In a small bowl sprinkle the yeast over 1 cup warm water and let it proof for 5 minutes, or until it is foamy.
In a large bowl combine well 4 cups of the flour with the mashed potatoes and the salt until the mixture resembles coarse meal, add the yeast mixture, and stir the dough until it is combined well.
Turn the dough out onto a floured surface and knead it, incorporating as much of the remaining 1/2 cup flour as necessary to prevent it from sticking, for 8 to 10 minutes, or until it is smooth and elastic.
Form the dough into a ball, put it in an oiled bowl, and turn it to coat it with the oil.
Let the dough rise, covered with plastic wrap, in a warm place for 1 1/2 hours, or until it is double in bulk.
While the dough is rising, in a small bowl stir together the garlic, the rosemary, and the oil and let the mixture stand, covered.
Turn the dough out into a well-oiled 15 1/2- by 10 1/2-inch jelly-roll pan, press it evenly into the pan, and let it rise, covered loosely, in a warm place for 45 minutes or less, or until it is almost double in bulk.
Using a mandoline or hand-held slicer cut the red potatoes into paper-thin slices, arrange the slices on the dough, overlapping them, and brush them with the oil mixture, discarding the garlic.
Sprinkle the focaccia with salt and pepper to taste and bake it in the bottom third of a preheated 400°F oven for 40 to 50 minutes, or until it is golden.
Let the focaccia cool in the pan on a rack and serve it warm or at room temperature.