Adamantly is how you do something when you will not change your mind. It is the adverb that belongs to adamant. “He adamantly refused.” “They adamantly denied it.” It loves verbs like refuse, deny, oppose, insist — verbs where someone is holding a line. It is a little formal; in easy talk people more often say “she just wouldn’t change her mind” or “he said no, and he meant it.” Though actually, “adamantly agree” sounds odd to careful ears. If you already agree, there is nothing to stand like a stone against. The word wants resistance: a no, a denial, a wall. Students sometimes clip -ly onto a strong adjective and then use it the way they use “very.” Save this one for the moment someone will not give way.