Online Talk 

We will see the definition and some examples of piecewise toric varieties, then show how they can be interpreted as quotients of certain polyhedral products. Next, we look at how their geometry can be studied via complexes of sheaves on an associated complex of polytopes.  This yields results such as a version of the Decomposition Theorem and identifying the weight and Leray filtrations on the cohomology groups of a piecewise toric variety.