Workshops: Homotopy Harnessing Higher Structures

OVERVIEW: This is a proposal seeking funding to support the travel and local expenses of early career US mathematicians to attend four workshops to be held in the second half of 2018 at the the Isaac Newton Institute for Mathematical Sciences in Cambridge, England. These workshops area associated with the semester program "Homotopy Harnessing Higher Structures". The funds will be used exclusively for travel support junior US based mathematicians who do not have funds from other sources to attend the events. Housing, local support, workshops speakers, and basic infrastructure will all be supplied by the Newton Institute. INTELLECTUAL MERIT: The last fifteen years have been a renaissance for algebraic topology. Old problems have been solved using new methods, new methods led to new ideas, new ideas to new problems, and new problems to new theorems. The result is a rebirth: established mathematicians have reinvented their research programs and there is an entire cohort of new researchers taking the field in new directions. In retrospect, we can see that this fundamental change occurred as the mathematicians in the field confronted and solved an array of related problems. In each case, there was enormous progress after the introduction and the study of higher homotopical structure. It wasn't enough to know some structure up to homotopy--we needed higher order information, such as knowledge of the homotopy type of the space of all possible solutions to some problem. In some sense this idea goes back to Kan and Quillen, but in this period we internalized and applied the methods, and the field was transformed. The program at the Newton Institute will assess what has been done, highlight what is working well, to develop the field, to give instructional and research workshops, and to provide a meeting place for researchers at all levels of the various branches of the field. There will be mathematicians in residence, but the larger community will be incorporated through four workshops of different emphases and flavors. It makes particular sense to hold this program and these workshops at the Newton Institute. This is an international field, with a balance of leading mathematicians across many countries. Cambridge is the perfect Crossroads to reinforce the essential networks of connections with the international research world. BROADER IMPACTS: We are requesting National Science Foundation funding solely to support the participation of junior research mathematicians from the United States, meaning graduate students, postdoctoral fellows, and junior faculty without other significant support. The direct impact of NSF funding will be the training and career development of up to 50 junior researchers, who will gain the opportunity to participate in a major program in Europe and, in particular, at the Newton Institute, which is a major international nexus in mathematics. A secondary impact is to further develop collaboration between emerging research groups in algebraic topology and derived algebraic geometry in the US and Europe, in particular the United Kingdom.