I am very interested in working with students who want to pursue an undergraduate or master's thesis. My training is broadly in applied mathematics with my doctoral research in the area of calculus of variations, differential geometry, continuum mechanics, and applied analysis. However, as an applied mathematician, I have a very broad range of interests. Below you will find descriptions of current and past projects I have mentored. Some of the descriptions have clickable images which link to more detailed descriptions of the project. In the past I used to include a list of potential projects I could mentor. However, in some sense I have found this to be constraining. Generally, if your interests lie within the intersection of science, mathematics, and modeling then I am more than happy to work with you to develop a project that overlaps with both of our interests. I would love to hear about your ideas!
Grace Hofmann: Noise Induced Tipping in the Wilson Cowan Equations
Grace is completing a master's thesis under my direction in which she is studying noise induced tipping in a piecewise linear version of the Wilson Cowan equations. In computational neuroscience, the Wilson–Cowan model describes the dynamics of interactions between populations of very simple excitatory and inhibitory neurons. The classic Wilson Cowan equations can exhibit limit cycle behavior, multiple steady states, and hysteresis. Using the Freidlin–Wentzell theory of large deviations, we are interested in computing the average transition time between metastable states when weak background noise is introduced into the model. We are also interested in determining the "most probable transition paths" between metastable states as depicted in the figure to the left.
John Turnage 2021-2022: Controllability of a Rolling Disk on Surfaces of Revolution
John completed a master's thesis under my direction in which techniques from differential geometry, Hamiltonian mechanics, and Lie algebras to understand the controllability of a rolling disk on surfaces of revolution. The challenge with this problem is that the governing constraints are non-holonomic and thus the dynamics of the disk evolve on a higher dimensional manifold than one would expect from simply counting equations. Specifically, since non-holonomic constraints generically correspond to an inexact diffential form, the system is non-integrable. The lack of integrability frees the disk to move in exotic directions generated by the Lie brackets of the disks standard flow directions. An example of this phenomenon is a parallel parking procedure in which a car moves orthgonal to its orientation. John is currently pursuing a Ph.D. in mathematics at the The University of Utah.
CURM 2021-2022: Mathematical Modeling of COVID-19 Dynamics
In partnership with Hwayeon Ryu at Elon University and with financial support for the Center for Undergraduate Research in Mathematics (CURM), three undergraduate students from Wake Forest and two undergraduate students from Elon University developed and analyzed mathematical models of infectuous diseases that incorporated human behavior. Specifically, the group studied the spread of an infectious disease on a network which adapts its topology in response to the disease, human behavior, contact tracing, and testing. The approach taken by the students was to develop and analyze a hybrid model bridging network models (mesocale) with ODE models (macroscale) and implementing an optimal control strategy to minimize the economic impact as well as the spread of the disease. The alumni from this program include graduates Minato Hiraoka who is pursuing a Ph.D. in applied mathematics at Northwestern University
and Sarah Ruth Nichols who is pursuing a Ph.D. in mathematics at Rice University
as well as rising seniors Malindi Whyte, Christopher Boyette, and Danielle DaSilva who subsequently participated in REU programs at The University of Minnesota
, Iowa State University
, and Indiana University-Purdue University Indianapolis
Hannah Scanlon 2019-2021: Spread of Infectious Diseases on Adaptive Networks
Hannah completed a senior in which she studied the spread of an infectious disease on an adaptive network. Specifically, to account for human behavior, Hannah constructed two models which allow individuals to pause connections depending on whether they are connected to infected. Her first model operates at the micro level and consists of probabilistically modeling the time evolution of the state of edges and nodes in the human network. The second model operates at the macro level and uses differential equations to model the mean density of susceptible and infected individuals as well as the density of the types of connections. Hannah is currently pursuing a Ph.D. in mathematics at Duke University.
Grace Hofmann 2020-2021: Spiral Waves in the Belousov–Zhabotinsky reaction
Grace completed a senior thesis in which she studied spatio-temporal patterns in systems of reaction diffusion equations. The image to the left illustrates the formation of spiral waves in fast-slow systems of reaction diffusion equations that arose in Graces's work. Grace's work combines techniques from bifurcation theory, geometric singular perturbation theory, partial differential equations, and numerical analysis to study the patterns that emerge in such systems. Beyond chemical reactions, Grace's work has application to epidemiology, morphogenesis, and climate change, to name a few. Grace is currently pursuing a masters degree in mathematics at Wake Forest University. Go Deacs!!!
