Michael J. Hopkins
Harvard University
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Primary Section: 11, Mathematics Membership Type:
Member
(elected 2010)
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Research Interests
As a mathematician I am interested in the way the conceptual framework of algebraic systems applies to problems in geometry, homotopy theory, number theory and mathematical physics. The main focus of my research is in the area of homotopy theory, which investigates the measurements of "shape" one can make, that are insensitive to continuous deformation of the underlying geometric structure. I have studied the theory of "nilpotence" in homotopy theory, and the relationship it creates between homotopy theory and algebraic geometry. Building on that my co-workers and I developed the theory of "topological modular forms" connecting the 19th theory of elliptic functions with the very modern abstractions of algebraic topology. Most recently I was part of a team that solved the longstanding "Kervaire invariant" problem, which describes obstructions to cutting up one shape and assembling the pieces into another.
