We are organizing the 10th Complex Systems REU (CSREU) in Summer 2024. This REU is a six week program designed to engage students in research problems in dynamical systems and pattern formation, motivated by applications to the physical sciences. We are seeking four students to work on two distinct projects in close collaboration with faculty mentor Arnd Scheel as well as graduate student mentors. Students will share an office in the School of Mathematics and have daily access to both faculty and graduate student mentors.
Potentially relevant fields of math include dynamical systems, numerical analysis and scientific computing, bifurcation theory, PDE theory, and harmonic analysis. The focus of the projects will be shaped by student interest, and prior background in these fields is not required. Some coursework in differential equations will be helpful. No prior research experience is required.
Pattern formation and complex systems. Pattern formation is the study of general mechanisms leading to the appearance of simple or complex spatial patterns. It is motivated by the existence of similar patterns in seemingly dissimilar systems (e.g. animal coat markings, vegetation patterns, phase separation problems, convection patterns).
Participants will receive a stipend of $3,600 as well as up to $2,400 for travel and living expenses. Students will need to be US citizens or permanent residents in order to receive funding. Women and underrepresented minorities are especially encouraged to apply.
I regularly offer research opportunities for undergraduate students. Please contact me if you are interested in doing research in the general area of dynamical systems, pattern formation, and nonlinear waves. Some students in past programs have been supported from other sources such as Undergraduate Research Scholarships at the University of Minnesota.
If you are interested in undergraduate research outside the scope of this NSF-sponsored program or do not qualify as an applicant, please contact me via
and be prepared to send
A list of upper level math courses you have completed or
are taking, along with your grades in these courses
A statement of your interest including your availability.
I will then arrange for a meeting to discuss frameworks and possible projects.
What to expect.
The setup will be similar to previous REU programs that took place in the summers 2009, 2011, 2012, 2014, 2015, 2017, 2018, 2020, and 2022; see the advertisement for past experiences 2009, 2011, 2012, 2014, 2017, 2018, 2020, and 2022.
What do students do after the REU?
Previous students have subsequently gone on to graduate school at Brown (2), CalTech, Columbia, Cornell (2), Georgia Tech, Harvard, Maryland, McGill, Moscow, NYU (2), Oxford, Princeton, Purdue U, U Chicago (2), UC Boulder, UC Irvine, UIUC, UMass, U Mich, Northwestern, U Minnesota (3), 7 received NSF Graduate Fellowships.
What happens in projects - two examples:
Self-organized sacrifice, or, the disappearance of vegetation patches
Vegetation in arid and semi-arid climates is organized in patches, benefiting from synergistic effects of higher density in the patches and saving overall water resources. We studied the effect of decreasing overall rainfall on such self-organized vegetation patterns in an idealized pattern-forming model. As a parameter, representing average rainfalls, decreases, a pattern slowly adapts until a moment of crisis when one or several patches disappear. We predict these moments of crisis and how many patches disappear in such a crisis. The results are somewhat universal across models. Mathematically, we study reduced ordinary differential equations with a slow parameter and a trivial equilibrium that undergoes subsequent pitchfork bifurcations as the parameter evolves. Intriguingly, the predictions are based on spatio-temporal resonances, commonly associated with instability in dispersive waves and plasma dynamics.
Bias and bounded confidence, or, the shifting of opinion in groups
How do people change their opinions? This simple question has led to decades of mathematical study, and one foremost class of models, called bounded confidence models, posit the following: people will talk, and compromise, but they will only compromise with those whose opinions are close to theirs. We study the addition of bias to one such model on a lattice. In particular, we look at how bias causes clusters of opinions to move. We find that with a certain form of bias called self-incitement, where people in larger same-opinion groups are more likely to shift their opinion, a cluster will move but retain its shape. We prove existence of such moving clusters for bias close to a critical value, and use numerical continuation to explore the whole parameter space. The proof in the subcritical regime relies on nonlocal center manifold expansions for the functional equation in a traveling frame. In the small bias regime, we also use ideas from geometric singular perturbation theory to get expansions for cluster speed. Notably, for other bias terms which do not display self-incitement, one does not see this movement - clusters either disperse or fail to propagate.
Publications. Results of previous programs have led to the following journal publications.
A. Asch, M. Avery, A. Cortez, A. Scheel
Slow passage through the Busse balloon -- predicting steps on the Eckhaus staircase.
O. Cannon, T. Bondurant, M. Whyte, A. Scheel
Shifting consensus in a biased compromise model (PDF)
SIAM J. Appl. Dyn. Sys., to appear
K. Chen, Z. Deiman, R. Goh, S. Jankovic, A. Scheel