純粋・応用数学・数学隣接分野（含むガロア理論）19

[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）19 (1002ﾚｽ)
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717: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/03/29(土) 15:40:29.16 ID:PbaCEUaz

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https://www.nobelprize.org/prizes/physics/2016/summary/
Nobel Prize in Physics 2016
The Nobel Prize in Physics 2016 was awarded with one half to David J. Thouless, and the other half to F. Duncan M. Haldane and J. Michael Kosterlitz "for theoretical discoveries of topological phase transitions and topological phases of matter"

https://mathoverflow.net/questions/251470/topology-and-the-2016-nobel-prize-in-physics
Topology and the 2016 Nobel Prize in Physics
I was very happy to learn that the work which led to the award of the 2016 Nobel Prize in Physics (shared between David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz) uses Topology. In particular, the prize was awarded "for theoretical discoveries of topological phase transitions and topological phases of matter".
Which topological concepts and results are involved in the work which led to the award of the 2016 Nobel Prize in Physics?
And a follow-up question:
Where should a topologist go to read about topological phases of matter and topological phase transitions?
edited Jun 15, 2020 at 7:27
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Mark Grant

<21> answer
In modern applications, an important role is played by the (N-dimensional and thus finite dimensional) projector the subspace of Hilbert space spanned by the eigenfunctions corresponding to he N lowest eigenvalues, again fibered over the Brillouin zone. Then one can use K-theory (and KO-theory in fact) related to this projector to classify the possible classes of Fermi surfaces (these are the "topological phases of matter", as eventually, when the perturbation becomes too strong even the discrete invariants can jump which then physically corresponds to a phase transition).
edited Oct 10, 2016 at 7:45
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http://rio2016.5ch.net/test/read.cgi/math/1725190538/717

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