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arXiv · 2312.05226

Derived Moduli Spaces of Nonlinear PDEs: Singular Propagations

Jacob Kryczka·Artan Sheshmani·Shing-Tung Yau
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Abstract

We construct a sheaf theoretic and derived geometric machinery to study nonlinear partial differential equations and their singular supports. We establish a notion of derived microlocalization for solution spaces of non-linear equations and develop a formalism to pose and solve singular non-linear Cauchy problems globally. Using this approach we estimate the domains of propagation for the solutions of non-linear systems. It is achieved by exploiting the fact that one may greatly enrich and simplify the study of derived non-linear PDEs over a space $X$ by studying its derived linearization which is a module over the sheaf of functions on the $S^1$-equivariant derived loop stack $\mathcal{L}X$.

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Jacob Kryczka, Artan Sheshmani, Shing-Tung Yau. 2023-12-08. Derived Moduli Spaces of Nonlinear PDEs: Singular Propagations. https://arxiv.org/abs/2312.05226

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