{"681607":{"#nid":"681607","#data":{"type":"news","title":"Anton Leykin Awarded Simons Fellowship","body":[{"value":"\u003Cp dir=\u0022ltr\u0022\u003E\u003Ca href=\u0022https:\/\/math.gatech.edu\/\u0022\u003ESchool of Mathematics\u003C\/a\u003E Professor\u0026nbsp;\u003Ca href=\u0022https:\/\/research.gatech.edu\/people\/anton-leykin\u0022\u003E\u003Cstrong\u003EAnton Leykin\u003C\/strong\u003E\u003C\/a\u003E has been awarded a prestigious\u0026nbsp;\u003Ca href=\u0022https:\/\/www.simonsfoundation.org\/mathematics-physical-sciences\/simons-fellows\/\u0022\u003ESimons Fellowship\u003C\/a\u003E for his proposal of applying nonlinear algebra to tackle one of the key mathematical questions of the 21st century.\u0026nbsp; Leykin is one of two mathematicians in the School awarded the Fellowship, and is joined by Associate Professor \u003Ca href=\u0022https:\/\/math.gatech.edu\/people\/benjamin-jaye\u0022\u003EBenjamin Jaye\u003C\/a\u003E.\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003EThe work could lead to new discoveries and a deeper understanding of how celestial bodies like planets, moons, and asteroids interact. The fellowship will fund one year of work, during which Leykin also plans to finish writing a book on nonlinear algebra for advanced undergraduate and beginning graduate students.\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003ELeykin explains that the mathematical problem \u2014\u0026nbsp;known as \u201cSmale\u0027s sixth problem\u201d for its position as number six on\u0026nbsp;\u003Ca href=\u0022http:\/\/www.smaleinstitute.com\/problem.html\u0022\u003Ethe list of questions for the 21st century\u003C\/a\u003E compiled by\u0026nbsp;\u003Ca href=\u0022https:\/\/www.mathunion.org\/imu-awards\/fields-medal\/fields-medals-1966\u0022\u003EFields Medalist\u0026nbsp;\u003Cstrong\u003EStephen Smale\u003C\/strong\u003E\u003C\/a\u003E \u2014 involves understanding the number of ways celestial bodies can be arranged in space so that they stay at relative equilibrium, growing neither further apart nor closer to each other as they orbit.\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003E\u201cPeople have been trying to solve this problem for more than two hundred years \u2014 since Euler and Lagrange \u2014 but even proving that the number of relative equilibria in an\u0026nbsp;\u003Cem\u003En\u003C\/em\u003E-body problem is\u0026nbsp;\u003Cem\u003Efinite\u003C\/em\u003E is extremely difficult,\u201d Leykin says. One reason for this? \u201cEven for small cases, (e.g.\u0026nbsp;\u003Cem\u003En=5\u003C\/em\u003E) the brute-force approach leads to an enormous amount of computation.\u201d\u0026nbsp;\u0026nbsp;\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003EEach of Leykin\u2019s initial experiments for the case\u0026nbsp;\u003Cem\u003En\u003C\/em\u003E=6 required a CPU year \u2014 the computational power equivalent to a single computer running for an entire year.\u0026nbsp;\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003EThis difficulty is partially what draws Leykin to the problem. \u201cWe use supercomputers to help with the computation time, but this isn\u2019t an area that AI and machine learning can advance,\u201d he explains. \u201cFor this type of problem, we need human intelligence \u2014 and even with our current technology, there are no easy solutions. It\u2019s a challenge, but that is what makes it interesting.\u201d\u003C\/p\u003E\u003Ch3\u003E\u003Cstrong\u003EStellar pathways\u003C\/strong\u003E\u003C\/h3\u003E\u003Cp dir=\u0022ltr\u0022\u003EImagine the Moon and Earth as two celestial bodies on a plane. Both exert gravitational force on each other \u2014 we can see the result as tides on Earth and the Moon\u2019s orbit. Now add the Sun and other planetary bodies to the plane: asteroids, satellites, and other planets. These bodies also exert gravitational force \u2014 a function of their masses and distances apart \u2014 creating a complex system of orbits and trajectories.\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003ESmale\u2019s sixth problem imagines a plane like this, with any number of celestial bodies arranged on it. The problem considers an arrangement of the bodies in a way that the gravitational forces balance, so that even while they are interacting and orbiting, none of the bodies travel further away or closer to each other.\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003E\u201cIt has been conjectured, but so far not shown, that the number of such configurations is finite,\u201d Leykin says. \u201cIt seems simple. Is it finite or infinite? But progress is minimal at the moment.\u0026nbsp; Several approaches settle the question for almost all values of\u0026nbsp;\u003Cem\u003En=5\u0026nbsp;\u003C\/em\u003Emasses with the case\u0026nbsp;\u003Cem\u003En=6\u003C\/em\u003E wide open, even for a non-special choice of masses.\u201d\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003ELeykin is taking a different approach than many researchers, leveraging a field of mathematics called tropical geometry, which simplifies the geometry of curved equations as straight lines.\u0026nbsp;\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003E\u201cWe\u0027re not trying to compute or describe the original solution manifold but rather replace it with its tropicalization, a combinatorial shadow which captures the finiteness aspect,\u201d he explains.