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This is a talk on my personal experience on research for students in Tsinghua High School.

Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosov’s deformation quantization of a symplectic manifold X and the Batalin-Vilkovisky (BV) quantization of a one-dimensional sigma model with target X. This model is a quantum field theory of AKSZ type and is quantized rigorously using Costello’s homotopic theory of effective renormalization. We show that Fedosov’s Abelian connections on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV integration produces a natural trace map on the deformation quantized algebra. This formulation allows us to exploit a (rigorous) localization argument in quantum field theory to deduce the algebraic index theorem via semi-classical analysis, i.e., one-loop Feynman diagram computations.

Discontinuous Galerkin (DG) methods combine features in finite element methods (weak formulation, finite dimensional solution and test function spaces) and in finite volume methods (numerical fluxes, nonlinear limiters) and are particularly suitable for simulating convection dominated problems, such as linear and nonlinear waves including shock waves. In this article we will give a brief survey of DG methods, emphasizing their applications in computational fluid dynamics (CFD). We will discuss essential ingredients and properties of DG methods, and will also give a few examples of recent developments of DG methods which have facilitated their applications in CFD.