Inspired by the LG/CY correspondence, we study the local index theory of the Schrodinger operator associated to a singularity defined on C-n by a quasi-homogeneous polynomial f. Under some mild assumption to f, we show that the small time heat kernel expansion of the corresponding Schrodinger operator exists and is a series of fractional powers of time t. Then we prove a local index formula which expresses the Milnor number of f by a Gaussian type integration. The heat kernel expansion provides the spectral invariants of f. Furthermore, we can define the torsion type invariants associated to a homogeneous singularity. The spectral invariants provide another way to classify the singularity.
 H. Fan, and H. Fang, Torsion type invariants of singularities, Vietnam Journal of Mathematics (celebrate Juergen Jost's 65 birthday), Published online,50 pages, 2021.
 H. Fan, T. Lan and Z. Yang, LG/CY Correspondence Between tt*Geometries, Comm. Math. Research, Vol.37,No.3,297-349, 2021.
Prof. Fan Huijun