{"id":6112,"date":"2025-02-14T10:50:10","date_gmt":"2025-02-14T10:50:10","guid":{"rendered":"https:\/\/site.itshrt.com\/worldnews\/whats-best-according-to-the-italian-mathematician-alessio-figalli\/"},"modified":"2025-02-14T10:50:10","modified_gmt":"2025-02-14T10:50:10","slug":"whats-best-according-to-the-italian-mathematician-alessio-figalli","status":"publish","type":"post","link":"https:\/\/site.itshrt.com\/worldnews\/whats-best-according-to-the-italian-mathematician-alessio-figalli\/","title":{"rendered":"What\u2019s Best, According to the Italian Mathematician Alessio Figalli"},"content":{"rendered":"<p> <script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-6606220950177433\"\r\n     crossorigin=\"anonymous\"><\/script>\r\n<!-- ItShrt World News -->\r\n<ins class=\"adsbygoogle\"\r\n     style=\"display:block\"\r\n     data-ad-client=\"ca-pub-6606220950177433\"\r\n     data-ad-slot=\"1882483372\"\r\n     data-ad-format=\"auto\"\r\n     data-full-width-responsive=\"true\"><\/ins>\r\n<script>\r\n     (adsbygoogle = window.adsbygoogle || []).push({});\r\n<\/script>\r\n<br \/>\n<\/p>\n<div data-testid=\"companionColumn-0\">\n<div class=\"css-53u6y8\">\n<p class=\"css-at9mc1 evys1bk0\">The words \u201coptimal\u201d and \u201coptimize\u201d derive from the Latin \u201coptimus,\u201d or \u201cbest,\u201d as in \u201cmake the best of things.\u201d Alessio Figalli, a mathematician at the university ETH Zurich, studies optimal transport: the most efficient allocation of starting points to end points. The scope of investigation is wide, including clouds, crystals, bubbles and chatbots.<\/p>\n<p class=\"css-at9mc1 evys1bk0\">Dr. Figalli, who was awarded the Fields Medal in 2018, likes math that is motivated by concrete problems found in nature. He also likes the discipline\u2019s \u201csense of eternity,\u201d he said in a recent interview. \u201cIt is something that will be here forever.\u201d (Nothing is forever, he conceded, but math will be around for \u201clong enough.\u201d) \u201cI like the fact that if you prove a theorem, you prove it,\u201d he said. \u201cThere\u2019s no ambiguity, it\u2019s true or false. In a hundred years, you can rely on it, no matter what.\u201d<\/p>\n<p class=\"css-at9mc1 evys1bk0\">The study of optimal transport was introduced almost 250 years ago by Gaspard Monge, a French mathematician and politician who was motivated by problems in military engineering. His ideas found broader application solving logistical problems during the Napoleonic Era \u2014 for instance, identifying the most efficient way to build fortifications, in order to minimize the costs of transporting materials across Europe.<\/p>\n<p class=\"css-at9mc1 evys1bk0\">In 1975, the Russian mathematician Leonid Kantorovich shared the <a class=\"css-yywogo\" href=\"https:\/\/www.nobelprize.org\/prizes\/economic-sciences\/1975\/kantorovich\/facts\/\" title=\"\" rel=\"noopener noreferrer\" target=\"_blank\">Nobel in economic science<\/a> for refining a rigorous mathematical theory for the optimum allocation of resources. \u201cHe had an example with bakeries and coffee shops,\u201d Dr. Figalli said. The optimization goal in this case was to ensure that on a daily basis every bakery delivered all its croissants, and every coffee shop got all the croissants desired.<\/p>\n<\/div>\n<aside class=\"css-ew4tgv\" aria-label=\"companion column\"\/><\/div>\n<div data-testid=\"companionColumn-1\">\n<div class=\"css-53u6y8\">\n<p class=\"css-at9mc1 evys1bk0\">\u201cIt\u2019s called a global wellness optimization problem in the sense that there is no competition between bakeries, no competition between coffee shops,\u201d he said. \u201cIt\u2019s not like optimizing the utility of one player. It is optimizing the global utility of the population. And that\u2019s why it\u2019s so complex: because if one bakery or one coffee shop does something different, this will influence everyone else.\u201d<\/p>\n<p class=\"css-at9mc1 evys1bk0\">The following conversation with Dr. Figalli \u2014 conducted at an event in New York City organized by the Simons Laufer Mathematical Sciences Institute and in interviews before and after \u2014 has been condensed and edited for clarity.<\/p>\n<p class=\"css-at9mc1 evys1bk0\"><strong class=\"css-8qgvsz ebyp5n10\">How would you finish the sentence \u201cMath is \u2026 \u201d? What is math?<\/strong><\/p>\n<p class=\"css-at9mc1 evys1bk0\">For me, math is a creative process and a language to describe nature. The reason that math is the way it is is because humans realized that it was the right way to model the earth and what they were observing. What is fascinating is that it works so well.<\/p>\n<p class=\"css-at9mc1 evys1bk0\"><strong class=\"css-8qgvsz ebyp5n10\">Is nature always seeking to optimize?