Here assuming is a linking word, not the -ing of a verb in progress. You use it when you take a possible situation as true for a moment, and then say what follows: Assuming it doesn't rain, we'll eat outside. It means “if we treat this as true…” and it often pairs with that (assuming that the train is on time). It is everyday and polite, a softer cousin of if. Speakers use it when the “if” is a working guess, not a wild dream. Though actually, assuming can also be an adjective meaning “too sure of yourself, a bit arrogant” — an assuming manner — but that use is older and rarer; students almost always meet the “if we take this as true” use. Put a comma when the assuming-clause comes first. And notice it does not prove the thing will happen; it only builds the next sentence on that guess.