{"id":270468,"date":"2023-11-22T20:22:24","date_gmt":"2023-11-22T14:52:24","guid":{"rendered":"https:\/\/forumias.com\/blog\/?p=270468"},"modified":"2023-11-22T20:22:24","modified_gmt":"2023-11-22T14:52:24","slug":"the-langlands-program","status":"publish","type":"post","link":"https:\/\/forumias.com\/blog\/the-langlands-program\/","title":{"rendered":"The Langlands Program"},"content":{"rendered":"

Source<\/strong>– This post is based on the article “Meet the Langlands Program, the world\u2019s biggest maths project | Explained<\/strong>” published in “The Hindu<\/strong>” on November 21, 2023.<\/p>\n

Why in the News?<\/h2>\n

Langlands Program provides a beautifully intricate set of connections between various areas of mathematics, pointing the way toward novel solutions for old problems in mathematics.<\/p>\n

What is the Langlands Program?<\/h2>\n\n\n\n\n\n\n\n
Aspect<\/strong><\/td>\nDetails<\/strong><\/td>\n<\/tr>\n
What is it?<\/strong><\/td>\n1. It is the world\u2019s biggest maths project <\/strong>and can be likened to building bridges across mathematical cultures with different objects and languages.
\n<\/strong>2. It is named after Robert Langlands<\/strong>, a mathematician at the Institute for Advanced Study in Princeton.
\n3. In 2018, he was awarded the Abel Prize<\/strong> for \u201chis visionary program connecting representation theory to number theory\u201d<\/td>\n<\/tr>\n
Objective<\/strong><\/td>\nTo find<\/strong> connections between <\/strong>two far-flung areas of mathematics – number theory and harmonic analysis.
\n<\/strong>– It seeks to do this by connecting every Galois group with automorphic functions<\/strong>, allowing mathematicians to investigate polynomial equations using tools from calculus.<\/td>\n<\/tr>\n
Initiation<\/strong><\/td>\n1967<\/td>\n<\/tr>\n
Significance of the program<\/strong><\/td>\n1. Solving old Maths problems<\/strong>– In 1994, Andrew Wiles and Richard Taylor applied Langlands\u2019 conjectures to prove Fermat\u2019s last theorem<\/strong>.
\n2. Creating new functions<\/strong>– It can help mathematicians create new automorphic functions from preexisting ones. Such efforts could be crucial to prove the Ramanujan conjectures<\/strong> (speculations), many of which remain unsolved.
\n3. Can evolve into new fields of mathematics<\/strong>–\u00a0For example, Geometric Langlands<\/strong>. It investigates connections between algebraic geometry and representation theory.
\n4. Multidisciplinary significance<\/strong>– Mathematicians have suggested connections between Geometric Langlands<\/strong> and physics, particularly for studying electromagnetic waves.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Note-<\/strong>
\n1. Number theory<\/strong>: It is the arithmetic study of numbers and the relationships between them. For example, the pythagoras theorem.<\/p>\n

2. Harmonic analysis<\/strong>: It is interested in the study of periodic phenomena. Unlike number theorists, who deal with discrete arithmetic’s (like integers), harmonic analysts deal with mathematical objects more continuous in nature (like waves).<\/p>\n

3. Galois group<\/strong>: In 1832, mathematician \u00c9variste Galois concluded that it is impossible to have a general formula to find the precise roots of polynomial equations and instead suggested focusing on symmetries between roots as an alternate route. Galois group is a collection of such symmetries of the roots of a polynomial equation.<\/p>\n

4. Automorphic functions<\/strong>: Functions that turn back into themselves when the variables are changed by some process are called automorphic functions. The sine function is a simple example.<\/p>\n

UPSC Syllabus- Science & Technology<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Source– This post is based on the article “Meet the Langlands Program, the world\u2019s biggest maths project | Explained” published in “The Hindu” on November 21, 2023. Why in the News? Langlands Program provides a beautifully intricate set of connections between various areas of mathematics, pointing the way toward novel solutions for old problems in… Continue reading The Langlands Program<\/span><\/a><\/p>\n","protected":false},"author":10367,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"aside","meta":{"jetpack_post_was_ever_published":false,"footnotes":""},"categories":[1566,1738,11210],"tags":[11872],"class_list":["post-270468","post","type-post","status-publish","format-aside","hentry","category-daily-factly-articles","category-science-and-technology-daily-factly-articles","category-personalities-in-news","tag-9pm-daily-factly","post_format-post-format-aside","entry"],"jetpack_featured_media_url":"","views":{"total":17,"cached_at":"","cached_date":1700676385},"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/posts\/270468","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/users\/10367"}],"replies":[{"embeddable":true,"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/comments?post=270468"}],"version-history":[{"count":0,"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/posts\/270468\/revisions"}],"wp:attachment":[{"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/media?parent=270468"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/categories?post=270468"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/tags?post=270468"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}