Learning Tropical geometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:25:15Z http://mathoverflow.net/feeds/question/84629 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84629/learning-tropical-geometry Learning Tropical geometry john 2011-12-31T07:01:34Z 2011-12-31T18:20:35Z <p>I'm interested in learning tropical geometry. But my background in algebraic geometry is limited. I know basic facts about varieties in affine and projective space, but nothing about sheaves, schemes etc.</p> <p>I wanted to know how much algebraic geometry does one need to understand the research literature in tropical geometry.</p> <p>What are the other subjects which I must know before starting tropical geometry? </p> http://mathoverflow.net/questions/84629/learning-tropical-geometry/84646#84646 Answer by Leandro Vendramin for Learning Tropical geometry Leandro Vendramin 2011-12-31T16:37:33Z 2011-12-31T16:37:33Z <p>I believe that only basic algebraic geometry is needed. </p> <p><a href="http://vod.mathnet.or.kr/sub1.php?key_s_title=Tropical+Geometry" rel="nofollow">This</a> on-line course on Tropical Geometry given by Bernd Sturmfels is based on <a href="http://www.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook.pdf" rel="nofollow">this</a> book, written by D. Maclagan and B. Sturmfels, and in the introduction they claim the following:</p> <blockquote> <p>We have attempted to make the first part of the book (Chapters 1–5) accessible to readers with a minimal background in algebraic geometry, say, at the level of the undergraduate text book Ideals, Varieties, and Algorithms by Cox, Little, and O’Shea.</p> </blockquote> http://mathoverflow.net/questions/84629/learning-tropical-geometry/84649#84649 Answer by Michael Joyce for Learning Tropical geometry Michael Joyce 2011-12-31T18:20:35Z 2011-12-31T18:20:35Z <p>Several years ago, I participated in a learning seminar in tropical algebraic geometry and collected several helpful survey articles. (This was before Maclagan and Sturmfels' book was written, which I suspect is excellent.)</p> <p>Anyway, here were some of the most helpful intro points for me: <a href="http://arxiv.org/abs/math/0408099" rel="nofollow">Tropical Mathematics</a>, <a href="http://arxiv.org/abs/math/0306366" rel="nofollow">First Steps in Tropical Geometry</a>, <a href="http://arxiv.org/abs/math/0601322" rel="nofollow">Tropical Algebraic Geometry</a>, <a href="http://arxiv.org/abs/0709.1049" rel="nofollow">Introduction to Tropical Geometry</a>, <a href="http://arxiv.org/abs/math/0304218" rel="nofollow">The Tropical Grassmannian</a>, <a href="http://arxiv.org/abs/math/0504390" rel="nofollow">The Number of Tropical Plane Curves Through Points in General Position</a>.</p> <p>Sturmfels, Speyer, and Gathmann all write very well, and Gathmann especially devotes considerable space to giving motivation for the field. Mikhalkin, of course, was the one who pioneered the idea of attacking challenging classical problems (such as counting the number of plane curves of genus \$g\$ and degree \$d\$ passing through \$3d + g - 1\$ points, which had just been solved by Capraso-Harris in the late 90s) using the tropical semifield.</p>