29 papers
Formation of trapped surfaces for the spherically symmetric Einstein-Yang-Mills system with non-trivial incoming data
Nikolaos Athanasiou, Puskar Mondal, Shing-Tung Yau
We establish a trapped surface formation theorem for the spherically symmetric Einstein Yang Mills equations in a double-null gauge. The theorem concerns characteristic initial dat…
Volume Stability for Hyperbolic Manifolds and Applications to General Relativity
Puskar Mondal, Shing-Tung Yau
The paper proves that for closed hyperbolic 3‑manifolds, any sequence of metrics with scalar curvature ≥ −6 whose volumes approach the hyperbolic volume must converge in the C⁰ sen…
Statistically Meaningful Geometry (SMG) Beyond the Euclidean Paradigm, with Application to Generative AI
Bing Cheng, Yi-Shuai Niu, Howell Tong +1
Conventional uniform convergence bounds and empirical risk minimization break down in massive over-parameterized models, such as large language transformers and biological sequence…
Semi-Global Existence and Trapped Surface Formation for the Einstein-Vlasov System
Nikolaos Athanasiou, Puskar Mondal, Shing-Tung Yau
In this article, we study the Einstein-Vlasov system for massless particles. Our main contribution is the provision of a novel method for controlling the Vlasov matter in a double…
Lagrangian correspondences of nonabelian Hodge type and shifted twistor structures
Jacob Kryczka, Yuuji Tanaka, Shing-Tung Yau
Classical nonabelian Hodge theory identifies Dolbeault and de Rham moduli spaces by providing a real-analytic isomorphism. In this paper, motivated by the Kapustin--Witten theory,…
Derived Moduli Spaces of Nonlinear PDEs: Singular Propagations
Jacob Kryczka, Artan Sheshmani, Shing-Tung Yau
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 micro…