Derived Moduli Spaces of Nonlinear PDEs I: Singular Propagations
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917695708987392 |
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| author | Kryczka, Jacob Sheshmani, Artan Yau, Shing-Tung |
| author_facet | Kryczka, Jacob Sheshmani, Artan Yau, Shing-Tung |
| contents | 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 $\mathcal{L}X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05226 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Derived Moduli Spaces of Nonlinear PDEs I: Singular Propagations Kryczka, Jacob Sheshmani, Artan Yau, Shing-Tung Algebraic Geometry Differential Geometry Primary: 14A20, 14A30, 35A27, 58A15, 58A20, Secondary: 18N65, 53D30 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 $\mathcal{L}X$. |
| title | Derived Moduli Spaces of Nonlinear PDEs I: Singular Propagations |
| topic | Algebraic Geometry Differential Geometry Primary: 14A20, 14A30, 35A27, 58A15, 58A20, Secondary: 18N65, 53D30 |
| url | https://arxiv.org/abs/2312.05226 |