By “toric topology”, we mean a branch of mathematics studying various topological spaces with torus symmetries.  One of the central objects in toric topology is the toric variety which has provided a fertile testing ground for general theories in different fields of mathematics such as algebraic geometry, representation theory and combinatorics. Due to their nice torus symmetries, one can expect several nice properties, one of which is the cohomological rigidity conjecture. It asks if the family of smooth toric varieties can be classified by their cohomology rings. In this talk, we will look at this conjecture in the context of singular toric varieties and introduce several recent works on the topology of singular toric varieties.
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