90年历史第三人！破解百年几何难题，35岁华裔女数学家王虹摘
字幕摘录
| 时间 | 英文 | 中文 |
|---|---|---|
| 0:01 | Some of the hardest problems in pure mathematics look simple at first. | 纯数学中的一些最难的问题起初看起来很简单. |
| 0:06 | Imagine holding a pen and slowly turning it so it points in every direction. | 想象一下拿着一支笔,慢慢地转动,这样它就指向每一个方向. |
| 0:11 | How much space does the needle need to turn? | 针头需要多少空间才能转动? |
| 0:15 | Your first guess might be that it sweeps out a solid sphere, | 你的第一个猜测可能是 它扫荡了一个坚实的球体, |
| 0:19 | but over a century ago, mathematicians discovered this simple puzzle | 但一个多世纪前 数学家发现了这个简单的谜题 |
| 0:23 | actually reveals the strange behavior of high-frequency waves. | 实际上揭示了高频波的奇怪行为. |
| 0:28 | This turned it into a major research quest. | 这使得它变成了一项重大的研究探索. |
| 0:32 | In February 2025, a paper appeared on Archive | 2025年2月,一篇论文出现在 Archive |
| 0:36 | that solved the three-dimensional Kakeya set conjecture. | 解决了三维卡基亚设定猜想. |
| 0:40 | Wang和她的合作者Joshua Zall 贴的证明是 |
展开字幕全文（195 条）
| 序号 | 英文 | 中文 |
|---|---|---|
| 1 | Some of the hardest problems in pure mathematics look simple at first. | 纯数学中的一些最难的问题起初看起来很简单. |
| 2 | Imagine holding a pen and slowly turning it so it points in every direction. | 想象一下拿着一支笔,慢慢地转动,这样它就指向每一个方向. |
| 3 | How much space does the needle need to turn? | 针头需要多少空间才能转动? |
| 4 | Your first guess might be that it sweeps out a solid sphere, | 你的第一个猜测可能是 它扫荡了一个坚实的球体, |
| 5 | but over a century ago, mathematicians discovered this simple puzzle | 但一个多世纪前 数学家发现了这个简单的谜题 |
| 6 | actually reveals the strange behavior of high-frequency waves. | 实际上揭示了高频波的奇怪行为. |
| 7 | This turned it into a major research quest. | 这使得它变成了一项重大的研究探索. |
| 8 | In February 2025, a paper appeared on Archive | 2025年2月,一篇论文出现在 Archive |
| 9 | that solved the three-dimensional Kakeya set conjecture. | 解决了三维卡基亚设定猜想. |
| 10 | Wang和她的合作者Joshua Zall 贴的证明是 | |
| 11 | Volume Estimates for Unions of Convex Sets | Convex 设备联盟的量估计 |
| 12 | and the Kakeya Set Conjecture in Three Dimensions. | 和Kakeya设定的三维猜想。 |
| 13 | Within days, mathematicians everywhere realized the significance of their work. | 几天之内,各地的数学家都意识到自己工作的重要性. |
| 14 | The impact was so strong that MIT's Larry Guth wrote a 21-page survey | 影响如此强烈,麻省理工学院的拉里·古斯写了一篇长达21页的调查 |
| 15 | to help others understand the proof. | 帮助他人理解证据 |
| 16 | By January 2026, Guth, Wang, and Zall released a shorter 47-page version | 到了2026年1月,古思,王和扎尔发行了较短的47页版本. |
| 17 | to make the main ideas easier to follow. | 使主要想法更容易遵循。 |
| 18 | 赖斯大学数学家Nets Katz说: | |
| 19 | This thing doesn't need hyping up. | 这东西不需要叫声 |
| 20 | Eyal Lubetzky, chair of mathematics at NYU's Courant Institute, | Eyal Lubetzky,纽约大学库兰特研究所数学系主任, |
| 21 | called it one of the top mathematical achievements of the 21st century, | 这被称为21世纪最优秀的数学成就之一, |
| 22 | and the Institute for Advanced Study noted the problem had | 高级研究所指出,问题在于: |
| 23 | eluded some of the greatest living mathematicians. | 躲过了一些最伟大的数学家 |
| 24 | Put simply, the theorem says that any Kakeya set in three-dimensional space, | 简而言之,定理说任何三维空间的卡基亚 |
| 25 | meaning any set that contains a line segment of length one in every direction, | 意思是每个方向都有一个线段, |
| 26 | must have both a Minkowski dimension and a Hausdorff dimension of exactly three. | 明考斯基的维度 和豪斯多夫的维度 必须是3 |
| 27 | Wang and Zall proved that there is a strict mathematical limit to how small a pattern can be | 王和扎尔证明,对于一个图案的大小,有一个严格的数学极限. |
| 28 | when you rotate a one-dimensional segment, like a pen, | 当你旋转一个单维段, 像笔, |
