Stability of functional and geometric inequalities and applications

Overview. Functional and geometric inequalities play a vital role across the calculus of variations, partial differential equations, and geometry. Given an inequality for which equality is attained and equality cases are characterized, a natural question is that of stability: Suppose a function almost achieves equality in the inequality. Then is it close, in a suitable sense, to an equality case? This proposal focuses on stability properties of various inequalities and especially their applications. Selected projects from the proposed research include: – stability of the Perelman entropy, which in turn implies regularity properties for limit spaces of Riemannian manifolds with lower bounds on scalar curvature. – stability of the Alt-Caffarelli-Friedman monotonicity formula, which has potential applications in spectral shape optimization problems and relates to a conjecture of Carleson in harmonic analysis. – a refined form of stability for the isoperimetric inequality, which will yield the best known bounds toward Choksi-Peletier’s conjecture on the liquid drop model in nuclear physics. Intellectual Merit. The past decade has seen an explosion of stability results for functional and geometric inequalities, but the potential applicability of this field has been underdeveloped thus far. A particularly significant feature of this proposal is its emphasis on applications, which are far-reaching and range from geometric analysis to harmonic analysis to mathematical physics. While the proposed projects are broad in scope, they share a potentially transformative underlying theme: employing stability as a tool for understanding regularity, structure of singularities, and characterization of minimizers in variational problems. The PI’s proposed approaches call for the integration and development of assorted techniques across geometric analysis and the calculus of variations; these methods are sufficiently robust to carry over to other variational problems with similar structure. Broader Impacts. The PI is dedicated to promoting the participation of women in STEM fields. As a postdoc at Northwestern, she volunteered as a mentor at the 2017 Graduate Research Opportunities for Women conference for undergraduates in mathematics, and established biweekly community-fostering lunches among female postdocs. She spoke at two women’s conferences in 2018 and will participate in the Women in Geometry workshop in Oaxaca in 2019. During her Ph.D. at UT Austin, the PI co-organized a program inviting distinguished female mathematicians to speak at UT, tutored a female high school student with disabilities, and organized an interactive Math Day for a class of first-grade girls. More broadly, the PI initiated a geometric analysis reading group at Northwestern in 2018, and on her website shares notes from her five part lecture on the Yamabe problem. This past summer, she co-organized a special session at the AIMS Conference on Dynamical Systems, Differential Equations, and Applications. She will continue to disseminate her research at conferences and make expository notes available on her website. Furthermore, the PI will continue her demonstrated commitment to mentorship and outreach at the K-12 level in addition to early-career stages, especially with respect to the community of women in mathematics.