Frobenius and spherical codomains and neighbourhoods

Abstract:
In this talk I will report on a joint work with Ciaran Meachan.
Given an exact functor between triangulated categories, one can try to compare its left and right adjoint. In some special cases, it is possible to enforce that both adjoints are equal (i.e. the functor becomes Frobenius) or differ by an autoequivalence (i.e. the functor becomes spherical or quasi-Frobenius).
I will present these constructions, which lead to the codomains and neighbourhoods of the title, and give also some examples of algebro-geometric origin.