Jonathan BlockZeng, Qingyun2023-05-222025-01-312022-09-172021-01-012022-09-17https://repository.upenn.edu/handle/20.500.14332/31917We study various problems arising in higher geometry using derived Lie $\infty$-groupoids and algebroids. We construct homotopical algebras for derived Lie $\infty$-groupoids and algebroids and study their homotopy-coherent representations. Then we apply these tools in studying singular foliations and their characteristic classes. Finally, we prove an $A_{\infty}$ de Rham theorem and higher Riemann-Hilbert correspondence for foliated manifolds.228 p.application/pdfQingyun Zengalgebraic geometryalgebraic topologycomplex analysisdifferential geometrydifferential topologynoncommutative geometryMathematicsDissertation/Thesis