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From the article: > The three Laureates’ use of topological concepts in physics was decisive for their discoveries. Topology is a branch of mathematics that de
by westoncb 10y ago
From the article:
> The three Laureates’ use of topological concepts in physics was decisive for their discoveries. Topology is a branch of mathematics that describes properties that only change step-wise.
I've never heard topology characterized that way. Typically I think of it as dealing with connection properties—sort of a more abstract geometry where objects can be considered the same even if their shape changes; identity is only defined by which parts are connected to which.
Could anyone elaborate on the 'step-wise' change aspect from the article?
- weerd 10y agoThat sounded odd to me also. Maybe they are referring to a tear/cut that would change an object's topology. Like taking a bite from a donut changes it from a torus to a sphere. That break is not a continuous change, more "step-wise".
- miles7 10y agoIn the context of physics, topological phases of matter often reveal themselves by causing certain quantities that are usually continuous to change in discrete steps instead. A key example is the electrical conductivity of certain quantum Hall effect systems, which jumps in discrete steps (is "quantized" in physics jargon) as an ever stronger magnetic field is applied to the system in a transverse direction. The connection to the donut/bagel thing more familiar in topology is that topology also deals with integers: a manifold can have 0,1,2,3... holes in it, that sort of thing. One key mathematical quantity that comes up in the works of Thouless and Haldane is the Chern number https://en.wikipedia.org/wiki/Chern_class https://en.wikipedia.org/wiki/Chern_class The reason the physics quantities jump is that the Chern numbers of certain electron bands change, and by definition a Chern number is an integer.