Core Viewpoint - A significant advancement in the Langlands Program has been achieved by four mathematicians, extending the connection between modular forms and Abelian varieties, which is considered a major step in the quest for a unified theory in mathematics [3][4][18]. Group 1: Background and Importance - The Langlands Program is regarded as one of the largest single projects in modern mathematics, linking number theory, algebraic geometry, and representation theory [4]. - The recent breakthrough demonstrates that ordinary Abelian varieties can correspond to a modular form, expanding the previous work done on elliptic curves [5][9]. Group 2: Research Process and Collaboration - The four mathematicians began their collaboration in 2016, aiming to follow the steps taken by Wiles and Taylor in their proof of Fermat's Last Theorem [20][21]. - They faced challenges in constructing modular forms due to the additional variables introduced by Abelian varieties, leading them to explore a weaker form of correspondence [22][24]. Group 3: Key Contributions and Findings - Chinese mathematician Pan Lue's previous research provided crucial insights that facilitated the breakthrough, particularly through his introduction of a differential operator and the relationship between local analytic vectors and modular forms [32][34]. - The team worked intensively for a week in a basement to refine Pan's methods, ultimately leading to the successful construction of modular forms applicable to ordinary Abelian varieties [36][40]. Group 4: Future Directions - The results of this research not only open new avenues for studying Abelian varieties but may also lead to new conjectures similar to the Birch and Swinnerton-Dyer conjecture [41]. - The mathematicians plan to collaborate with Pan Lue to extend their findings to non-ordinary Abelian varieties, expressing confidence in their future explorations [43].

Core Viewpoint - The article discusses a significant breakthrough in mathematics, where four mathematicians extended the modularity theorem from one-dimensional elliptic curves to the more complex two-dimensional abelian surfaces, marking a revolutionary step towards a unified theory in mathematics [5][14][46]. Group 1: Historical Context - The proof of Fermat's Last Theorem by Andrew Wiles in 1994 was a monumental event in mathematics, resolving a problem that had persisted for over 350 years [9][10]. - Wiles' proof revealed a deep connection between elliptic curves and modular forms, providing a powerful method for mathematicians to explore properties of elliptic curves through their corresponding modular forms [11][12][13]. Group 2: Recent Breakthrough - In February 2023, a team of four mathematicians proved that a large class of abelian surfaces has corresponding modular forms, extending the modularity theorem significantly [16][45]. - The team members include Frank Calegari, George Boxer, Toby Gee, and Vincent Pilloni, who collaborated to tackle a problem previously considered nearly impossible [14][16][30]. Group 3: Implications and Future Directions - This breakthrough is expected to provide new tools for solving unresolved problems in number theory, similar to how the proof of the modularity of elliptic curves opened new research avenues [20][46]. - The mathematicians aim to prove that all types of abelian surfaces satisfy the modularity condition, which could lead to further discoveries in the field [20][46].

鱼羊 一水 发自 凹非寺 量子位 | 公众号 QbitAI 如果有这么一个人，写下这样的复杂公式，并声称是受女神梦中启发所得，大家伙儿通常会送他两个字： 民科 。 但当这个人一生中数千次写下类似的数学公式和命题，并在此后的100年间，不断地被证实正确，那么就只有一个可能—— 他是拉马努金。 之所以再度火爆，是因为直到今天，数学界还不断有最新发现，在验证他当年留下的"谜题"。 拉马努金， 一位全数学界公认的神人 ，被认为是数学史上最伟大的天才之一： 没有接受过正统数学教育 ，在印度挂科到本科学位都没拿到，却凭借自己惊人的数学直觉征服数学大师G.H.哈代，使得剑桥大学三一学院的 大门破例向他打开。 32岁就英年早逝，职业搞数学的时间只有短短6年，但他的数学笔记至今仍是传奇—— 留下了近4000个公式，很多都在后来被证明正确 。 他的恩师哈代甚至开玩笑说，自己对数学最大的贡献就是发现了拉马努金： 和拉马努金的交往是我一生中唯一的浪漫事件。 △ 中间为拉马努金，最右为G.H.哈代 直到今天，后辈数学家们仍在追赶着拉马努金的步伐。 "梦中女神的启示" 拉马努金传奇故事的构成要素之一，是他独特的做数学的方式。 简单来说就 ...