<\/strong><\/p>\n<p class=\"css-at9mc1 evys1bk0\">Nature is naturally an optimizer. It has a minimal-energy principle \u2014 nature by itself. Then of course it gets more complex when other variables enter into the equation. It depends on what you are studying.<\/p>\n<p class=\"css-at9mc1 evys1bk0\">When I was applying optimal transport to meteorology, I was trying to understand the movement of clouds. It was a simplified model where some physical variables that may influence the movement of clouds were neglected. For example, you might ignore friction or wind.<\/p>\n<\/div>\n<aside class=\"css-ew4tgv\" aria-label=\"companion column\"\/><\/div>\n<div data-testid=\"companionColumn-2\">\n<div class=\"css-53u6y8\">\n<p class=\"css-at9mc1 evys1bk0\">The movement of water particles in clouds follows an optimal transport path. And here you are transporting billions of points, billions of water particles, to billions of points, so it\u2019s a much bigger problem than 10 bakeries to 50 coffee shops. The numbers grow enormously. That\u2019s why you need mathematics to study it.<\/p>\n<\/div>\n<aside class=\"css-ew4tgv\" aria-label=\"companion column\"\/><\/div>\n<div data-testid=\"companionColumn-3\">\n<div class=\"css-53u6y8\">\n<p class=\"css-at9mc1 evys1bk0\"><strong class=\"css-8qgvsz ebyp5n10\">What about optimal transport captured your interest?<\/strong><\/p>\n<p class=\"css-at9mc1 evys1bk0\">I was most excited by the applications, and by the fact that the mathematics was very beautiful and came from very concrete problems.<\/p>\n<p class=\"css-at9mc1 evys1bk0\">There is a constant exchange between what mathematics can do and what people require in the real world. As mathematicians, we can fantasize. We like to increase dimensions \u2014 we work in infinite dimensional space, which people always think is a little bit crazy. But it\u2019s what allows us now to use cellphones and Google and all the modern technology we have. Everything would not exist had mathematicians not been crazy enough to go out of the standard boundaries of the mind, where we only live in three dimensions. Reality is much more than that.<\/p>\n<p class=\"css-at9mc1 evys1bk0\">In society, the risk is always that people just see math as being important when they see the connection to applications. But it\u2019s important beyond that \u2014 the thinking, the developments of a new theory that came through mathematics over time that led to big changes in society. Everything is math.<\/p>\n<\/div>\n<aside class=\"css-ew4tgv\" aria-label=\"companion column\"\/><\/div>\n<div data-testid=\"companionColumn-4\">\n<div class=\"css-53u6y8\">\n<p class=\"css-at9mc1 evys1bk0\">And often the math came first. It\u2019s not that you wake up with an applied question and you find the answer. Usually the answer was already there, but it was there because people had the time and the freedom to think big. The other way around it can work, but in a more limited fashion, problem by problem. Big changes usually happen because of free thinking.<\/p>\n<p class=\"css-at9mc1 evys1bk0\"><strong class=\"css-8qgvsz ebyp5n10\">Optimization has its limits. Creativity can\u2019t really be optimized.<\/strong><\/p>\n<p class=\"css-at9mc1 evys1bk0\">Yes, creativity is the opposite. Suppose you\u2019re doing very good research in an area; your optimization scheme would have you stay there. But it\u2019s better to take risks. Failure and frustration are key. Big breakthroughs, big changes, always come because at some moment you are taking yourself out of your comfort zone, and this will never be an optimization process. Optimizing everything results in missing opportunities sometimes. I think it\u2019s important to really value and be careful with what you optimize.<\/p>\n<p class=\"css-at9mc1 evys1bk0\"><strong class=\"css-8qgvsz ebyp5n10\">What are you working on these days?<\/strong><\/p>\n<p class=\"css-at9mc1 evys1bk0\">One challenge is using optimal transport in machine learning.<\/p>\n<p class=\"css-at9mc1 evys1bk0\">From a theoretical viewpoint, machine learning is just an optimization problem where you have a system, and you want to optimize some parameters, or features, so that the machine will do a certain number of tasks.<\/p>\n<p class=\"css-at9mc1 evys1bk0\">To classify images, optimal transport measures how similar two images are by comparing features like colors or textures and putting these features into alignment \u2014 transporting them \u2014 between the two images. This technique helps improve accuracy, making models more robust to changes or distortions.