data sets to help describe the mortgage market of the 2000s decade at a national level. This summer, my team's task was to focus this analysis on North Carolina.
We first had to find ways to use the data present in the Home Mortgage Disclosure Act (HMDA) data set to help identify predatory practices because we did not have access to the proprietary data used previously. Usually predatory practices can be identified by looking at delinquent loans or repetitive refinancing loans for the same property. But the HMDA data set was lacking in many ways, and did not allow for either of those more typical approaches. Instead, we used indicators like the rate spread (difference between its APR and a survey-based estimate of current APRs on comparable prime mortgage loans) and denial rates to help identify racial disparity. Our analysis showed that black applicants were denied loans at almost twice the rate as white applicants across the state, and received significantly higher rate spreads when they were able to obtain loans, even when accounting for applicant income or the value to income ratio for the loan. My students presented a poster of our results at the end of the summer.
Women in Professional Hierarchies
After graduation, I worked on a collaboration attempting to model how the distribution of women at each level of a professional hierarchy changes over time, which led to a paper `Mathematical Model of Gender Bias and Homophily in Professional Hierarchies.’ In our model, we examined the role that cultural and psychological factors, such as promotion bias and homophily (self-seeking), play in the ascension of women through these hierarchies. Incorporating these factors into a model that accounts for the proportion of women at each level in a hierarchy resulted in a system of nonlinear differential equations. This system exhibits a rich range of dynamics, and, importantly, indicates that gender parity is not inevitable. We collected data on the gender fractionation over time of a number of different professional hierarchies, mainly examining the progression of women through different fields in academia. We used this data to verify our model, and quantify the degree of homophily and bias in each hierarchy. For this project, I helped in the derivation of the original model, and contributed many of the numerical results.
Nonlocal Population Dynamics (2014-2018)
My graduate work focused on migratory traveling wave behaviors in models of population dynamics governed by nonlocal species interactions. Nonlocality here refers to interactions between individuals that occur over a distance. To model this, we considered the weighted spatial average of the population, which is represented by a convolution of the population against a kernel function. The addition of these convolutions leads to nonlinear integro-partial differential equations that can be analytically intractable. My research focused on the formation of patterned states in these models and how they interact with the traveling wave, species migration problem.
My first project led to the paper `Traveling Waves in a Nonlocal, Piecewise Linear Reaction-Diffusion Population Model.' The model considered small populations that are governed by local natural growth and decay. When the population increases beyond a threshold, growth was instead controlled by nonlocal competition. This piecewise linear assumption, along with a specific choice of kernel function, allowed us to reduce the integro-partial differential equation to a system of algebraic equations, which enabled an analytic characterization of traveling waves in the presence of nonlocality.
My second project involved a three-species food chain system in the context of biological control, and led to the paper `Biological Control with Nonlocal Interactions.' The ecological system in question consisted of a crop, a pest infesting the crop, and an artificially introduced superpredator designed to devour the pest. The goal of the model was to examine the possibility of biological control, where the pest was fully eliminated. In my research, I extended this model to a system of integro-partial differential equations to allow for species mobility and superpredator nonlocality. I found that resurgences of the pest species can occur in certain parameter regimes, such as when the pest is highly mobile relative to the superpredator. I was able to identify parameter regimes where robust control is attainable.
Future Interest in Social Justice:
Both my Des Moines traffic project and the gerrymandering projects have the potential to generate research into the foreseeable future. In the traffic project we continue to receive updates, and more complete data, and are making connections with other possible sources of data. Further, some of my collaborators are of national prominence and are willing to help me make the necessary connections outside of Iowa so that we could study patterns in other cities. In gerrymandering, there remain promising avenues of exploration with the question of identifying communities of interest. I also recently led an exploratory project looking at electoral fairness in Thailand’s multi-party system which can help to clarify the use of these computational techniques for a fundamentally different electoral system.
