Geometry, analysis, and morphogenesis: Problems and prospects
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- by Marta Lewicka and L. Mahadevan;
- Bull. Amer. Math. Soc. 59 (2022), 331-369
- DOI: https://doi.org/10.1090/bull/1765
- Published electronically: May 23, 2022
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Abstract:
The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics, and mathematics. How might these shapes be predicted, and how can they eventually be designed? We review our current understanding of this problem, which brings together analysis, geometry, and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we examine the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we revisit known rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the (scaled) thickness becomes vanishingly small and the local curvature can become large. Along the way, we discuss open problems that include those in mathematical modeling and analysis along with questions driven by the allure of being able to tame soft surfaces for applications in science and engineering.References
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Bibliographic Information
- Marta Lewicka
- Affiliation: University of Pittsburgh, Department of Mathematics, 139 University Place, Pittsburgh, Pennsylvania 15260
- MR Author ID: 619488
- Email: lewicka@pitt.edu
- L. Mahadevan
- Affiliation: School of Engineering and Applied Sciences, and Departments of Physics, and Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138
- MR Author ID: 340014
- ORCID: 0000-0002-5114-0519
- Email: lmahadev@g.harvard.edu
- Received by editor(s): April 12, 2021
- Published electronically: May 23, 2022
- Additional Notes: The first author was partially supported by NSF grant DMS 2006439. The second author was partially supported by NSF grants BioMatter DMR 1922321, MRSEC DMR 2011754, and EFRI 1830901
- © Copyright 2022 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 59 (2022), 331-369
- MSC (2020): Primary 35-XX, 49-XX, 53-XX, 74-XX, 92-XX
- DOI: https://doi.org/10.1090/bull/1765
- MathSciNet review: 4437801