Nick Corak 2019-2020: Stability and Formation of Hurricanes
Nick completed a master's thesis under my direction that studied a low dimensional dynamical system model for the formation of a hurricane. With increasing sea surface temperatures it is postulated that the intensity of hurricanes will rise. Nick explored the formation and stability of hurricanes with slowly rising sea surface temperatures. In particular, using tools from calculus of variations, stochastic differential equations, and dynamical systems, Nick studied so called "noise induced" tipping events in which random fluctuations in sea surface tempature drive low intensity hurricanes to transition to high intensity super storms. Nick is currently pursuing a Ph.D. in physics at Wake Forest University. Go Deacs!!!
Kevin Buck 2019-2020: Spread of Infectious Diseases, Vaccination and Opinion Formation
Kevin completed a senior thesis with me that developed and analyzed a spatio-temporal model for the spread of infectious diseases with vaccinations. Kevin coupled the vaccination rate to opinion dynamics on whether children should be vaccinated or not. This opinion evolves depending on neighbors opinions, infection density, as well as societal influence. Using this model, Kevin gained a more complete understanding on how to best educate the population about the necessity of vaccination. Kevin is currently pursuing a Ph.D. in mathematics at Indiana University.
Elizabeth Dicus (2019-2020): Mathematical Modeling of Sex Trafficking
Elizabeth Dicus completed a senior thesis with me in which she developed and analyzed a mathematical model of sex trafficking. Her approach was to use a compartmental model with various populations recruited into different compartments through interaction. In her model, the recruitment rate into the prostitution class as well as the active customers depends on the current supply and demand for prostitues. By using this type of recruitment model coupled with a simple economic impetus, Elizabeth was able to understand the efficacy of government intervation such as increased incarceration rates, forced rehabilation, etc. Elizabeth is currently a data analyst at Heyrick Research
, an organization focused on disrupting human traficking.
Addie Harrison (2019-2020): Reaction Diffusion Equations and Pattern Forming Systems
Addie Harrison completed a senior thesis with me studying the formation and stability of patterns in systems of reaction diffusion equations. The underlying mechanism that leads to such patterns is the Turing bifurcation in which globally patterned states emerge with a dominant (spatial) wave number. The picture on the left is an example of such a pattern that arrised in Addie's work. While Addie's work was originally motivated by its applications to chemistry, reaction diffusion equations appear in applications ranging from morphogenesis in biology, ecology and the spread of epidemics to, name a few. Addie is currently persuing a Ph.D. in applied mathematics at the University of Arizona. Go WILDCATS!!!
Max Rezek (2018-2019): The Formation of Singularities in Non-Euclidean Plates
Max completed a Master's thesis under my direction that studied pattern formation in thin elastic sheets. In his thesis, he showed that the periodic patterns observed in leaves, torn elastic sheets, etc. naturally arise as low energy deformations of the sheet. He did this by explicitly constructing isometric immersions of hyperbolic geometries in the small slopes regime for algebraic and exponentially decaying metrics. His results were obtained using a priori
analysis as well as through rigorous justification using gamma-convergence methods. Max is currently persuing a Ph.D. in mathematics at the University of Arizona. Go WILDCATS!!!
Addie Harrison (2018): Human Visual Tracking and Image Formation
Addie worked with me during the summer of 2018 on a project studying image tracking in the human vestibulo-ocular system. Specfically, Addie created a model that couples image formation on the retina with eye and head motion to track an object moving in space. This work is applicable to image stabilization technology in cameras as well understanding the neuroscience of human visualization and depth perception.
Hanwen Wang (2018-2019): Most Probable Tipping Events in Stochastic Differential Equations
Hanwen worked with me on a project that studied most probable tipping events in stochastic differential equations. Specifically, Hanwen developed a numerical gradient descent method for finding the most probable transition path between basins of attraction for low dimensional dynamical systems. Hanwen applied his techniques to a consumer-resource model and compared his numerical results with Monte Carlo simulations. Hanwen is currently pursuing a Ph.D. in applied mathematics at the University of Pennsylvania.
