강의일시: 2026년 6월 12일 14시~16시

장소: 129동 104호

Title: Cohomology of contact loci and the arc-Floer conjecture 

Speaker: Quy Thuong LÊ (Vietnam National University)

Abstract: I will discuss an interesting and surprising relationship between symplectic geometry and singularity theory. This was questioned by Seidel, Denef-Loeser, Mclean, and then conjectured by Budur, de Bobadilla, Hong Duc Nguyen, and myself (JDG 2022, recently called the arc-Floer conjecture) that there are isomorphisms between the Floer cohomology of monodromy powers and the singular cohomology of iterated contact loci. 
In this talk, I will provide a brief review of the motivation behind the conjecture. Floer first introduced the Floer cohomology in his proof of the Arnold conjecture in symplectic geometry. In 2019, Mclean proved that the Floer cohomology groups are invariants of links, and that the multiplicity and the log canonical threshold are also invariants of such links. This is done by constructing a spectral sequence converging to the fixed point Floer cohomology whose first page is explicitly described in terms of a log resolution. Using Denef-Loeser’s theory on the geometry of arc spaces, we obtained a stratification of the contact loci, which induces a cohomological spectral sequence, whose first page, up to a shift, coincides with that of McLean's spectral sequence. 
We checked the conjecture is true at the degree being the multiplicity of an isolated hypersurface singularity. Inspired by it, one conjectures that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of their tangent cones are homotopy equivalent. More recently, the arc-Floer conjecture is proved true for plane curve singularities by de la Bodega and de Lorenzo Poza (JDG 2026), and for semi-homogeneous singularities by de Lorenzo Poza and Huang (arxiv 2025).