### Abstract: Breakthrough in Contact and Symplectic Geometry by Tsinghua University and Collaborators On June 12, 2024, Tsinghua University's Yau Mathematical Sciences Center (YMSC) announced a significant breakthrough in the fields of contact and symplectic geometry. Assistant Professor Honghao Gao, in collaboration with Roger Casals from the University of California, Davis, has made groundbreaking progress on the classification of Lagrangian fillings, a critical problem in low-dimensional symplectic geometry. **Background:** The classification of Lagrangian fillings for Legendrian links has been a focal point in symplectic geometry. In 1996, Yakov Eliashberg and Leonid Polterovich provided the first complete classification for Legendrian trivial knots, setting a foundational result in this area. Since then, mathematicians have made various advancements, constructing numerous examples of Lagrangian fillings. A prevailing conjecture was that the number of Lagrangian fillings for a fixed Legendrian link is always finite. However, in 2022, Gao and Casals disproved this conjecture, demonstrating that a large class of Legendrian links can admit infinitely many Lagrangian fillings. Their work, titled "Infinitely Many Lagrangian Fillings," was published in the first issue of the *Annals of Mathematics* in 2022. **Recent Developments:** In their latest research, Gao and Casals have further explored the relationship between Lagrangian fillings and cluster algebra seeds. They have established a surjective correspondence, showing that every cluster algebra seed can be induced by a Lagrangian filling. This achievement is a crucial step towards the complete classification of Lagrangian fillings and significantly enhances the understanding of the geometric properties of low-dimensional symplectic manifolds. **Methodology:** The team's approach involves aligning algebraic cluster mutations with geometric Lagrangian surgeries. While algebraic operations can be performed freely, repeated geometric operations without constraints often lead to the creation of immersion points, which halt the process. To circumvent this issue, the researchers introduced a potential function on quivers, which tracks the intersections during geometric operations. By applying appropriate Hamiltonian isotopy transformations, they managed to avoid the formation of immersion points, thereby ensuring a one-to-one correspondence between algebraic and geometric operations. **Implications:** This work not only resolves a long-standing conjecture but also deepens the connection between symplectic geometry and cluster algebra. The ability to map each cluster algebra seed to a Lagrangian filling provides a powerful tool for classifying these fillings and understanding the underlying symplectic structures. The findings have been published in the May 2024 issue of *Inventiones Mathematicae* under the title "A Lagrangian Filling for Every Cluster Seed." **References:** - Gao, H., & Casals, R. (2022). Infinitely many Lagrangian fillings. *Annals of Mathematics*, 195(1), 207-282. - Gao, H., & Casals, R. (2024). A Lagrangian filling for every cluster seed. *Inventiones Mathematicae*, 240(1), 1-68. **Links:** - [Paper: A Lagrangian filling for every cluster seed](https://link.springer.com/article/10.1007/s00222-024-01268-y) **Contributors:** - **Honghao Gao**: Assistant Professor at the Yau Mathematical Sciences Center, Tsinghua University. - **Roger Casals**: Professor at the University of California, Davis. **Editorial Team:** - **Contributor**: Yau Mathematical Sciences Center - **Editor**: Huashan Li - **Reviewer**: Ling Guo This breakthrough by Gao and Casals represents a significant leap forward in the study of contact and symplectic geometry, offering new insights and methodologies that are likely to influence future research in this field.