A doughnut-shaped surface, perhaps with more than one hole - or none at all, can be cut into triangles, and if we record how the cuts were done, we can put the surface back together by gluing. Alternatively, we can create a dual surface, where we make new triangles with a vertex for each of the original ones, and sides corresponding to the touching data of the original triangles. Poincaré's amazing theorem from 1895 tells us that the dual shape is deformable to the original, regardless of the way cuts were done. Since its discovery, this duality result has been improved on and generalized in many different areas of mathematics. In homotopy theory one not only studies objects up to deformations, but also keeps track of deformations between them as well as coherence data, all in a streamlined way. Along with its recent augmentation into derived algebraic geometry, homotopy theory has become a unifying ground for numerous mathematical concepts, including duality.
The PI will work with her collaborators to explore two duality contexts in which homotopical and arithmetic information are intertwined. One of those involves establishing a homotopical extension of a classical result of Poitou and Tate about duality in the cohomology of number fields, as well as investigating the implications of such an extension to questions in arithmetic. The other involves understanding duality for some of the basic objects in so-called chromatic homotopy theory, whereby one organizes structural and computational information in homotopy according to periodicity properties. This would be a homotopical enhancement of a cohomological duality property of the chromatic Galois groups known as Morava stabilizer groups, and is based on seminal work of Gross and Hopkins. Both of these project goals may involve further developing the foundations for profinite group actions on profinite objects.
This 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.
