{"656894":{"#nid":"656894","#data":{"type":"event","title":"PhD Defense by Marc H\u00e4rk\u00f6nen","body":[{"value":"\u003Cp\u003EMarc H\u0026auml;rk\u0026ouml;nen\u003Cbr \/\u003E\r\n(Advisor: Prof. Anton Leykin)\u003Cbr \/\u003E\r\nwill defend a doctoral thesis entitled\u003Cbr \/\u003E\r\nDual representation of polynomial modules with applications to partial\u003Cbr \/\u003E\r\ndifferential equations\u003Cbr \/\u003E\r\nOn\u003Cbr \/\u003E\r\nFriday, April 15 2022 at 11:00 a.m.\u003Cbr \/\u003E\r\nSkiles 006, or online\u003Cbr \/\u003E\r\nhttps:\/\/gatech.zoom.us\/j\/95997197594?pwd=RDN2T01oR2JlaEcyQXJCN1c4dnZaUT09\u003Cbr \/\u003E\r\nAbstract\u003Cbr \/\u003E\r\nIn 1939, Wolfgang Gr\u0026ouml;bner proposed using differential operators to represent ideals in a\u003Cbr \/\u003E\r\npolynomial ring. Using Macaulay inverse systems, he showed a one-to-one correspondence\u003Cbr \/\u003E\r\nbetween primary ideals whose variety is a rational point, and finite dimensional vector spaces\u003Cbr \/\u003E\r\nof differential operators with constant coefficients. The question for general ideals was left\u003Cbr \/\u003E\r\nopen. Significant progress was made in the 1960\u0026#39;s by analysts, culminating in a deep result\u003Cbr \/\u003E\r\nknown as the Ehrenpreis-Palamodov fundamental principle, connecting polynomial ideals and\u003Cbr \/\u003E\r\nmodules to solution sets of linear, homogeneous partial differential equations with constant\u003Cbr \/\u003E\r\ncoefficients.\u003Cbr \/\u003E\r\nThis talk aims to survey classical results, and provide new constructions, applications, and\u003Cbr \/\u003E\r\ninsights, merging concepts from analysis and nonlinear algebra. We offer a new formulation\u003Cbr \/\u003E\r\ngeneralizing Gr\u0026ouml;bner\u0026#39;s duality for arbitrary polynomial ideals and modules and connect it to the\u003Cbr \/\u003E\r\nanalysis of PDEs. This framework is amenable to the development of symbolic and numerical\u003Cbr \/\u003E\r\nalgorithms. We also study some applications of algebraic methods in problems from analysis.\u003Cbr \/\u003E\r\nCommittee\u003Cbr \/\u003E\r\n\u0026bull; Prof. Anton Leykin, School of Mathematics (advisor)\u003Cbr \/\u003E\r\n\u0026bull; Prof. Josephine Yu, School of Mathematics\u003Cbr \/\u003E\r\n\u0026bull; Prof. Grigoriy Blekherman, School of Mathematics\u003Cbr \/\u003E\r\n\u0026bull; Prof. Matthew Baker, School of Mathematics\u003Cbr \/\u003E\r\n\u0026bull; Prof. Dr. Jonas Hirsch, Mathematisches Institut, Universit\u0026auml;t Leipzig\u003C\/p\u003E\r\n","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"Dual representation of polynomial modules with applications to partial differential equations"}],"uid":"27707","created_gmt":"2022-03-31 20:49:42","changed_gmt":"2022-03-31 20:49:42","author":"Tatianna Richardson","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2022-04-15T12:00:00-04:00","event_time_end":"2022-04-15T14:00:00-04:00","event_time_end_last":"2022-04-15T14:00:00-04:00","gmt_time_start":"2022-04-15 16:00:00","gmt_time_end":"2022-04-15 18:00:00","gmt_time_end_last":"2022-04-15 18:00:00","rrule":null,"timezone":"America\/New_York"},"extras":[],"groups":[{"id":"221981","name":"Graduate Studies"}],"categories":[],"keywords":[{"id":"100811","name":"Phd Defense"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[{"id":"1788","name":"Other\/Miscellaneous"}],"invited_audience":[{"id":"78761","name":"Faculty\/Staff"},{"id":"78771","name":"Public"},{"id":"78751","name":"Undergraduate students"}],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}