I'm posting for a friend in the interest of anonymity.
For context, they're in the third year of a US mathematics PhD, and are currently without an advisor following the failure of an 18 month academic relationship with a highly prominent senior faculty member. What can be said of their academic relationship was that it often consisted of generally unhealthy behaviours (e.g. frequent extended yelling) from the advisor. The decision to terminate the relationship was taken by the advisor, and the official position is that the cause of its failure was personal incompatibility and diverging research interests. The advisor officially is available to provide strong recommendation letters to potential alternative advisors.
My friend intends to remain in the PhD program, but the problem is that there are no other professors with a similar research interest at their institution. They currently have a new internal co-advisor, and the department is generally on board with supporting an external arrangement. The alternative would be to start from scratch in a radically different research area at a somewhat late stage in their degree.
The question is, how should they go about cold-emailing potential advisors from other universities? Their interim advisor has suggested that the most important aspect of such an email would be addressing the double red flags associated with having an previous academic relationship fail and their stated research incompatibility with any of the many other members of their department. My friend believes that it is necessary to clearly address the situation (in neutral terms without negative language) in such an email, and that not accounting for their work over the past 18 months is not an option.