User mathieu dutour sikiric - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T17:29:58Z http://mathoverflow.net/feeds/user/8883 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37136/classification-of-finite-groups-of-isometries Classification of finite groups of isometries Mathieu Dutour Sikiric 2010-08-30T09:33:42Z 2013-02-01T00:54:02Z <p>Consider the problem of classifying the finite groups of isometries of R^n. --For n=2 it is cyclic and dihedral groups. --For n=3 they are well known, probably from Kepler and are related to ade-classification. --For n=4 we can get them by taking the universal cover of SO(4) which is isomorphic to SU2 x SU2, though I do not know where the classification is available.</p> <p>But my main question is for dimension n=5 and above. Does anybody knows the state of the art? A reference would be most helpful. Note that the finite subgroups of GLn(Z) are classified for n&lt;=10.</p> <p>Mathieu</p> http://mathoverflow.net/questions/85803/perfect-forms-equivalent-to-minkowski-extreme-forms/89273#89273 Answer by Mathieu Dutour Sikiric for Perfect forms equivalent to Minkowski extreme forms Mathieu Dutour Sikiric 2012-02-23T11:35:04Z 2012-02-23T11:35:04Z <p>no, this cannot happen.</p> http://mathoverflow.net/questions/85021/enumerating-perfect-lattices/89165#89165 Answer by Mathieu Dutour Sikiric for Enumerating Perfect Lattices Mathieu Dutour Sikiric 2012-02-22T08:14:40Z 2012-02-22T08:14:40Z <p>Voronoi Algorithm is about optimal if you are interested in enumerating all perfect form. The problem is that usually one is interested in enumerating the ones with some specific property (high packing density, etc.) but there is no way that I know to restrict the enumeration.</p> <p>The algorithm works by finding an initial perfect form, finding the adjacent perfect form, testing for isomorphism and finishing when all perfect forms have been enumerated. There are three key algorithm in the computation: testing for isomorphism, flipping the form to the adjacent one and computing facets of the perfect cone. Of those the first two are usually ok and the last one is the one that causes problems.</p> <p>The best is probably to use an empirical approach of running the enumeration and seeing how much form it finds. Using this in dimension 9, 500000 perfect forms were found and it is likely that the number is much higher hence uncomputable. If the number is not too large, then it may be possible to prove that the list is complete by solving those hard dual description problems.</p> http://mathoverflow.net/questions/6890/generalizations-of-the-birkhoff-von-neumann-theorem/51750#51750 Answer by Mathieu Dutour Sikiric for Generalizations of the Birkhoff-von Neumann Theorem Mathieu Dutour Sikiric 2011-01-11T11:11:21Z 2011-01-11T11:11:21Z <p>The remaining case is actually H4. For this case the conjecture is false, since I found 1063 orbits of facets. In the same paper, it is claimed that for F4 the convex hull is given by the Birkhoff tensors. But this is false since I found another orbit defined by a matrix of rank 3. See details <a href="http://www.liga.ens.fr/~dutour/ConvCoxeter/index.html" rel="nofollow">there</a></p>