Summer Program

Geometrical Singularities and Singular Geometries

"Singularity is almost invariably a clue. The more and featureless and commonplace a "crime" is, the more difficult it is to bring it home." Sherlock Holmes

Description:

The interplay between geometry and physics in the material world is a rich source of questions close to experience, amenable to experiments and open to a range of theoretical approaches, from scaling and asymptotics to computation and geometric/topological methods. The breakup of a drop or the coalescence of two, the collapse of a bubble or the birth of one, the crumpling and wrinkling of a sheet of paper or of the continental crust, the tearing of a sheet of plastic or plasma, the pinching of a tube of myelin or that of a magnetic flux tube, are all examples of extreme events that involve the strongly inhomogeneous interactions (in space and time) between length and time scales that are are normally well separated. Almost all of these events can also be observed and quantified experimentally without too much of a fuss.

Each of these examples has a strong geometrical signature associated with a singular or near singular event. Both the pinching of a drop or the internal tearing of a sheet of plastic or plasma involve a change in topology that accompanies a divergent curvature, while the crumpling of paper involves the formation of nearly singular geometrical structures that accompany the focusing of stress and energy. In all these examples, localized low-dimensional geometric structures influence and are influenced by the behavior in the bulk, and their structure and evolution may aptly be termed the study of geometric singularities and singular geometries.

A two week workshop at the IMA will focus on what we know and what we would like to know about these types of singular structures; with the first week focusing on fluid singularities, and the second on elastic singularities. We also hope to have an intense mid-week mini-tutorial on singularities in geometric field theories: examples from gravitation to condensed matter.

LIST OF CONFIRMED PARTICIPANTS