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I am reading a book on algebraic geometry and I google some keywords, eventually come up with this post in terry tao blog:


I think I got good intuition on the different thoughts on a group, but not this one:

(6) Dynamic: A group represents the passage of time (or of some other variable(s) of motion or action) on a (reversible) dynamical system.

Can anyone explain to me how a group represents the passage of time? I can only think of Noether's theorem on conservation law when combining the concept of time and algebra.

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I'm not sure if that's what he meant, but (the action of) the semigroup of positive reals is often used to denote some kind of passage of time, like in the case of the heat semigroups. You could have the negative reals too if you allowed reversal of time, though that wouldn't be quite as well defined as in the case of usual heat semigroup. ;) –  tomasz Jan 4 at 16:17
Googling "dynamic group" brings up lots of relevant stuff. "Dynamic" is the operative modifier here for the connection with time. –  rschwieb Jan 4 at 16:28
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1 Answer

up vote 4 down vote accepted

It's hard to say what Terry meant exactly unless you ask him. Nevertheless, it seems quite probable that he is talking about flows on manifolds (or more general kind of spaces but let's stick with this).

A reversible flow on a manifold $M$ is a family of diffeomorphisms $\phi_t:M \to M, t \in {\bf R}$ that obey the rule of composition $\phi_t \circ \phi_s = \phi_{s+t}$. In other words, flow is an action of the group $({\bf R}, +)$ on the manifold. That this describes dynamics is immediate: for any $x \in M$, $\phi_t(x)$ is a curve that shows how $x$ moves under the flow.

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OK. We think the action of a group is to make the time go forward or backward. I think get this point. If we think of a group (as passage of time) like this, then the time defined by a group may not be a straight line. What I mean it that the time could be a recurrence of time $\mathbb{Z}/n\mathbb{Z}$ (going forward for some time and eventually I will get back to the origin) or I have different way to go forward (or backward) in time like $\mathbb{D}_4$ –  wonghang Jan 5 at 6:49
I can also define a time passage look like $\mathbb{R}/\mathbb{Z}$ so that when I go forward with time, I will never meet the points I visited. –  wonghang Jan 5 at 7:03
@wonghang: precisely. Another often occurring group is integers that occurs in discrete dynamics. You obtain it by simply taking any map $f$ on a given space and iterating it and its inverse. Although to be fair, most dynamics are not reversible, maps not invertible, so one uses natural numbers instead and moves from groups to monoids or semigroups. But dynamics coming from physics usually are reversible (with the notable exception of heat equation). –  Marek Jan 5 at 11:19
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