Langlands program

We investigate three of 2024’s biggest breakthroughs in mathematics, including a better way to pack spheres in high dimensions, a new way to avoid forming patterns of numbers, and an 800-page proof of the so-called geometric Langlands conjecture. Read about more math breakthroughs from this year at Quanta Magazine: https://www.quantamagazine.org/the-year-in-math-20241216/ 00:04 High-Dimensional Sphere Packing Problem The sphere-packing problem is simple to state: How do you arrange identical spheres to fill as much volume as possible without overlapping? In three dimensions, you can arrange spheres in a pyramid-shaped pile, the way oranges get stacked in a grocery store. But what about in higher dimensions? In April, Quanta reported on the first significant advance on this version of the sphere-packing problem in 75 years. The result improved on the efficiency of previous packings, while making use of a novel approach: Rather than packing spheres in a nice, organized way, the mathematicians used graph theory to pack spheres in a very disorderly fashion. https://www.quantamagazine.org/to-pack-spheres-tightly-mathematicians-throw-them-at-random-20240430/ PAPERS - "The sphere packing problem in dimension 8" https://arxiv.org/abs/1603.04246 -" A new lower bound for sphere packing" https://arxiv.org/abs/2312.10026 05:10 Inevitable Patterns in Big Number Sets Three graduate students proved a better estimate of the largest size that sets can reach before they must inevitably contain arithmetic progressions, which are evenly spaced patterns of numbers. The work, which deals with how order inevitably emerges from disorder in mathematics, marks the first progress on “Szemeredi’s problem” in decades. https://www.quantamagazine.org/grad-students-find-inevitable-patterns-in-big-sets-of-numbers-20240805/ PAPERS - "On Some Sequences of Integers" https://academic.oup.com/jlms/article-abstract/s1-11/4/261/842087? - " On sets of integers containing k elements in arithmetic progression" http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27132.pdf - "Improved Bounds for Szemerédi's Theorem" https://arxiv.org/abs/2402.17995 09:33 Monumental Geometric Langlands Proof In May, a team of nine mathematicians announced a major breakthrough. They had proved what is called the geometric Langlands conjecture — a central component of a broader research program to build a “grand unified theory” of mathematics. The proof, which totaled more than 800 pages and marked the culmination of 30 years of work, was, in the words of one mathematician, a “crowning achievement.” https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/ PAPERS - "Proof of the geometric Langlands conjecture" https://people.mpim-bonn.mpg.de/gaitsgde/GLC/ #math #bestof2024 #mathematics #combinatorics --------- - VISIT our Website: https://www.quantamagazine.org - LIKE us on Facebook: https://www.facebook.com/QuantaNews - FOLLOW us Twitter: https://twitter.com/QuantaMagazine Quanta Magazine is an editorially independent publication supported by the Simons Foundation: https://www.simonsfoundation.org/