## REU Opportunities - next project in Summer 2024, dates TBD!

### Arnd Scheel

 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. Congratulations to the students from 2022 for successful graduate school applications and 3(!) NSF GRFP awards! What happens in projects - two examples: Growing Stripes, With and Without Wrinkles This project fit into a longstanding effort to understand pattern formation in growing domains. We studied a partial differential equation model in which pattern formation is driven at an interface - to the left of the interface, patterns can form, but to the right, they cannot. The central question was: how does the speed of the interface determine the features of the pattern left behind in its wake? We were specifically interested in answering this question in the presence of a mechanism known as the zigzag instability, which itself selects angles of stripe patterns. We formally reduced this problem to studying an equation for the angle of the pattern itself, which ultimately becomes an ordinary differential equation, allowing for some explicit analysis. We found that oblique stripe patterns, at an angle to the interface, exist for small speeds. From this starting point, we continued these oblique stripes numerically, and found that they terminate in a bifurcation which gives way to periodic wrinkling of stripe patterns. We finished the project in collaboration with Ryan Goh (an alum of this REU, now assistant professor at Boston University!) and computed a 3D bifurcation diagram which systematically encodes the structure of these stripe patterns depending on the interface speed. See figure on the left for emerging patterns.    Pinning and depinning: from periodic to chaotic and random media In this project, we studied propagation of waves on discrete lattices, modeling for instance electrical impulses in nerve fibers. Our work focused on the phenomenon of pinning, in which for certain values of system parameters, waves are unable to propagate, becoming stuck or "pinned". As a parameter is increased past some critical value, the wave is suddenly able to propagate, and so undergoes depinning. The central goal in our project was to understand how the speed of the wave depends on system parameters near this depinning threshold, particularly in complex media. We modeled this complexity by considering media which are generated by discrete dynamical systems. In this way, we built a framework that captures transitions from constant media to periodic media and then to chaotic, highly complex media. This framework also presented an abundance of concrete examples for numerical study. We demonstrated that depinning asymptotics are universal: the dependence of the wave speed on the parameter near the depinning threshold is determined only by a measure of the underlying dimension of the medium. See figure on the right for depinning thresholds in a cat-map medium. Publications. Results of previous programs lead to the following journal publications. O. Cannon, T. Bondurant, M. Whyte, A. Scheel Shifting consensus in a biased compromise model (PDF) K. Chen, Z. Deiman, R. Goh, S. Jankovic, A. Scheel Strain and defects in oblique stripe growth (PDF) Multiscale Model. Simul. 19 (2021), 1236-1260 M. Avery, C. Dedina, A. Smith, A. Scheel Instability in large bounded domains — branched versus unbranched resonances (PDF) Nonlinearity 34 (2021), 7916-7937 M. Avery, R. Goh, O. Goodloe, A. Milewski, A. Scheel Growing stripes, with and without wrinkles (PDF) [Moduli space movie with delicate arch, Supplementary materials] SIAM J. Appl. Dyn. Sys. 18 (2019), 1078-1117 N. Ankney, M. Avery, T. Khain, A. Scheel Pinning and Depinning: from periodic to chaotic and random media (PDF) Chaos 29, 013127 (2019) R. Samuelson, Z. Singer, J. Weinburd, A. Scheel Advection and autocatalysis as organizing principles for banded vegetation patterns (PDF) J. Nonlinear Science 29 (2019), 255-285 P. Flynn, Q. Neville, A. Scheel Self-organized clusters in run-and-tumble processes (PDF) DCDS-S 13 (2020), 1187-1208 R. Goh, R. Beekie, D. Matthias, J. Nunley, A.Scheel Universal wavenumber selection laws in apical growth (PDF) Phys. Rev. E 94 (2016), 022219. T. Anderson, G. Faye, A. Scheel, D. Stauffer Pinning and unpinning in nonlocal systems (PDF) J. Dyn. Diff. Eqns.28 (2016), 897-923.. C. Browne, A. Dickerson. Mentors: G. Faye, A. Scheel. Coherent structures in scalar feed-forward chains (PDF) SIURO 7 (2014), 306-329. K. Bose, T. Cox, S. Silvestri, P. Varin. Mentors: M. Holzer, A. Scheel. Invasion fronts and pattern formation in a model of chemotaxis in one and two dimensions. (PDF) SIURO 6 (2013), 228-245. M. Kotzagiannidis, J. Peterson, J. Redford, A. Scheel, Q. Wu Stable pattern selection through invasion fronts in closed two-species reaction-diffusion systems. (PDF) In RIMS Kokyuroku Bessatsu B31 (2012), Far-From-Equilibrium Dynamics, eds. T. Ogawa, K. Ueda , pp 79-93. R. Goh, S. Mesuro, A. Scheel Spatial wavenumber selection in recurrent precipitation (PDF) SIAM J. Appl. Dyn. Sys. 10 (2011), 360-402. R. Goh, S. Mesuro, A. Scheel Coherent structures in reaction-diffusion models for precipitation (PDF) Special volume on "Precipitation patterns in reaction-diffusion systems", Research Signpost (2010), 73-93.

Pictures from the REU participants 2012,2015,2017,2018,2020,2022