Is Algebraic Geometry really natural? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-22T08:48:36Z http://mathoverflow.net/feeds/question/86973 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86973/is-algebraic-geometry-really-natural Is Algebraic Geometry really natural? Victor 2012-01-29T19:04:13Z 2012-01-29T19:39:51Z <p>Dear All!</p> <p>I recently had a conversation with one mathematician who reckons that all sorts of combinatorial results are nothing compared to the things done in the algebraic geometry. As I do not have any expertise in AG, I have the following question:</p> <p>Is AG really that important?</p> <p>Surfing over the book of Hartshorne, one may get a feel that this is just some bunch of definitions with the hope to be able to do some big problems. First steps of AG, as far as I have been told, indeed lead to solutions, but is AG still natural, or just a trend in Maths (for producing Fields medalists).</p> <p>To get things clear, I have nothing against AG, but it really got me upset that some mathematicians do not count such things as results about the random graph as Maths.</p> http://mathoverflow.net/questions/86973/is-algebraic-geometry-really-natural/86974#86974 Answer by Dustin Clausen for Is Algebraic Geometry really natural? Dustin Clausen 2012-01-29T19:15:22Z 2012-01-29T19:15:22Z <p>Algebraic geometry is the study of spaces defined by polynomial equations. So on the one hand it provides examples of interest for geometers and topologists, and on the other hand it lets algebraists and number theorists take a geometric perspective on their subjects.</p> http://mathoverflow.net/questions/86973/is-algebraic-geometry-really-natural/86978#86978 Answer by Charles Matthews for Is Algebraic Geometry really natural? Charles Matthews 2012-01-29T19:39:51Z 2012-01-29T19:39:51Z <p>Algebraic geometry has been an important part of mathematics since Descartes, who pretty much invented it. In other words it is part of 17th century mathematics, like calculus. It happens that there is more left to do in algebraic geometry than in calculus (in terms Newton, who worked on cubic curves, would understand); and that some problems that are apparently in calculus, for example indefinite integration by closed formulae, are really better understood geometrically, as became apparent in the 19th century. We are still working out where the ideas of Hilbert and Poincaré leave us in expressing methods and solutions in algebraic geometry, in fact. A very active 20th century didn't settle that.</p>