| 29 | in every direction within a three-dimensional space. | 在三维空间的每一个方向 |
| 30 | Experts agree this is truly a once-in-a-century proof. | 专家一致认为,这确实是一个世纪一度的证据。 |
| 31 | The 3D case is far more important than the 2D case, which was solved in the 1970s, | 3D案比1970年代破获的2D案重要得多, |
| 32 | because adding a third dimension brings much more complex geometry, especially with skew lines. | 因为加上第三个维,带来了更复杂的几何学,尤其是用skew线. |
| 33 | In three dimensions, tubes can point in many directions and still overlap in complex ways, | 在三个维度上,管子可以指向许多方向,并且仍然以复杂的方式重叠, |
| 34 | allowing them to pack much more densely than on a flat surface. | 让他们比平坦的表面 更密集的包裹 |
| 35 | The breakthrough quickly made Hong Wang its young author, | 突破很快使洪旺成为了年轻的作者, |
| 36 | one of the most promising mathematicians of the coming decades. | 未来几十年最有前途的数学家之一. |
| 37 | She was already well-known, having won the 2022 Miriam Mirzakhani New Frontiers Prize | 赢得2022年米里亚姆·米尔扎哈尼新疆奖 她已经是名人了 |
| 38 | for her work in restriction theory and local smoothing conjectures before solving the Kakeya conjecture. | 在解开卡基亚猜想之前 在限制理论和局部平滑猜想方面的工作 |
| 39 | But the 2025 proof took her to a new level. | 但2025年的证明让她进入了一个新的水平. |
| 40 | After publishing it, she received the 2025 Salem Prize, the 2025 Ostrowski Prize, | 出版后,她获得了2025年萨利姆奖,2025年奥斯特罗夫斯基奖, |
| 41 | the 2025 ICCM Gold Medal, and the 2026 Association for Women in Mathematics Sadowski Prize. | 2025年ICCM金质奖章,以及2026年数学界妇女协会萨多夫斯基奖. |
| 42 | She is widely regarded as the leading contender for the 2026 Fields Medal, | 她被普遍认为是2026年菲尔兹奖章的主要竞争者, |
| 43 | the closest thing mathematics has to a Nobel Prize in July 2026. | 数学最接近2026年7月的诺贝尔奖 |
| 44 | There is no official shortlist for the Fields Medal, | 菲尔兹奖章没有正式的短名单, |
| 45 | and the International Mathematical Union keeps its decisions secret, | 以及国际数学联合会 保密的决定, |
| 46 | so any talk about the prize is just speculation. | 所以,任何关于奖金的谈话都是猜测。 |
| 47 | But Wang's acumen, which combines a rare mix of intuition, persistence, and technical know-how, | 但王敏敏,结合了罕见的 直觉,坚持,技术诀窍, |
| 48 | makes her one of the most deserving candidates. | 让她成为最有资格的候选人之一 |
| 49 | You might be wondering, doesn't the long-standing focus on a spinning needle seem strange or even pointless? | 你可能想知道, 长期关注旋转的针头 是不是看起来很奇怪甚至毫无意义? |
| 50 | Why spend so much time and effort on it? | 为什么要花这么多时间和精力去研究它? |
| 51 | Who cares? | 谁在乎? |
| 52 | The reason is that the Kakeya conjecture, in fact, | 原因是卡基亚猜想,事实上, |
| 53 | forms the foundation for a whole series of important problems in harmonic analysis. | 构成整系列口琴分析中重要问题的基础. |
| 54 | The saga began in 1917 when Japanese mathematician Soichi Kakeya asked his original needle question. | 萨迦起源于1917年日本数学家卡基亚素一(英语:Soichi Kakeya)问他最初的针头问题. |
| 55 | Imagine rotating a needle in every direction. | 想象一下每个方向旋转一根针. |
| 56 | You might think the needle would cover a solid ball of space in 3D or a circle in 2D. | 你可能认为针会覆盖一个3D的固态球或2D的圆圈. |
| 57 | But mathematicians found a surprising trick. | 但是数学家发现了一个令人惊讶的诡计. |
| 58 | By moving the needle in careful sliding steps, you can create a set that is incredibly thin. | 通过在细心的滑动步骤中移动针头, 你可以创建出一个非常薄的集合。 |
| 59 | While the original 2D volume question was famously answered by Abram Besikovitch in 1928, | 虽然最初的2D卷问题在1928年被艾布拉姆·贝西科维奇(Abram Besikovitch)有名地回答, |
| 60 | who shockingly proved the area could be pushed arbitrarily close to zero, | 他们令人震惊地证明,该地区可能会被任意推至接近零, |
