I've been thinking about the equiangular lines (or SIC-POVM) conjecture, and my conclusion is that the best means of attack would be through some kind of fixed point theorem -- I'm thinking specifically of geometric fixed point theorems, like Brouwer's. So my (rather vague) questions are:

1) is there some good survey article or classification for fixed point theorems?
2) are there fixed-point theorems which are related to actions of groups on geometric spaces?
3) has anybody tried this idea?

Added: In response to Joe's comment below, let me note that while the motivation is from quantum information theory, the equiangular lines conjecture is a purely classical geometry problem (see my comment below). The conjecture is really intriguing: numerical constructions of sets of equiangular lines have been found up to dimension 67, at which point the computer time required exceeded the patience of the investigators. However, only a handful of these numerical solutions have been shown to be rigorously correct by finding corresponding algebraic numbers. See this recent paper.

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    $\begingroup$ Have you tried the references at en.wikipedia.org/wiki/Fixed_point_theorem ? $\endgroup$ – Qiaochu Yuan Jul 7 '10 at 18:05
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    $\begingroup$ @Peter: Can the equiangular lines conjecture be extracted from the SIC-POVM context and understood without 'grokking' all the quantum mechanics, or is it instead intimately entangled within that context? $\endgroup$ – Joseph O'Rourke Jul 8 '10 at 1:56
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    $\begingroup$ @Joseph: The equiangular lines conjecture can be understood completely classically. In complex Euclidean space $\mathbb{C}^d$, are there $d^2$ lines such that the absolute value of the angles between any pair of them is equal (a matrix calculation shows that that the absolute value of the inner product of any pair must be $1/\sqrt{d+1}$ ). $\endgroup$ – Peter Shor Jul 8 '10 at 13:25
  • $\begingroup$ I wish you the best of luck... I don't know that anyone has tried such an approach before, so it would be great if this leads to a solution. $\endgroup$ – Steve Flammia Jul 8 '10 at 13:34
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    $\begingroup$ @Peter: Can you formulate the conjecture as a fixed point problem ? $\endgroup$ – jjcale Dec 3 '13 at 19:09

The book "Fixed point theory" by Dugundji and Granas is a nice reference. The headers of the sections in the book give some kind of classification of fixed point theorems.

  • results based on compactness
  • order theoretic results
  • results based on convexity
  • Borsuk theorem and topological transitivity
  • homology and fixed points
  • Leray-Shauder degree and fixed point index

Part VI of the bibliography is really extensive and contains a finer classification of fixed point theorems.

  • $\begingroup$ Thanks ... this is exactly the kind of thing I was looking for. $\endgroup$ – Peter Shor Jul 8 '10 at 12:51

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