Person: Hopkins, Michael
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Publication Comparing Dualities in the K(n)-local Category(Cambridge University Press, 2021-10-31) Goerss, Paul G.; Hopkins, MichaelIn their work on the period map and the dualising sheaf for Lubin–Tate space, Gross and the second author wrote down an equivalence between the Spanier–Whitehead and Brown–Comenetz duals of certain type n-complexes in the K(n)-local category at large primes. In the culture of the time, these results were accessible to educated readers, but this seems no longer to be the case; therefore, in this note we give the details. Because we are at large primes, the key result is algebraic: in the Picard group of Lubin–Tate space, two important invertible sheaves become isomorphic modulo p.Publication Topological Quantum Field Theories from Compact Lie Groups(American Mathematical Society, 2010) Freed, Daniel; Hopkins, Michael; Lurie, Jacob; Teleman, ConstantinIt is a long-standing question to extend the definition of 3-dimensional Chern-Simons theory to one which associates values to 1-manifolds with boundary and to 0-manifolds. We provide a solution in case the gauge group is a torus. We also develop from different points of view an associated 4-dimensional invertible topological field theory which encodes the anomaly of Chern-Simons. Finite gauge groups are also revisited, and we describe a theory of "finite path integrals" as a general construction for a certain class of finite topological field theories. Topological pure gauge theories in lower dimension are presented as a warm-up.Publication Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant(American Mathematical Society (AMS), 2022-02-23) Lin, Jianfeng; Shi, XiaoLin Danny; Xu, Zhouli; Michael, Hopkins; Hopkins, MichaelIn studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin ( 2 ) \operatorname {Pin}(2) -equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the Pin ( 2 ) \operatorname {Pin}(2) -equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem.
We prove our theorem by analyzing maps between certain finite spectra arising from B Pin ( 2 ) B\operatorname {Pin}(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the j j -based Atiyah–Hirzebruch spectral sequence.