Advances in Combinatorial and Geometric Rigidity (15w5114)
Arriving in Banff, Alberta Sunday, July 12 and departing Friday July 17, 2015
OBJECTIVESA series of gatherings of people in rigidity, including three previous BIRS 5 day workshops, and two companion 2 day BIRS workshops have played essential roles in the development of a collaborative research community with ongoing exchanges, sharing of methods, conceptual development, and results. Each such gathering has generated new collaborations, new conjectures, and new results, while also consolidating other recent work into a form that invited new people to apply and extend the refined methods and results, and solve new problems. Our primary objective is to continue this trajectory. The workshop discussions will connect to the five threads above, while leaving space of new areas to emerge in the next 15-20 months. These threads have been highlighted to support the following larger objectives:- Maintain the momentum from recent contributions to the field, either in solving a long open problem (e.g., the characterization of global rigidity as a rank condition, by Gortler, Healy and Thurston  and subsequent activity in global and universal rigidity; or the solution to the so-called molecular conjecture by Katoh and Tanigawa ; or in opening a fresh new direction, with extended methods (e.g. the series of results on rigidity in the presence of symmetry and periodicity).- Strengthening effective connections between and within the 5 topics. There is significant overlap among the five threads mentioned above, visible in the text above and within papers that incorpo- rate multiple threads. In addition, methods from one area, such as 1, are applicable to themes such as 2,3,4. As new sets of constraints are introduced, the community searches for associated decompo- sitions, inductive constructions, algorithms. When infinitesimal rigidity has been characterized the questions of global and universal rigidity are also posed. In turn, results and methods from these extensions give insight into still unsolved problems in other areas. Typically each of the invited researchers works on at least 3 out of the 5 topics. The area is ripe for a broader and deeper theory and unification of techniques. - Introduce new problems and associated new entrants who bring new perspectives and methods to the larger community. Over the last 6 years, the work on global rigid- ity (e.g. semidefinite programming), on symmetry (representation theory), on periodic structures (group actions), and most recently on infinite frameworks (operator theory), has brought in new people and new methods. We anticipate this workshop will bring in one or two new connections and associated communities.- Provide a supportive environment for new collaborations to develop, and new problems to be posed and shared with the larger community. In particular, provide support to younger researchers (a recent influx) to establish collaborations, and leadership in the area, and further widen the international base for the work.TIMELINESSThe subject is expanding quickly, and will evolve further between this proposal and a possible 2015 workshop. Key to the prior successful workshops of this group at BIRS has been the flexibility to focus on topics and new results that are on the participants minds as we gathered. At the last workshop, it was particularly successful to have all participants give a short (5 minute) talk the first day about a problem or question they want to work on and discuss. Within hours, people were connecting over meals to talk about ways new perspectives and alternative methods could help address the questions raised. As a companion to these short talks, several key survey talks each giving an overview of a key area (as highlighted in the first point above) provided a common ground for ongoing work, including posing of new synthesizing questions and debates about refined definitions and methods. This has proven effective in ensuring that the talks, group discussions, and working sessions are timely, and have ongoing impact.