Does Applied Researcher reduce Deflect cost?

Yes, Applied Researchers reduces Deflect costs.

Like Vex, Cheerless, Applied Researchers applies a discount to the total cost of spells — "Your spells cost [1] [A] less." Deflect is a mandatory additional cost added to the total cost during finalization (Rule 356.2.a.2), and total cost discounts are applied after all additional costs are summed (Rule 356.4). This means the discount offsets the Deflect Power cost.

The "minimum of [1]" restriction applies only to the Energy component of the discount (Rule 356.4.e), so it doesn't interfere with reducing the Power cost imposed by Deflect. As established in FAQ #9237 and #10609, this is the same interaction that lets Vex reduce Deflect costs — and Ezreal, Prodigy does not, because his discount targets optional additional costs only, not the total cost.