Student Projects

I am very interested in working with students who want to pursue an undergraduate or master's thesis. As an applied mathematician I have a very broad range of interests. Below you will find descriptions of potential projects I could mentor as well as current and past projects. Some of the descriptions have clickable images which link to more detailed descriptions of the project. Feel free to browse the projects and if you find something that interests you feel free to contact me or stop by my office. If you have an interesting idea for a project that is not listed but you are intersted in working with me please contact me, I would love to hear about your ideas!

Potential Projects

## Non-Euclidean Elastic Sheets

I have a long running project studying the patterns formed by swelling thin elastic sheets. The so called non-Euclidean model of elasticity is a variational model of such sheets. In this model the pattern results from the system realizing a local minimizer of a functional which is the sum of strong in-plane contributations and a weak out-of-plane bending term. In particular, the hyperbolic geometries illustrated to the left result from the system reaching an isometric immersion of hyperbolic geometries in Euclidean space. I would be happy to discuss this project with students who have some background in differential geometry and analysis.

## Modeling Singularities in Cloth

There is a theorem due to Chebychev that you cannot cover a sphere with a single cut of cloth. In more general settings this geometric fact implies that any form fitting piece of clothing must contain a seam, i.e. a singularity. The figure on the left is an example of such a singularity (image reference: cermics.enpc.fr/~massony/img/sphere_catenoid.png). I am interested in working with a student in understanding how to cloth general surfaces with a minimial number of singularities. In particular there is an elastic energy associated with the fibers of the cloth and the covering should minimize this elastic energy. This project will apply techniques from differential geometry, partial differential equations, and numerical optimization.

## Motion by Mean Curvature Through a Junction

Motion by mean curvature of a network is a natural spatio-temporal model for the dynamics of phase boundaries. However, when the network meets at a junction the evolution through the singularity is not unique. In the figure to the left, I present one possibility for how the system can flow through the junction. I am intersted in working with a student to classify all motions through the junction and determine criterion for which patterns are selected. This project will apply techniques from ordinary and partial differential equations as well numerical simulations.

## Tipping Points in Stochastic Differential Equations

I am interested in understanding the transition between metastable states for stochastic perturbations of dynamical systems. A well known example of such a system is the thickness of Arctic sea ice. The efect of the noise is to introduce a probability for transitioning between stable states in the system. For noise sufficiently small, typical solution trajectories follow an Ornstein-Uhlenbeck (OU) process about the deterministic orbits. However, for times when the barrier height
between the stable states is on the order of the standard deviation of the OU process the probability of
transitioning can drastically increase. This phenomenon has been coined "stochastic resonance" and
can lead to dramatic changes in the dynamics of the system. As part of an ongoing project I am interested in understanding so called early warning signs for tipping. This project will apply techniques from ordinary differential equations, probability and calculus of variations.

Current Projects

## Nick Corak: Stability and Formation of Hurricanes

For Nick's Master's thesis he is studying a low dimensional dynamical system developed by Kerry Emanuel that models the formation of a hurricane. This model couples the interaction between the velocity and moisture of a hurricane. With increasing sea surface temperatures it is postulated that the intensity of hurricanes will rise. Nick is currently exploring the formation and stability of hurricanes with slowly rising sea surface temperatures. In particular, Nick is studying 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.

## Kevin Buck: Spread of Infectious Diseases, Vaccination and Opinion Formation

Kevin is completing a senior thesis with me that is developing and analyzing a spatio-temporal model for the spread of infectious diseases with vaccinations. Kevin is coupling 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 we hope to understand how to best educate the population about the necessity of vaccination.

## Elizabeth Dicus: Mathematical Modeling of Sex Trafficking

Elizabeth Dicus is completing a senior thesis with me that is developing and analyzing a mathematical model of sex trafficking. Her approach is to use a compartmental model with populations consisting of potential prostitutes, active prostitutes, prostitutes undergoing treatment, reformed prostitutes, potential customers, active customers, incarcerated customers, and an obstaining popultion. In her model populations are recruited into different compartments through interaction with the other populations. Moreover, 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, we hope to understand the efficacy of government intervation such as increased incarceration rates, forced rehabilation, etc.

## Addie Harrison: Reaction Diffusion Equations and Pattern Forming Systems

Addie Harrison is completing 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 arrises in the Belousov-Zhabotinsky (BZ) chemical reaction. 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.

Past Projects

## 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 Master's degree 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 is currently persuing a Ph.D. 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 .**