duality, characterised by a number called the CY dimension. Much work

has been carried out on understanding positive CY triangulated categories, especially in the

context of cluster-tilting theory. Even though CY dimension is usually

considered to be a positive (or fractional) number, there are natural

examples of CY triangulated categories where this”dimension” or parameter is negative,

for example, stable module categories of self-injective algebras. Therefore, negative CY

triangulated categories constitute a class of categories that warrant

further systematic study. In this talk, we will give a brief

survey regarding what is so far known about the structure of negative

CY triangulated categories, and we will focus on an example given by

triangulated categories generated by spherical objects.