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The minimal width of deep neural networks that can approximate all continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ in the topology of uniform convergence on compact subsets is investigated, suggesting an intriguing connection between the approximation theory of neural networks and differential topology.
For continuous viscosity solutions $u$ of fully nonlinear uniformly elliptic equations $F(D^2 u)=0$, the work of Nadirashvili--Tkachev--Vl\u{a}du\c{t} shows that $u$ need not be $C^2$ in dimensions $n\ge 5$. It is therefore natural to ask how large the set $\operatorname{Sing}(u)$, consisting of points at which $u$ has...
Nirenberg's theorem gives interior $C^{2,\alpha}$ regularity for continuous viscosity solutions of fully nonlinear uniformly elliptic equations $F(D^2u)=0$ in dimension two. In dimensions five and higher, the work of Nadirashvili--Tkachev--Vl\u{a}du\c{t} shows that such solutions need not be $C^2$. We construct a famil...
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