게 겹치다

overlap, coincide

An adverb in "-게" plus 겹치다, "overlap, lie on top of one another". Physically it is things stacked — everyone puts a hand forward and they stack on top of each other. More often here it is timing or territory coinciding, and the adverb marks how uncanny the coincidence is: 미묘하게 subtly, 기가 막히게 astonishingly, 절묘하게 exquisitely, 우연치 않게 not by chance. Two scheduling lines use it for a clash of dates, which is the everyday use.