Milnor fibers and Milnor monodromies are the most basic invariants of the hypersurface singularities.
I will explain the way to study them, with mixed Hodge modules.
The cohomologies of Milnor fibers have canonical mixed Hodge structures.
If a singular point is an isolated singular point, their weight filtrations are the monodromy weight filtrations, which have the data of the Jordan normal forms of the Milnor monodromies.
To compute them, we express the mixed Hodge structures of the cohomologies of the Milnor fibers in terms of nearby cycle sheaves as mixed Hodge modules.
Then, thanks to the power of the functorial properties of the mixed Hodge modules, we can compute the mixed Hodge structures and the Milnor monodromies of the cohomologies of the Milnor fibers.
In this talk, first, I will introduce the definition of the Milnor fibrations, Milnor fibers and Milnor monodromies of hypersurface singular points.
Second, I will briefly explain the basic notions of mixed Hodge structures and mixed Hodge modules.
Finally, I will demonstrate how to compute the mixed Hodge structures and the Milnor monodromies of the cohomologies of Milnor fibers, with nearby cycle sheaves as mixed Hodge modules.