John A. Gemmer

Clare is completing a master's thesis under my direction in which we are studying the metastability of stationary patterns for the Swift-Hohenberg equation with white noise. Originally derived from the equations for thermal convection, the Swift-Hohenberg equation is a fourth partial differential equation that is a canonical example of pattern forming system. Moreover, the Swift-Hohenberg equation is variational in the sense that it is the gradient flow for an underlying energy functional. Clare is adapting the Freidlin-Wentzell theory of large deviations to this system in order to understand most probable transition paths between patterned states. For systems of ordinary differential equations, it is well known that for gradient systems the most probable transition paths correspond to the time reversed dynamics however no such result holds in the stochastic PDE setting. The figure to the left illustrates the evolution of the one-dimensional Swift-Hohenberg equation with space-time white noise.

Emily is working on a project under my direction in which we are studying the dynamics of the sine-Gordon equation with periodic forcing and random fluctuations. The sine-Gordon equation is an example of an integrable system with traveling soliton and stationary "breather" solutions. Our goal is to understand the metastability of such solutions as function of the amplitude and frequency of the periodic forcing as well as the strength of the noise. It is well known that under periodic forcing this system can exhibit quasiperiodicity and spatiotemporal chaos depending on the amplitude and frequency of the periodic forcing. In the non-chaotic regimes, Emily is working on determining if noise can cause this system to transition between various soliton/breather solutions. We are also interested in applying the techniques we learn to understand soliton solutions to Einstein's vacuum equations in general relativity.

Shelby is working on a project under my direction in which we are studying the problem of phase shaping the modulus of a complex valued function of two variables into a desired target pattern. In this setting, the modulus corresponds to the intensity of an image in physical space while the argument corresponds to the spatial phase of the signal. This is a classic problem in optics in which a complex valued signal is propagated using the linear Schrodinger equation and is closely associated with the problem of phase retrieval from two intensity measurements. In the classic setting, the intensity profiles can be related via Fourier transforms which allows for computationally efficient algorithms to be developed using the Fast Fourier Transform. Shelby is working on understanding how to develop computationally efficient algorithms for systems in which the underlying evolution equation is weakly nonlinear and hence the Fast Fourier Transform is not applicable.

The Arctic sea ice is anticipated to melt away to a seasonally or perennially ice-free state by the end of the century. Kelsy completed an honor's thesis (co-directed by Dr. Kaitlin Hill at Saint Mary's University San Antonio) in which she studied the most probable manner which the Arctic sea ice may melt. To do so, she applied the Freidlin-Wentzell the theory of large deviations to the Eisenman and Wettlaufer Arctic energy balance model to compute most probable transition paths. Kelsy compared these results with Monte Carlo simulations which were used to obtain noise induced tipping events from which the distribution of escape times and the expected transition time could be computed. Kelsy is currently pursuing a Ph.D. in applied mathematics at The University of California Davis.

In the classic chaos game on a convex polygon, a sequence of points is iteratively created by first starting with an initial random point. Second, a new point is generated by randomly selecting a vertex and generating a new point on the line connecting the vertex and the previous iteration. The new point is chosen to lie a fixed fraction of the distance between the previous point and the random vertex. The process is then repeated using the new point as the initial point. This algorithm is well known to generate fractal patterns depending on the fixed fractional difference. For example, when the fraction is a half and the polygon is a triangle, the algorithm generates the Sierpinski triangle pattern. In Jack's thesis he generalized the chaos game to non-convex domains and provided numerical evidence that the on non-convex domains there is still fractal like structure in which the iterates are attracted to. Jack is currently pursuing a masters degree in operations research at Cornell University.

As people increasingly rely on social media for news and information, personalized online content plays a significant role in determining or reinforcing personal opinions. To better understand this phenomenon, Ashley developed a coupled system of ordinary and partial differential equations which model the interaction between social media influencers (ODEs) and the density of public opinion (PDE). Specifically, the influencers were modeled by a (reverse) gradient system in which they moved to maximize their influence while the public opinion evolved by a conservation law with a flux being driven by the pull of the influencers. Assuming the dynamics of the social media influencers move on a much faster time scale than the option of the population, Ashley was able to derive necessary and sufficient conditions for the system to become polarized in the presence of two influencers, i.e. concentrate in measure around the influencer opinions. Ashley is currently pursuing an interdisciplinary Ph.D. in Social and Engineering Systems and Statistics at The Massachusetts Institute of Technology with the financial support of a NSF-GRFP.

Understanding how people compromise to a consensus or polarize to a fragmented state is important to preserving a well-functioning society. Malindi studied the problem of determining which mechanisms could lead to consensus/polarization by adapting the bounded confidence model to a dynamic network in which people only compromise with those they are connected with and break connections if opinions differ too much. Specifically, Malindi compared the effects the network had on the original model, including party formation, through the derivation of ordinary differential equations that modeled the average dynamics at the network level. She also studied the network and how its topology changed as opinions evolved, specifically the formation of clusters which signified the formation of a party. Malindi is currently pursuing a Ph.D. in applied mathematics at at Brown University with the financial support of a NSF-GRFP.

Grace completed a master's thesis under my direction in which she studied 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, Grace used tools for Hamiltonian system to compute "most probable transition paths" between metastable states as depicted in the figure to the left. Grace is currently a high school instructor at Georgetown Visitation Preparatory School.

John completed a master's thesis under my direction in which techniques from differential geometry, Hamiltonian mechanics, and Lie algebras were used 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 differential 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 orthogonal to its orientation. John is currently pursuing a Ph.D. in mathematics at the The University of Utah.

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 infectious 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 (mesoscale) 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, respectively.

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 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 Grace'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 went on to complete a masters degree in mathematics at Wake Forest University. Go Deacs!!!

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 temperature 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 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 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 sex workers. By using this type of recruitment model coupled with a simple economic impetus, Elizabeth was able to understand the efficacy of government intervention such as increased incarceration rates, forced rehabilitation, etc. Elizabeth went on to become a data analyst at Heyrick Research, an organization focused on disrupting human trafficking.

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 arose 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 pursuing a Ph.D. in applied mathematics at the University of Arizona. Go WILDCATS!!!