A new direction for my work in social justice research is examining the fairness of jury composition. A fundamental aspect of the United State’s judicial system is the right to a fair and impartial trial by a jury of one’s peers. This right encompasses a ‘fair cross-section requirement,’ which is effectively the idea that the pool of possible jurors needs to be sufficiently diverse so as to represent the community involved in the case. The requirement itself derives principally from the Sixth Amendment right to an impartial jury and the Fourteenth Amendment’s Equal Protection Clause. Under the seminal case of the Duren v. Missouri a defendant making a fair cross-section argument based on race must establish, among other things, that representation in the jury pool is not fair and reasonable in relation to the number of such persons of that race in the population, and that the under-representation of the group results from a systematic exclusion of the racial group in the jury selection process. As with many things the devil is in the details, and practical roadblocks have all but ended such challenges in many states.
For a defendant to argue a fair cross-section claim, they must present evidence that a distinctive group is being excluded from the jury selection process. So knowledge of what groups are being included in the jury selection process is critical to any such evidence. And unfortunately, a lack of data is not considered evidence of exclusion. The biggest obstacle to fair cross-section challenges currently is a lack of available data on the demographic makeup of juries. A recent report from a group at Berkeley found that 31 states simply do not collect demographic jury data on race and ethnicity and that list includes 8 of the 11 states with the largest Black populations in the country. Although a large group of jurisdictions do not collect race and ethnicity data on their juries, they do still collect some sort of data: usually in the form of names, addresses, or zip codes of the potential jurors for a given day/week. This is not the racial data needed for a fair cross-section claim, but this type of data can be used as a proxy for race in some cases, and can be used to impute the racial data - inferring the race of the individual based on their name or home address.
The use of imputed data, while helping overcome the lack of proper demographic jury data, does come with its own challenges. The most significant is the need for the collection and cleaning of the data, the application of sophisticated statistical methods, and the communication of those statistical results to an audience of primarily laypeople (judges, lawyers, policymakers, advocates). My ultimate goal for this future project is to create a ‘road-map’ – a document to help guide advocates through this process of data collection, analysis, and reporting – and prepare it for wide dissemination through my connections in the Iowa/Nebraska NAACP. However, there are many remaining research challenges in collecting, imputing, and analyzing the necessary data. In order to address these research questions, I have a series of sub-projects that together would form the basis of my road-map, and help lead to its completion. Notably, Iowa is actually on the cutting edge of this field because of nearly unique legal victories by my collaborators at the NAACP. Furthermore, since Iowa is one of the states that collects demographic jury data, it can act very effectively as a test-case for the tools and analysis I hope to develop. This series of sub-projects performing a complete analysis in Iowa, exploring the use of imputed data in Iowa, examining the presence and quality of data in other states, and preparing a data toolkit with supporting case studies, will be a major focus of my research over the next several years.
Future Interest In Population Dynamics
While my research interests in recent years have branched out into a wide variety of different projects, my interest in populations dynamics abides.
One promising avenue of research is related to comparing the nonlocal population systems with real world data. The inclusion of nonlocality in models of population dynamics is often justified with arguments that a scarcity of resources will force competition between individuals over larger distances, as they all search for the shared, scarce resource. This means that the species these models are typically applied to tend to be few in numbers and live in remote environments, making them difficult to study. Nonlocal behaviors, however, can also arise when the species involved are highly mobile so that two individuals can interact over large distances. This means that some migratory bird species might be good candidates for comparison, and it will be interesting to examine the available data on various migrations.
Another area of interest is the study of these migratory problems in higher dimensions. A problem of interest would be when a large, planar wave hits some obstacle (such as a migratory herd maneuvering around a large lake or a city). Boundary interactions are known to give rise to patterned states in these ecological models, so the interactions with the obstacle could provide some interesting dynamics. In numerical simulations, it would be necessary to ensure that the artificially imposed boundary around the simulated domain does not cause spurious patterns to form. One approach might be to introduce artificial damping in an extended layer around the obstacle, with the artificial domain boundaries determined by the unobstructed planar wave solution.
A third area of interest would be to examine a nonlocal model where the extent of the nonlocality itself varied in space, i.e., the kernel function would be dependent on the spatial variable(s). Since nonlocality is often used as a means of representing a scarcity of a shared resource, this type of model could represent a situation where the relative abundance/scarcity of a resource was not uniformly distributed in the environment.
