2023/12/08 by Kryczka, Jacob, Sheshmani, Artan, Yau, Shing-Tung
#14A30 #35A27 #53D30 #58A15 #58A20 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Primary: 14A20 #Secondary: 18N65
paper · doi:10.48550/arxiv.2312.05226
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 S1-equivariant derived loop stack LX.