Double Bubbles Minimize

The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 120 degrees.Comment: 57 pages, 32 figures. Includes the complete code for a C++ program as described in the article. You can obtain this code by viewing the source of this articl

Morsifications and mutations

We describe and investigate a connection between the topology of isolated singularities of plane curves and the mutation equivalence, in the sense of cluster algebra theory, of the quivers associated with their morsifications.Comment: Major revision and expansion. Several new results. 80 pages, 66 figure

CFT, BCFT, ADE and all that

These pedagogical lectures present some material, classical or more recent, on (Rational) Conformal Field Theories and their general setting ``in the bulk'' or in the presence of a boundary. Two well posed problems are the classification of modular invariant partition functions and the determination of boundary conditions consistent with conformal invariance. It is shown why the two problems are intimately connected and how graphs -ADE Dynkin diagrams and their generalizations- appear in a natural way.Comment: Lectures at Bariloche, Argentina, January 2000. 36 pages, 4 figure