This is a proposal to provide travel support for U.S. participants in the conference "Cube complexes and groups", which will be held at the University of Copenhagen in Copenhagen, Denmark from 7-11 July 2014. (More information on the conference can be found at https://sites.google.com/site/cubescph2014/) The conference focuses on an exciting and relatively new collection of mathematical results and techniques, which have recently proved useful in resolving longstanding problems. The conference will feature lectures on current research on these and related topics, and also three short courses aimed at early-career researchers interested in working in this field. Accordingly, this proposal is largely aimed at providing early-career mathematicians and graduate students the means to travel from the U.S. to Denmark to attend the lectures and short courses and to collaborate with their European counterparts.
The conference is on cubical geometry, broadly construed -- the study of nonpositively-curved cube complexes and median spaces and their applications in group theory and low-dimensional topology. The lectures will cover many different aspects of the topic, including its relationship with geometry, topology, and discrete mathematics.
In the early 1980s, the mathematician William Thurston posed a list of questions that mapped out a plan for understanding the whole (infinite and shockingly varied) world of 3-dimensional shapes. The last four of these questions were answered in a dramatic fashion in 2012 by work of Ian Agol (of UC Berkeley) and his collaborators, relying on extensive recent work of Dani Wise (of McGill University) and his collaborators. The key idea was to replace the 3-dimensional object in question (which can be a bizarre and tricky thing) with a much more organized object called a "cube complex". Free lunches being no more common in mathematics than anywhere else, one pays for this extra organization: the cube complex usually has many, many dimensions. To answer Thurston's questions about (possibly wild) 3-dimensional objects, one first reduces them to questions about (less wild, but high-dimensional) cube complexes and then answers these, taking advantage of their highly organized structure. The theory of cube complexes -- so-called "cubical geometry" -- was popularized in the mathematical community by this application to the realization of Thurston's rather radical vision, but it is interesting in its own right and has applications beyond low-dimensional topology (the subfield of mathematics in which one studies things like 3-dimensional shapes). Cube complexes arise readily in any context where some collection of things can be divided in half in numerous different ways. Accordingly, these objects have been studied, under various guises, in many fields. For example, cube complexes have arisen in robotics and even mathematical biology, where they can be used to model the possible evolutionary histories of a species. There are also connections to computer science, especially to the study of systems in which several computations are to run simultaneously in an efficient way. Finally, cubical geometry has important and far-reaching applications in group theory, the mathematical study of symmetry (these are the applications most closely related to low-dimensional topology). Given the recent burst of activity and continued importance of cubical geometry, it is vital that research results in this field be disseminated throughout the relevant parts of the mathematical community, and especially that graduate students and junior researchers be exposed to these new ideas. One important means of accomplishing this goal is by holding mathematical conferences, which bring together researchers with a common interest to attend formal lectures and to have informal discussions. A well-executed mathematics conference contributes to the scientific growth of participants and provides a venue for collaborative work by researchers who are ordinarily separated by considerable geographic distance. The goal of the project that is the subject of this report was to bring a delegation of graduate students and early-career mathematicians from the US to the University of Copenhagen, in Denmark, for one week in July 2014, to participate in a conference on cubical geometry and related ideas. The conference included mini-courses, of 3 lectures each, from established experts in the field (from McGill University, Canada; Oxford University, UK; and the University of Wisconsin). These mini-courses were designed to be accessible to graduate students and to provide useful background for beginning researchers. The conference also featured six more advanced lectures on cutting-edge research from a diverse range of mathematicians from US and European universities. Finally, the conference provided a venue for informal discussions between the participants, including active work on ongoing international research collaborations. The actual expenses of operating the conference, and the facilities, were provided by the University of Copenhagen. National Science Foundation funding was used to pay travel and lodging expenses for the US participants, most of whom were graduate students or postdoctoral researchers with limited funding. NSF funding also paid travel and lodging expenses for the US conference speakers. (The ideas that formed the main themes of this conference, and their relationship to the recent exciting developments in topology, are the subject of an excellent article by Erica Klarreich that appeared in Scientific American and that the interested reader can find here: www.scientificamerican.com/article/getting-into-shapes/)
