{"625237":{"#nid":"625237","#data":{"type":"news","title":"Georgia Tech Mathematician Solves Cartan-Hadamard Conjecture","body":[{"value":"\u003Cp\u003ESchool of Mathematics\u0026nbsp;Professor\u0026nbsp;\u003Ca href=\u0022https:\/\/people.math.gatech.edu\/~ghomi\/\u0022\u003EMohammad Ghomi\u003C\/a\u003E and \u003Ca href=\u0022https:\/\/mathematics.jhu.edu\/directory\/joel-spruck\/\u0022\u003EJoel Spruck\u003C\/a\u003E of Johns Hopkins University this week \u003Ca href=\u0022https:\/\/arxiv.org\/abs\/1908.09814\u0022\u003Eposted in ArXiv\u003C\/a\u003E their proposed solution to the Cartan-Hadamard Conjecture, a long-standing problem in Analysis and Geometry, which is concerned\u0026nbsp;with a generalization of the classical\u0026nbsp;\u003Ca href=\u0022https:\/\/en.wikipedia.org\/wiki\/Isoperimetric_inequality\u0022\u003Eisoperimetric inequality\u003C\/a\u003E.\u003C\/p\u003E\r\n\r\n\u003Cp\u003EIn Euclidean space, to enclose a given volume with the smallest possible perimeter, one should use a\u0026nbsp;ball.\u0026nbsp;Comparing the volume of the ball to its surface area yields a numerical measure of the efficiency of this\u0026nbsp;enclosure which lies at the base of a range of important inequalities in mathematics and physics.\u0026nbsp;There\u0026nbsp;are more general models of space than the Euclidean framework, however. Some of the most natural and attractive ones are those with\u0026nbsp;nonpositive curvature, which are known as Cartan-Hadamard manifolds. The\u0026nbsp;simplest examples of Cartan-Hadamard manifolds include potato chips and coral reefs. \u0026nbsp;Cartan-Hadamard\u0026nbsp;manifolds aim to include more area in less space, like waves of a\u0026nbsp;petunia flower.\u003C\/p\u003E\r\n\r\n\u003Cp\u003EThe exploration of the isoperimetric inequality in Cartan-Hadamard manifolds can be traced back to work of Andre\u0026nbsp;Weil, a student of Hadamard at the time, who\u0026nbsp;addressed the two dimensional case in 1926. Around fifty years later,\u0026nbsp;Thierry Aubin, Misha Gromov, Yuri Burago and Victor Zalgaller conjectured that, in all dimensions, Cartan-Hadamard manifolds should satisfy the same isoperimetric inequality as in Euclidean\u0026nbsp;space. Prior to Ghomi and Spruck\u0026rsquo;s work, the only other known cases where in dimensions 3 and 4, due to Bruce Kleiner in 1992 and Chris Croke in 1984 respectively.\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EThe proposed proof settles the conjecture in all dimensions. As a result several other well-known\u0026nbsp;inequalities in Analysis and Mathematical Physics may now be generalized to spaces of nonpositive\u0026nbsp;curvature as well. These include the Sobolev inequalities, which are fundamental in theory of functions,\u0026nbsp;and the Faber-Krahn inequality, which dates back to a conjecture of Lord Rayleigh in 1877 concerning\u0026nbsp;the spectrum of the Laplace operator, or the theory of sound.\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EMohammad Ghomi completed his Ph.D. in 1998 at Johns Hopkins University under the direction of Spruck. As a\u0026nbsp;geometer, Ghomi works on \u003Ca href=\u0022http:\/\/people.math.gatech.edu\/~ghomi\/Papers\/op.pdf\u0022\u003Eclassical\u0026nbsp;problems\u003C\/a\u003E\u0026nbsp;involving curves and surfaces in Euclidean space and,\u0026nbsp;more generally, Riemannian submanifolds. He is interested in all aspects of mathematics that touch upon\u0026nbsp;these objects, ranging from differential geometry and topology to real algebraic geometry and\u0026nbsp;combinatorics.