We will survey recent developments dealing with characterization of absolutely almost simple algebraic groups having the same isomorphism/isogeny classes of maximal tori over the field of definition. These questions arose in the analysis of weakly commensurable Zariski-dense subgroups. While there are definitive results over number fields (which we will briefly review), the theory over general fields is only emerging. We will formulate the existing conjectures, outline their potential applications, and report on recent progress. Joint work with A. Rapinchuk and I. Rapinchuk.
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