Understanding singularity formation is an important topic in the study of geometric flows. Since Gage-Hamilton-Grayson’s foundational results, it has largely been unknown how singularities of curve shortening flow form in higher codimensions. In this talk, I will present my recent results that in n dim Euclidean space, any curve with a one-to-one convex projection onto some 2-plane develops a Type I singularity and becomes asymptotically circular under curve shortening flow. As a corollary, an analog of Huisken’s conjecture for curve shortening flow is confirmed, in the sense that any closed immersed curve in n dim Euclidean space can be perturbed in n+2 dim Euclidean space to a closed immersed curve which shrinks to a round point under curve shortening flow.

The Gross–Zagier formula relates the first derivative of the L-function for PGL(2) to (arithmetic) intersection numbers on the modular curve. It plays a central role in proving known cases of the Birch–Swinnerton-Dyer conjecture. In their celebrated work, Yun and Zhang established a function-field analogue of this formula, replacing the modular curve by moduli spaces of PGL(2)-Shtukas. New phenomena arise in this setting: higher derivatives of L-functions can also be expressed in terms of intersection numbers. However, due to the lack of properness of moduli spaces of Shtukas in higher-dimensional cases, extending this formula to higher dimensions has remained open for many years.

In this talk, I will present a higher-dimensional version of the Yun–Zhang formula. The proof uses tools from Geometric Langlands theory and aspects of the relative Langlands duality proposed by Ben-Zvi, Sakellaridis, and Venkatesh. This is joint work with Shurui Liu.