Talk:Problems on Fredholm sections of polyfold bundles, perturbations, and implicit function theorems

My original code for the solution to the linearization problem included a drop-down box with the solution but, for some reason, the polyfold wiki doesn't want to display it, even though the code works just fine on the mediawiki sandbox page. The original code is as follows if anybody wants to figure out how to make this work:

<div class="NavFrame"> <div class="NavHead">'''Solution'''</div> <div class="NavContent" style="display: none;" align="left"> Note that, since <math>u(s,t)\equiv p</math>, we have that <math>\mathcal{E}_u\cong H^2(T^2,T_p\mathbb{R}^{2n})\cong H^2(T^2,\mathbb{R}^{2n})</math> as well as <math>\partial_su=0</math> and <math>\partial_tu=0</math> so the linearization of <math>\sigma:u\mapsto(u,\overline{\partial}_Ju)</math> is given by <div style="text-align: center;"> <math>\begin{align} D_u\overline{\partial}_J:&\xi\mapsto\left.\frac{d}{d\varepsilon}\overline{\partial}_J(u+\varepsilon\xi)\right|_{\varepsilon=0}\\&=\left.\frac{d}{d\varepsilon}(\partial_s(u+\varepsilon\xi)+J(u+\varepsilon\xi)\partial_t(u+\varepsilon\xi))\right|_{\varepsilon=0}\\&=\left.\frac{d}{d\varepsilon}((\varepsilon\partial_s\xi)+\varepsilon J(u+\varepsilon\xi)\partial_t\xi)\right|_{\varepsilon=0}\\&=\partial_s\xi+\left.\frac{d}{d\varepsilon}(\varepsilon J(u+\varepsilon\xi)\partial_t\xi)\right|_{\varepsilon=0}\\&=\partial_s\xi+\left.\left(J(u+\varepsilon\xi)\partial_t\xi+\varepsilon\frac{d}{d\varepsilon}(J(u+\varepsilon\xi)\partial_t\xi)\right)\right|_{\varepsilon=0}\\&=\partial_s\xi+J(u)\partial_t\xi=(\partial_s+J(u)\partial_t)\xi \end{align}</math> </div> which is to say that <math>D_u\overline{\partial}_J=\partial_s+J\partial_t</math>, where <math>J=J(u)</math> is constant. In other words, the linearization of <math>\sigma</math> at a constant map <math>u(s,t)\equiv p</math> is a Cauchy-Riemann operator. It then suffices to show that <math>D_u\overline{\partial}_J</math> is <math>sc</math>-Fredholm but such Cauchy-Riemann operators are Fredholm as maps <math>\mathcal{B}_0\to\left(\mathcal{E}_u\right)_0</math>, regularizing by elliptic regularity, and bounded at each level, all from the standard theory. Thus, by Lemma 3.6 of [//arxiv.org/abs/1209.4040?title= Fredholm notions in scale calculus and Hamiltonian Floer theory], by Katrin Wehrheim, <math>D_u\overline{\partial}_J</math> is <math>sc</math>-Fredholm. </div> </div>