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Recently, Mahesh Kakde and I proved a conjectural formula for Brumer-Stark units in terms of specializations of the Eisenstein cocycle. The mechanism of the proof involves relating the formula to a conjecture of Gross on the behavior of Brumer-stark units under the Artin reciprocity map in towers of number fields. A key idea is to expand the auxiliary set S used in the construction. This technique, which often goes by the name of "horizontal Iwasawa theory," was introduced by Wiles and used to great effect by Taylor-Wiles. We will conclude by describing a plan for future research with Kakde and Spiess in which we hope to prove my conjectural formula with Spiess for the characteristic polynomial of Gross' regulator matrix for CM fields using horizontal methods.