Brady Gales (2017-2019): Regularization of the Head Injury Criterion Functional
Brady worked with me on a project studying the so called head injury criterion (HIC) score functional. This functional attempts to quantify the chance of severe head injuries in terms of a functional defined on an accleration curve during an accident. Brady's work on this project has shown that the HIC score is fundamentally ill-posed in the sense that minimizers consist of accleration profiles with discontinuous jumps. In practice these jumps would lead to catastrophic injuries in the individual. Brady's work focused on introducing and studying regularized versions of the HIC score. Brady is currently persuing a Ph.D. in applied mathematics at the University of Arizona. Go WILDCATS!!!
Dylan King (2017-2018): Energy Driven Pattern Formation
Dylan King worked on a project studying energy driven pattern formation in two spatial dimensions. Specifically, Dylan studied the gradient flow of a proxy model for the energy of a two phase system. The image on the left illustrates the crystalline structure of an austenite-martensite alloy. Dylan's work centered on understanding on how such patterns can arrise as stable equilibrium of an appropriate gradient flow.
Jessica Zanetell (2017-2018): Tipping Points in Stochastically Perturbed Filippov Systems
Jessica completed a Master's thesis under my direction that studied tipping events in stochastic differential equations (SDES) with piecwise smooth drift. Specifically, Jessi extended the theory of large deviations to piecewise smooth systems to quantify most probable tipping events in the limit of vanishing noise strength for this type of system. This project was motivated by recent energy flux models of arctic sea ice thickness. Jessi went on to earn a master's degree in applied mathematics at the University of Arizona. Go WILDCATS!!!
Elizabeth Wallace (2017-2018): Wolves vs. Sheep
For her senior thesis Elizabeth studied a discrete version of the predator prey model on a square lattice with periodic boundary conditions. In particular, Elizabeth developed a discrete dynamical system which models the time evolution of the wolves and sheep. Similar to Conway's game of life, the dynamical system consists of a simple set of rules. However, even with these simple rules the system can exhibit complex behavior that qualitatively replicates the dynamics of a predator prey system. Elizabeth is currently working as a consultant for Workday
Christopher Grimm (2015-2016): Brachistochrone Problem in an Inverse Square Field
While I was a postdoc at Brown, undergraduate student Christopher Grimm worked with me on a project to understand brachistochrone curves in inverse square gravitational fields. Chris's work on this project used tools from calculus of variations, analysis, and numerical optimization to solve this problem. Currently Chris is pursuing a Ph.D. in computer science at the University of Michigan
Ragna Eide (2015-2016): Spread of Infectious Diseases on Adaptive Networks
While I was a postdoc at Brown, Ragna Eide completed an undergraduate thesis studying the spread of infectious diseases on adaptive networks. Ragna's thesis combined techniques from dynamical systems, probability theory and modeling to solve this problem. In particular, Ragna developed a macroscopic system of differential equations that closely modeled the mean of the underlying stochastic problem. Ragna went on to complete a Master's degree in mathematics at Oxford University UK.
Ekaterina Kryuchkova (2014-2016): Modeling Crowd Dynamics
Ekaterina Kryuchkova completed an honor's thesis with me at Brown. Ekaterina developed and analyzed a model of pedestrian dynamics in a crowded room. In particular, this model was designed to analyze how pedestrians evacuate a crowded room during a panic situation. This model was compared with real world data from experiments a virtual reality laboratory at Brown. Currently Ekaterina is a Ph.D. student in mathematics at Cornell University
Mackenzie Simper (2015): Metastable Systems
Mackenzie Simper worked with me during Summer 2015 on an REU project studying rare events in stochastic differential equations. Specifically, Mackenzie and I studied the escape problem for non-gradient systems. Using the Freidlin-Wentzell theory of large deviations we studied the most likely paths of escape in the limit of vanishing noise strength. Mackenzie won the AWM Alice T. Shafer prize for her work, was a Churchill scholar at Cambridge University UK
and is currently a mathematics Ph.D. student at Stanford University .