Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2312.05226

Derived Moduli Spaces of Nonlinear PDEs: Singular Propagations

Jacob Kryczka·Artan Sheshmani·Shing-Tung Yau
View original source ↗Read source document ↗Export metadata ↓

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 $\mathcal{L}X$.

Explore related subjects

math.AGmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jacob Kryczka, Artan Sheshmani, Shing-Tung Yau. 2026-06-22. Derived Moduli Spaces of Nonlinear PDEs: Singular Propagations. https://arxiv.org/abs/2312.05226

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Shrinking dynamic on multidimensional tropical series

Let $Ω\subset\mathbb R^n$ be a compact convex domain. An $Ω$-tropical series is a nonnegative, concave, integral-slope, piecewise-affine function on $Ω$ that vanishes on $\partialΩ$. For a finite set $P\subsetΩ^\circ$, we study the least such function above prescribed initial data whose corner locus contains $P$. It is obtained by repeatedly applying one-point shrinking operators $G_p$. We prove that every fair order of these operators stabilizes after finitely many nontrivial steps. We also describe an event-driven implementation that records the lowest monomials at each point and updates only affected watcher lists. Finally, we show that, on every compact subset of $Ω^\circ$, the resulting dynamics can be approximated by a finite path whose intermediate tropical hypersurfaces have only mild singularities on that compact set; equivalently, the corresponding local cells of the dual regular subdivision contain no lattice points other than their vertices.

math.AG↗
arXiv2026

A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus

Let $C$ be an algebraically closed perfectoid field over $\mathbb{Q}_p$ with the ring of integer $\mathcal{O}_C$ and the infinitesimal thickening $\Ainf$. Let $\mathfrak X$ be a semi-stable formal scheme over $\mathcal{O}_C$ with a fixed flat lifting $\widetilde{\mathfrak X}$ over $\Ainf$. Let $X$ be the generic fiber of $\mathfrak{X}$ and $\widetilde X$ be its lifting over $\BdRp$ induced by $\widetilde{\mathfrak X}$. Let $\MIC_r(\widetilde X)^{{\rm H}\text{-small}}$ and $\rL\rS_r(X,\BBdRp)^{{\rm H}\text{-small}}$ be the $v$-stacks of rank-$r$ Hitchin-small integrable connections on $X_{\et}$ and $\BBdRp$-local systems on $X_{v}$, respectively. In this paper, we establish an equivalence between these two stacks by introducing a new period sheaf with connection $(\calO\bB_{\dR,\pd}^+,\rd)$ on $X_{v}$.

math.AG↗
arXiv2026

A refinement of the coherence conjecture of Pappas and Rapoport

The coherence conjecture of Pappas and Rapoport, proved by Zhu, asserts the equality of dimensions for the global sections of a line bundle over a spherical Schubert variety in the affine Grassmannian and those of another line bundle over a certain union of Schubert varieties in a partial affine flag variety. We refine this equality of dimensions to an isomorphism of representations. The comparison is established by introducing a parahoric Bruhat-Tits group scheme $\mathcal{G}$ over the affine line, ramified at 0. We further strengthen this comparison by equipping any line bundle on the global Schubert variety of $\mathcal{G}$ with a unique equivariant structure under the global jet group scheme. As an application, we obtain new relations among affine Demazure modules.

math.AG↗