I will first give an introduction to front propagations, Hamilton-Jacobi equations, and homogenization theory. I will then describe some recent progress in periodic homogenization of Hamilton-Jacobi equations. We show that the optimal rate of convergence is $O(\varepsilon)$ in the convex setting. We then give a minimalistic explanation that the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts in two dimensions (if time permits). Joint works with Wenjia Jing and Yifeng Yu.
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