| 61 | mathematically it is possible to make a shape with zero volume that still contains a line segment pointing in every possible direction. | 在数学上,可以制作一个形状,其零体积仍然包含指向每个可能方向的线段。 |
| 62 | Over time, the focus changed from volume to dimension, leading to the concept of a Kakeya set. | 随着时间推移,焦点从体积变为维度,导致卡基亚集的概念. |
| 63 | In higher mathematics, volume and dimension mean different things. | 在更高的数学中,体积和维度意味着不同的事物. |
| 64 | Volume is just the amount of space an object occupies. | 量只是物体占用的空间。 |
| 65 | Dimension, like Hausdorff or Minkowski dimension, describes how complex or rough a shape is, or how it scales. | 维度,如Hausdorff或Minkowski维度,描述一个形状有多复杂或粗糙,或者它如何缩放. |
| 66 | To picture fractal dimension, think of a block of Swiss cheese or a sponge. | 为了描绘分形的维度 想想瑞士奶酪或海绵 |
| 67 | You can keep making tiny holes until almost no material remains, so the volume is small. | 你可以继续制造小孔,直到几乎没有物料留下,所以体积很小. |
| 68 | But the sponge still stretches in three directions, length, width, and depth, so its structure is still fully 3D. | 但海绵仍以三个方向延伸,长度,宽度和深度,因此其结构仍然完全为3D. |
| 69 | The Kakeya conjecture says that even if a Kakeya set is built in such a tricky way that it has zero volume, | 卡基亚猜想中说,即使卡基亚集的建造方式如此狡猾,其体积为零, |
| 70 | its fractal dimension still matches the full dimension of the space. | 其分形维度仍然与空间的全部维度相符。 |
| 71 | In 3D, that means its dimension is still 3. | 在3D中,这意味着它的维度仍然是3. |
| 72 | This is the heart of the paradox that Wang and Zoll solved. | 这是王和佐尔解决的矛盾的核心. |
| 73 | A shape can have almost no volume, but still keep its full fractal dimension. | 一个形状几乎没有体积,但仍保留其完整的分形维度. |
| 74 | This question became central because researchers saw it as the base of a tower of conjectures. | 这个问题成为中心,因为研究者视之为猜想塔的基础. |
| 75 | The Kakeya conjecture is the foundation, with the restriction conjecture above it, | 卡基亚猜想是基础 上面有限制猜想 |
| 76 | then the Bach-Norish conjecture, and finally the local smoothing conjecture at the top. | 然后是巴赫 -诺瑞希猜想 最后是当地最平滑的猜想 |
| 77 | Each of these big conjectures depends on the one below it. | 每一个大猜想都取决于下面的猜想. |
| 78 | If the Kakeya conjecture had turned out to be false, if there was a super-monster counterexample in 3D, | 如果卡基亚猜想是虚假的 如果3D有超级怪兽反例 |
| 79 | the whole structure of theories about wave equations and the Fourier transform could have fallen apart. | 关于波的方程和傅里叶变迁的整个理论结构 可能已经崩溃了 |
| 80 | Wang and Zoll showed that in 3D, no matter how cleverly you arrange your thin needles to overlap, | Wang和Zoll在3D显示 无论你如何巧妙地安排你的细针重叠 |
| 81 | the resulting fractal shape still has three dimensions. | 由此产生的分形形状仍然有三个维度。 |
| 82 | It seems strange, but it is true. | 这看起来很奇怪,但这是真的。 |
| 83 | You can make a cage with zero volume, but it will always be three-dimensional. | 你可以做一个没有体积的笼子,但它总是三维的. |
| 84 | Now that the Kakeya conjecture has been proven in 3D, | 现在,卡基亚猜想 已经证明在3D, |
| 85 | many other tough questions above it seem much more likely to be solved. | 上面的许多其他棘手问题似乎更有可能得到解决。 |
| 86 | It's not just about the solution, it's more like how different parts of mathematics fit together. | 这不仅仅是解决方案,更像是数学的不同部分如何融合在一起. |
| 87 | Hongwang was born in 1991 in Guilin, a city in China's Guangxi region known for its lush, surreal karst mountains. | 洪旺于1991年出生于中国广西地区的一个城市桂林,以茂盛,超现实的喀斯特山著称. |
| 88 | The landscape is so beautiful that a famous proverb says, | 景色如此美丽 以至于一个著名的谚语说, |
| 89 | I'd rather be born in Guilin than be a god. | 宁生于桂林而不为神. |
| 90 | Raised in a household with parents who were both secondary school teachers, she skipped multiple grades. | 在父母都是中学教师的家庭中长大的,她跳过多个年级。 |
