Mathematics Colloquium

Abstract: Given quasi-homogeneous polynomial $W$,  we can study it in two different areas of mathematics. Namely, we can set $W=0$ to define a hypersurface $X_W$ of Weight projective space or we can compute its Jacobian ring $C[x_1, \cdots, x_n]/\partial W$. The later is the subject of singularity theory or Landau-Ginzburg model. An old theorem said that the middle cohomology of $X_W$ can be computed using Jacobian ring. Motivated by physics, we can attach a range of invariants to $X_W$ as well as the Landau-Ginzburg side of $W$. The effort to connect two subject leads to Landau-Ginzburg/Calabi-Yau correspondence, a famous duality from physics. In the talk, I will survey some of developments about this duality.