Fixed point theorems and equiangular lines - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T21:34:49Z http://mathoverflow.net/feeds/question/30894 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30894/fixed-point-theorems-and-equiangular-lines Fixed point theorems and equiangular lines Peter Shor 2010-07-07T14:20:52Z 2010-07-08T13:09:21Z <p>I've been thinking about the <a href="http://en.wikipedia.org/wiki/SIC-POVM" rel="nofollow">equiangular lines (or SIC-POVM) conjecture</a>, 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:</p> <p>1) is there some good survey article or classification for fixed point theorems?<br> 2) are there fixed-point theorems which are related to actions of groups on geometric spaces?<br> 3) has anybody tried this idea?</p> <p><em>Added:</em> 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 <a href="http://arxiv.org/abs/0910.5784" rel="nofollow">this recent paper</a>.</p> http://mathoverflow.net/questions/30894/fixed-point-theorems-and-equiangular-lines/30933#30933 Answer by coudy for Fixed point theorems and equiangular lines coudy 2010-07-07T18:28:45Z 2010-07-07T18:28:45Z <p>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.</p> <ul> <li>results based on compactness</li> <li>order theoretic results</li> <li>results based on convexity</li> <li>Borsuk theorem and topological transitivity</li> <li>homology and fixed points</li> <li>Leray-Shauder degree and fixed point index</li> </ul> <p>Part VI of the bibliography is really extensive and contains a finer classification of fixed point theorems.</p>