\u003C\/p\u003E\r\n","summary":null,"format":"limited_html"}],"field_subtitle":[{"value":"Mohammad Ghomi partnered with Joel Spruck for the solution"}],"field_summary":[{"value":"\u003Cp\u003ESchool of Mathematics\u0026nbsp;Professor\u0026nbsp;Mohammad Ghomi and Joel Spruck of Johns Hopkins University recently posted ArXiv their proposed solution tothe Cartan-Hadamard Conjecture, a long-standing problem in Analysis and Geometry.\u003C\/p\u003E\r\n","format":"limited_html"}],"field_summary_sentence":[{"value":"Mohammad Ghomi and Joel Spruck solve the Cartan-Hadamard Conjecture."}],"uid":"34518","created_gmt":"2019-08-27 18:40:37","changed_gmt":"2019-08-30 20:34:35","author":"sbarone7","boilerplate_text":"","field_publication":"","field_article_url":"","dateline":{"date":"2019-08-27T00:00:00-04:00","iso_date":"2019-08-27T00:00:00-04:00","tz":"America\/New_York"},"extras":[],"hg_media":{"625330":{"id":"625330","type":"image","title":"slide_ghomi2","body":null,"created":"1567086520","gmt_created":"2019-08-29 13:48:40","changed":"1567087511","gmt_changed":"2019-08-29 14:05:11","alt":"","file":{"fid":"238075","name":"slide_ghomi5.png","image_path":"\/sites\/default\/files\/images\/slide_ghomi5.png","image_full_path":"http:\/\/www.tlwarc.hg.gatech.edu\/\/sites\/default\/files\/images\/slide_ghomi5.png","mime":"image\/png","size":202613,"path_740":"http:\/\/www.tlwarc.hg.gatech.edu\/sites\/default\/files\/styles\/740xx_scale\/public\/images\/slide_ghomi5.png?itok=pZyKiu1E"}},"606949":{"id":"606949","type":"image","title":"Mohammad Ghomi","body":null,"created":"1528819565","gmt_created":"2018-06-12 16:06:05","changed":"1528819565","gmt_changed":"2018-06-12 16:06:05","alt":"","file":{"fid":"231512","name":"2018 Mohammad Ghomi.sq250.jpg","image_path":"\/sites\/default\/files\/images\/2018%20Mohammad%20Ghomi.sq250.jpg","image_full_path":"http:\/\/www.tlwarc.hg.gatech.edu\/\/sites\/default\/files\/images\/2018%20Mohammad%20Ghomi.sq250.jpg","mime":"image\/jpeg","size":50498,"path_740":"http:\/\/www.tlwarc.hg.gatech.edu\/sites\/default\/files\/styles\/740xx_scale\/public\/images\/2018%20Mohammad%20Ghomi.sq250.jpg?itok=-N8WtBpf"}},"625352":{"id":"625352","type":"image","title":"Joel Spruck","body":null,"created":"1567108137","gmt_created":"2019-08-29 19:48:57","changed":"1567108137","gmt_changed":"2019-08-29 19:48:57","alt":"","file":{"fid":"238083","name":"spruck2016.jpg","image_path":"\/sites\/default\/files\/images\/spruck2016.jpg","image_full_path":"http:\/\/www.tlwarc.hg.gatech.edu\/\/sites\/default\/files\/images\/spruck2016.jpg","mime":"image\/jpeg","size":24176,"path_740":"http:\/\/www.tlwarc.hg.gatech.edu\/sites\/default\/files\/styles\/740xx_scale\/public\/images\/spruck2016.jpg?itok=RUHHFtmf"}}},"media_ids":["625330","606949","625352"],"related_links":[{"url":"http:\/\/people.math.gatech.edu\/~ghomi\/","title":"Prof. Ghomi\u0027s Homepage"}],"groups":[{"id":"1279","name":"School of Mathematics"},{"id":"1278","name":"College of Sciences"}],"categories":[],"keywords":[{"id":"182160","name":"_for_math_site_; Mohammad Ghomi"},{"id":"182161","name":"Cartan Hadamard Conjecture"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[],"invited_audience":[],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}