<\/p>\n<p class=\"css-at9mc1 evys1bk0\">These are very high-dimensional phenomena. You are trying to understand objects that have many features, many parameters, and every feature corresponds to one dimension. So if you have 50 features, you are in 50-dimensional space.<\/p>\n<\/div>\n<aside class=\"css-ew4tgv\" aria-label=\"companion column\"\/><\/div>\n<div data-testid=\"companionColumn-5\">\n<div class=\"css-53u6y8\">\n<p class=\"css-at9mc1 evys1bk0\">The higher the dimension where the object lives, the more complex the optimal transport problem is \u2014 it requires too much time, too much data to solve the problem, and you will never be able to do it. This is called the curse of dimensionality. Recently people have been trying to look at ways to avoid the curse of dimensionality. One idea is to develop a new type of optimal transport.<\/p>\n<p class=\"css-at9mc1 evys1bk0\"><strong class=\"css-8qgvsz ebyp5n10\">What\u2019s the gist of it?<\/strong><\/p>\n<p class=\"css-at9mc1 evys1bk0\">By collapsing some features, I reduce my optimal transport to a lower-dimensional space. Let\u2019s say three dimensions is too large for me and I want to make it a one-dimensional problem. I take some points in my three-dimensional space and I project them onto a line. I solve the optimal transport on the line, I compute what I should do, and I repeat this for many, many lines. Then, using these results in dimension one, I try to reconstruct the original 3-D space by a sort of gluing together. It is not an obvious process.<\/p>\n<p class=\"css-at9mc1 evys1bk0\"><strong class=\"css-8qgvsz ebyp5n10\">It kind of sounds like the shadow of an object \u2014 a two-dimensional, square-ish shadow provides some information about the three-dimensional cube that casts the shadow.<\/strong><\/p>\n<p class=\"css-at9mc1 evys1bk0\">It is like shadows. Another example is X-rays, which are 2-D images of your 3-D body. But if you do X-rays in enough directions you can essentially piece together the images and reconstruct your body.<\/p>\n<p class=\"css-at9mc1 evys1bk0\"><strong class=\"css-8qgvsz ebyp5n10\">Conquering the curse of dimensionality would help with A.I.\u2019s shortcomings and limitations?<\/strong><\/p>\n<p class=\"css-at9mc1 evys1bk0\">If we use some optimal transport techniques, perhaps this could make some of these optimization problems in machine learning more robust, more stable, more reliable, less biased, safer. That\u2019s the meta principle.<\/p>\n<\/div>\n<aside class=\"css-ew4tgv\" aria-label=\"companion column\"\/><\/div>\n<div data-testid=\"companionColumn-6\">\n<div class=\"css-53u6y8\">\n<p class=\"css-at9mc1 evys1bk0\"><strong class=\"css-8qgvsz ebyp5n10\">And, in the interplay of pure and applied math, here the practical, real-world need is motivating new mathematics?<\/strong><\/p>\n<p class=\"css-at9mc1 evys1bk0\">Exactly. The engineering of machine learning is very far ahead. But we don\u2019t know why it works. There are few theorems; comparing what it can achieve to what we can prove, there is a huge gap. It is impressive, but mathematically it is still very difficult to explain why. So we cannot trust it enough. We want to make it better in many directions, and we want mathematics to help.<\/p>\n<\/div>\n<aside class=\"css-ew4tgv\" aria-label=\"companion column\"\/><\/div>\n<p><script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-6606220950177433\"\r\n     crossorigin=\"anonymous\"><\/script>\r\n<!-- ItShrt World News -->\r\n<ins class=\"adsbygoogle\"\r\n     style=\"display:block\"\r\n     data-ad-client=\"ca-pub-6606220950177433\"\r\n     data-ad-slot=\"1882483372\"\r\n     data-ad-format=\"auto\"\r\n     data-full-width-responsive=\"true\"><\/ins>\r\n<script>\r\n     (adsbygoogle = window.adsbygoogle || []).push({});\r\n<\/script>\r\n<br \/>\n<br \/><a href=\"https:\/\/www.nytimes.com\/2025\/02\/14\/science\/mathematics-figalli-optimal-transport.html\">Source link <\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The words \u201coptimal\u201d and \u201coptimize\u201d derive from the Latin \u201coptimus,\u201d or \u201cbest,\u201d as in \u201cmake the best of things.\u201d Alessio<\/p>\n","protected":false},"author":1,"featured_media":6113,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[39],"tags":[2431,213,2430,848,188],"class_list":["post-6112","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-technology","tag-alessio","tag-artificial-intelligence","tag-figalli","tag-mathematics","tag-your-feed-science"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What\u2019s Best, According to the Italian Mathematician Alessio Figalli - World News<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/site.itshrt.com\/worldnews\/whats-best-according-to-the-italian-mathematician-alessio-figalli\/\" \/>\n<meta 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