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!
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 intersted 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.
Brady Gales: Regularization of the Head Injury Criterion Functional
Brady is working 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 is focused on introducing and studying regularized versions of the HIC score.
Dylan King: Energy Driven Pattern Formation
Dylan King is working on a project studying energy driven pattern formation in two spatial dimensions. Specifically, Dylan is studying 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 centers on understanding on how such patterns can arrise as stable equilibrium of the appropriate gradient flow.
Jessica Zanetell: 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 a University Fellow persuing a Ph.D. in applied mathematis at the University of Arizona. Go WILDCATS!!!
Elizabeth Wallace: 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.
Christopher Grimm: 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: 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: 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: 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.