A single round soap bubble is the least-area way to enclose a given volume of air, as was proved in 1884 by Schwarz. A double soap bubble is the least-area way to enclose and separate two given volumes of air, as was proved in 2000 as the culmination of a decade of work by many, including Williams faculty and students. Because it is hard to control ahead of time the complicated ways ("singularities") in which pieces of soap film theoretically might come together, the study of such physical problems had to wait for the development of a more general and inclusive kind of geometry, now known as Geometric Measure Theory. (These same tools can be applied to all kinds of singularities from fractures in materials to black holes in the universe.) Evaluation will be based primarily on classwork, homework, and exams. Prerequisites: Mathematics 211 and either Mathematics 301 or 305.