Workshops: Homotopy Harnessing Higher Structures

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This award supports travel for junior research mathematicians from the United States to participate in four workshops to be held at the Isaac Newton Institute for Mathematical Science in Cambridge, England. These workshops, all part of the semester program on Homotopy Harnessing Higher Structures are "Higher Structures in Homotopy Theory" (July 2-6, 2018), "Equivariant and Motivic Homotopy Theory" (August 13-17, 2018), "Derived Algebraic Geometry and Chromatic Homotopy Theory" (September 24-28, 2018), and "Manifolds" (December 3-7, 2018). The last fifteen years have seen 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. In each case, there was enormous progress after the introduction and the study of higher structures. The larger program at the Newton Institute will assess what has been done, highlight what is working well, develop the field, give instructional and research workshops, and 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 the four workshops.This award will be used solely to support the workshop participation of US graduate students, postdoctoral fellows, and junior faculty without other significant support. The direct impact of NSF funding will be the raining and career development of up to 30 junior researchers, who will gain the opportunity to participate in a major program 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. Each of the four workshops has a different emphasis and flavor. The first, on higher structures, is focused on the foundations and applications of foundations, and is expressly aimed at researchers new to the field. The second, on equivariant and motivic homotopy theory, is set to be a major research conference on a vital and extremely active area in the field. The third, on derived algebraic geometry and chromatic homotopy theory, is a blend of an emerging topic and a classical area; younger researchers have been important in this new mixture of fields. Finally, the algebraic topology of manifolds has become an intellectual crossroads for homotopy theory, derived geometry, and mathematical physics; this workshop is intended as cross-disciplinary dialog celebrating a wealth of developments. It is the expressed aim of all of these workshops to incorporate new voices not simply as audience participants, but as speakers as well. The website of the workshops can be found at https://www.newton.ac.uk/event/hhh/workshopsThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.