The classical isoperimetric inequality gives an amazing relationship between surface area and volume. If \(E\subset \mathbb{R}^n\) is a set of finite perimeter with \(|E|<\infty\) then the isoperimetric inequality states \begin{equation}\label{Isoperimetric Inequality}\tag{1.1} P(E) \geq n|B_1|^{1/n}|E|^{(n-1)/n} \end{equation} where \(B_1\) is the unit ball and \(P(E)\) is the distributional perimeter of \(E\); when \(E\) has piecewise smooth boundary this coincides with the usual perimeter \(\mathcal{H}^{n-1}(\partial E)\). Moreover, equality is attained in \eqref{Isoperimetric Inequality} if and only if \(|E\Delta B_r(x_0)|=0\) for some \(x_0\in \mathbb{R}^n\) and \(r>0\). One can then ask "If equality is almost attained in \eqref{Isoperimetric Inequality} then is \(E\) almost a ball in some sense?" Such a phenomenon is known as stability. To make this question more precise we define the isoperimetric deficit, denoted \(\delta(E)\) as \begin{equation}\label{Isoperimetric Deficit}\tag{1.2} \delta(E):=\frac{P(E)}{n|B_1|^{1/n}|E|^{(n-1)/n}}-1, \end{equation} which, owing to \eqref{Isoperimetric Inequality} is non-negative in general and zero if and only if \(E\) is essentially a ball. This measures the deviation in attaining equality for the isoperimetric inequality. To measure how close \(E\) is to a ball we introduce a asymmetry index. There are various asymmetry indexes, but the most common is the Fraenkel asymmetry, denoted \(\alpha(E)\) and defined as \begin{equation}\label{Fraenkel Asymmetry}\tag{1.3} \alpha(E):=\inf_{x\in \mathbb{R}^n} \left\lbrace \frac{|E\Delta B_r(x)|}{|E|} \ \bigg| \ |B_r|=|E| \right\rbrace. \end{equation} As \(\delta\) is a scale invariant quantity, it detects deviation in the shape of \(E\) from being a ball, and we expect it to control the Fraenkel asymmetry. Indeed this is the exact result in [FMP08], where the authors prove the following sharp quantitative stability: There exists a universal constant \(C(n)>0\) such that for any set of finite perimeter \(E\subset \mathbb{R}^n\) with \(|E|<\infty\) \begin{equation}\label{Sharp Quantitative Stability}\tag{1.4} \alpha(E)^2 \leq C\delta(E). \end{equation} Inequality \eqref{Sharp Quantitative Stability} is sharp in the sense that the exponent \(2\) cannot be improved, and quantitative in that it provides an explicit relationship between \(\delta(E)\) and \(\alpha(E)\), rather than a qualitative statement like "If \(\{E_k\}_{k=1}^{\infty}\) is a sequence of sets of finite perimeter such that \(\delta(E_k)\to 0\) then \(\alpha(E_k)\to 0\)." More recently, Fusco and Julin in [FJ13], using the selection principle introduced in [CL12] improved on \eqref{Sharp Quantitative Stability} by adding a term controlling the oscillation, \begin{equation}\label{Strong Sharp Quantitative Stability}\tag{1.5} \alpha(E)^2+\beta(E)^2 \leq C\delta(E), \end{equation} where \(\beta(E)\) is the oscillation index defined by \begin{equation}\label{Oscillation Index}\tag{1.6} \beta(E) = \inf_{y\in \mathbb{R}^n} \left\lbrace \frac{1}{n|B_1|^{1/n}|E|^{(n-1)/n}}\int_{\partial^* E} 1 -\left\langle \frac{x-y}{|x-y|},\nu_E(x) \right\rangle \ d\mathcal{H}^{n-1}(x) \right\rbrace \end{equation} Inequality \eqref{Strong Sharp Quantitative Stability} is also sharp.

Over Summer 2020 I participated in UChicago's REU and worked with Dr. Robin Neumayer on topics in convex analysis. I was introduced to optimal mass transport, including the Monge and Kantorovich formulations, as well as some basic existence results of optimal plans. Afterwards, I learned how to apply mass transport theory to solve isoperimetric problems. We investigated a hypothesis I had involving the anisotropic perimeter in dimension n=2, and eventually proved it. Following this, I studied convex bodies and mixed volumes. Using this theory, I was able to give an alternative proof of the result. Finally, I gave a conjecture involving a way to compute the Minkowski sum of convex sets solely in terms of their support functions.

In Summer 2019 I worked with Liam Mazurowski and Dr. Andre Neves as part of UChicago's 2019 REU. I learned about stable minimal surfaces, including the variational formulas, log-cutoff trick, and Sharp's compactness theorem. At the end, I gave a proof of a more basic compactness theorem using small curvature estimates.

From August 2018 to May 2020 I worked on several directed reading projects under the guidance of Dr. Luca Di Cerbo. I first learned basic differential geometry of curves and surfaces, then learned substantial Riemannian geometry. My senior thesis involved calculating variations of Christoffel symbols, components of the Riemann curvature tensor, and the volume element.

From January 2017 to May 2020 I worked in Dr. Alexander Angerhofer's physical chemistry lab. I analyzed the electronic structure of Oxalate Decarboxylase (OxDC) using an electron paramagnetic resonance (EPR) spectrometer. Learned to synthesize and purify protein. In studing OxDC, I used a process called redoxcycling to oxidize and reduce a central manganese; the changes in oxidation states can readily be measured using EPR. In addition, I spent a semester learning graduate-level EPR theory.

Talks:

1. Minimal Surfaces and the Variational Formulas, Lecture for UF's graduate Riemannian geometry course, April 2020.
2. Topological Groups, Lecture for UF's graduate topology course, April 2020.
3. Minimal Surfaces and Min-Max Theory, UChicago REU, Aug. 2019.