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I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with essential topology, general relativity, Hamilton–Jacobi equation, and see why in integrable systems the motion takes place on an N torus and in general fills the surface. Also, know that even the reduced three-body problem could have a chaotic, highly irregular dynamical evolution.

Referring to what is reported in the news you can see that specifically one of her discoveries show that even the potentially chaotic dynamic of the three-body problem follows some deeply geometric laws. What we consider unpredictable, in the interaction of Sun, Moon, and Earth, is still to some extent predictable.

Is it possible for you to explain this restriction on phase space using what I now know and understand, I mean without moduli spaces and ergodic theory?

In addition I greatly appreciate it if you please suggest good and accessible resources for a beginner in moduli spaces and ergodic theory assuming physics level not pure mathematics understanding of the essential topology often used in general relativity.

  • i would not say i am an expert on the field and i will only explain what i get from the wikipedia article, it is like that the (phase-)space of all possible configurations (that have same conditioons/restrictions), has a structure which provides information about a single solution as well (in very simple terms) – Nikos M. Aug 15 '14 at 15:10
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    Note that, if you wait a few days and this question hasn't received much attention, you might consider cross-posting to MathOverflow. – Semiclassical Aug 15 '14 at 15:12
  • in this sense each solution can be typical of the ensemble of solutions (as in ergodic theory and entropy). In entropy each sequence (solution) has same overall properties (like same entropy) – Nikos M. Aug 15 '14 at 15:12
  • entropy can be seen as a global (algebraic) (invariant) property of a system ensemble (space). Then one can use machinery from algebraic topology and express these things in these terms. Hope this can be useful to you – Nikos M. Aug 15 '14 at 15:18
  • Thanks @NikosM. greatly appreciate it if you please suggest good and accessible resources for a beginner in moduli spaces and ergodic theory –  Aug 15 '14 at 15:20
  • For ergodic theory and topology there are several classic texbooks (mathoverrflow) and math.se, for moduli spaces i suggest ask an algebraic topologist for best information – Nikos M. Aug 15 '14 at 15:21
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    @semiclassical : please don't send questions like this to MO, where it will be quickly closed. – Andy Putman Aug 16 '14 at 15:24

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