For simple syrup: Put sugar and 2 tsp. water in a glass bowl and microwave on full power for 10 to 15 seconds, or until sugar is dissolved.
Drop blackberries, pineapple, and mint leaves into a sturdy glass and crush to a pulp with the end of a wooden spoon. Add the simple syrup, lime juice, and rum.
Pour the contents into an old-fashioned glass and then back into first glass; repeat until thoroughly mixed, then pour back into old-fashioned glass. Add ice cubes and seltzer; stir. Garnish with sprig of mint and lime wedge.