\u0026nbsp;\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003ELeykin\u2019s method has already found success for the case\u0026nbsp;\u003Cem\u003En=5\u003C\/em\u003E. \u201cA recent paper proved that for five bodies, if the masses are general enough, there are a finite number of relative equilibria,\u201d Leykin shares. \u201cUsing our approach, we were able to reproduce the result for five celestial bodies in a simpler way.\u201d\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003E\u201cOur goal now is to collect more evidence by solving the problem for six bodies,\u201d he adds. \u201cIf this helps lead us to a general solution to the problem as a whole \u2014 that would be great.\u201d\u003C\/p\u003E\u003Ch3\u003E\u003Cstrong\u003ESpace \u2018storage spots\u2019\u003C\/strong\u003E\u003C\/h3\u003E\u003Cp dir=\u0022ltr\u0022\u003EWhile the project is theoretical, it could lead to a greater understanding of celestial mechanics.\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003ELeykin is collaborating on a separate but related project with aerospace departments around the country. \u201cWe\u0027re working to understand the trajectories of a massless spacecraft, assuming it is primarily affected by gravitation of the Moon and the Earth,\u201d he shares.\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003EThe 18th century mathematics developed for this type of problem, a restricted three-body problem, could help teams use the gravitational pull of the Earth and Moon to place a small spacecraft near a\u0026nbsp;\u003Ca href=\u0022https:\/\/science.nasa.gov\/resource\/what-is-a-lagrange-point\/\u0022\u003ELagrangian point\u003C\/a\u003E \u2014 a space \u201cstorage spot\u201d where it would remain stationary relative to the Earth.\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003E\u201cYou can place something at a Lagrangian point, and it will stay stationary relative to the system,\u201d Leykin explains. \u201cIt\u0027s a way to place things so they don\u0027t move.\u201d For example, in the Sun-Earth system, the\u0026nbsp;\u003Ca href=\u0022https:\/\/science.nasa.gov\/mission\/webb\/\u0022\u003EJames Webb Space Telescope\u003C\/a\u003E was placed at one of these points, where it conveniently stays in Earth\u2019s shadow \u2014 avoiding the bright light and heat of the Sun, Earth, and Moon.\u003C\/p\u003E\u003Cp dir=\u0022ltr\u0022\u003E\u201cSmale\u2019s sixth problem is about acquiring more theoretical knowledge,\u201d Leykin adds. \u201cIf we discover something on the theoretical front, it can be of practical importance for applied scientists and designing missions for exploratory spacecraft going far into the solar system.\u201d\u003C\/p\u003E","summary":"","format":"limited_html"}],"field_subtitle":"","field_summary":[{"value":"\u003Cp\u003E\u003Cem\u003ELeykin is among two Georgia Tech mathematicians to receive the prestigious award. The Fellowship will support one year of research, during which he aims to tackle a key celestial mechanics problem using nonlinear algebra and tropical geometry.\u0026nbsp;\u003C\/em\u003E\u003C\/p\u003E","format":"limited_html"}],"field_summary_sentence":[{"value":"Leykin is among two Georgia Tech mathematicians to receive the prestigious award. The Fellowship will support one year of research, during which he aims to tackle a key celestial mechanics problem using nonlinear algebra and tropical geometry. "}],"uid":"35599","created_gmt":"2025-04-04 17:01:25","changed_gmt":"2025-04-24 13:18:51","author":"sperrin6","boilerplate_text":"","field_publication":"","field_article_url":"","location":"Atlanta, GA","dateline":{"date":"2025-04-14T00:00:00-04:00","iso_date":"2025-04-14T00:00:00-04:00","tz":"America\/New_York"},"extras":[],"hg_media":{"676755":{"id":"676755","type":"image","title":"Leykin\u0027s work could lead to new discoveries and a deeper understanding of how celestial bodies like planets, moons, and asteroids interact. (Credit: Adobe Stock)","body":"\u003Cp\u003ELeykin\u0027s work could lead to new discoveries and a deeper understanding of how celestial bodies like planets, moons, and asteroids interact. (Credit: Adobe Stock)\u003C\/p\u003E","created":"1743786241","gmt_created":"2025-04-04 17:04:01","changed":"1743786354","gmt_changed":"2025-04-04 17:05:54","alt":"Leykin\u0027s work could lead to new discoveries and a deeper understanding of how celestial bodies like planets, moons, and asteroids interact. (Credit: Adobe Stock)","file":{"fid":"260585","name":"AdobeStock_CelestialMechanics.jpeg","image_path":"\/sites\/default\/files\/2025\/04\/04\/AdobeStock_CelestialMechanics.jpeg","image_full_path":"http:\/\/hg.gatech.edu\/\/sites\/default\/files\/2025\/04\/04\/AdobeStock_CelestialMechanics.jpeg","mime":"image\/jpeg","size":5456813,"path_740":"http:\/\/hg.gatech.edu\/sites\/default\/files\/styles\/740xx_scale\/public\/2025\/04\/04\/AdobeStock_CelestialMechanics.jpeg?itok=m-gwYgLq"}}},"media_ids":["676755"],"groups":[{"id":"1278","name":"College of Sciences"},{"id":"1279","name":"School of Mathematics"}],"categories":[{"id":"135","name":"Research"}],"keywords":[{"id":"192249","name":"cos-community"},{"id":"192252","name":"cos-planetary"},{"id":"193733","name":"_for_math_site_manual_feed_"},{"id":"173647","name":"_for_math_site_"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[],"invited_audience":[],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[{"value":"\u003Cp\u003EWritten by Selena Langner\u003C\/p\u003E\u003Cp\u003EContact: \u003Ca href=\u0022mailto: jess.hunt@cos.gatech.edu\u0022\u003EJess Hunt-Ralston\u003C\/a\u003E\u003C\/p\u003E","format":"limited_html"}],"email":[],"slides":[],"orientation":[],"userdata":""}}}