| 91 | By the age of 16, she scored a 6.53 on the grueling gaokao exam, China's equivalent of the SATs, | 到了16岁,她以6.53分的成绩考上了令人毛骨悚然的高考,中国相当于SAT, |
| 92 | and placed her in the top 1% or higher of examinees in her province and entered the prestigious Peking University, or PKU. | 并把她安排在她所在省份的考试成绩最高的1%或更高,并进入了著名的北京大学,或PKU. |
| 93 | But she did not follow the usual path of Chinese International Mathematical Olympiad competitors who often become breakout math stars. | 但她并没有遵循中国国际数学奥林匹克竞赛对手的通常道路,这些竞争对手经常成为突围数学明星. |
| 94 | In fact, she started university at the School of Earth and Space Sciences at Peking University, rather than in mathematics. | 实际上,她开始就读于北京大学地球与空间科学学院,而不是数学专业. |
| 95 | Her decision to switch from earth and space sciences to mathematics after a year was a turning point. | 她决定一年后从地球和空间科学转向数学是一个转折点. |
| 96 | Wang has talked about how the transition was initially difficult. | 汪精卫曾谈到过渡最初是如何困难的. |
| 97 | She often struggled, and her grades were not great at first. | 她经常挣扎,起初成绩不高. |
| 98 | Peking University was where Wang truly became a researcher. | 北京大学是汪真成为研究员的地方. |
| 99 | She earned her bachelor's degree from PKU in 2011 after transferring to the School of Mathematical Sciences. | 2011年她转入数学科学学院后获得了PKU的学士学位. |
| 100 | At PKU, she realized how different research is from exams. | 在PKU,她意识到研究与考试有多不同. |
| 101 | Exams are neat problems with clear answers, meant to be solved quickly. | 考试是有明确答案的整洁问题,意在迅速解决. |
| 102 | In research, you might spend a long time trying many messy, failed ideas before finding one that works. | 在研究中,你可能会花很长的时间 尝试许多混乱,失败的想法 然后再找到一个有效的。 |
| 103 | PKU gave her tough mathematical training, but it never made things feel too easy. | PKU给了她艰难的数学训练,但从未让事情变得太容易. |
| 104 | She realized she needed strong mental resilience as much as intelligence. | 她意识到她需要强大的精神适应能力 和智能一样多. |
| 105 | She learned this in the dorms and libraries of Beijing, | 她在北京的宿舍和图书馆学的 |
| 106 | where she changed from a student learning facts to a researcher learning to handle the stress of unsolved problems. | 在那里,她从一个学生学习事实变成了一个研究者学习,以应对未决问题的压力. |
| 107 | After graduating from PKU, Wang went to France, earning a master's from Université Paris-Saclay by 2014. | 从PKU毕业后,王前往法国,到2014年在巴黎萨克莱大学获得硕士学位. |
| 108 | Her time in Paris was full of growth and deep self-doubt. | 她在巴黎的时间充满了成长和深刻的自我怀疑. |
| 109 | In interviews, Wang shared how she became so unsure about her path in France that she stopped doing mathematics for about six months. | 在接受采访时,王 shared讲述了她如何对自己在法国的路径变得如此不肯定,以至于她停止了数学工作约六个月. |
| 110 | During this break, she seriously considered switching to architecture. | 在此期间,她认真考虑了转向建筑. |
| 111 | She was interested in how architects use computers and spatial thinking to turn ideas into designs. | 她对建筑师如何使用计算机和空间思维将想法转化为设计感兴趣. |
| 112 | What brought her back to mathematics was her motto, follow interest and feeling. | 让她回到数学界的,是她的座右铭,跟随兴趣和感觉. |
| 113 | With her commitment reaffirmed, Wang crossed the Atlantic to begin her doctoral studies at the Massachusetts Institute of Technology in 2014. | 在承诺的重申下,王跨大西洋于2014年开始在麻省理工学院攻读博士学位. |
| 114 | There, she was paired with Larry Guth, a professor whose mentorship would deeply influence her mathematical style and trajectory. | 在那里,她与拉里·格斯(Larry Guth)配对,这位教授的导师将深深影响她的数学风格和轨迹. |
| 115 | Under Guth, she focused on Fourier analysis and restriction theory, completing her PhD thesis in 2019. | 在Guth下,她专注于傅里叶分析和限制理论,于2019年完成了她的博士论文. |
| 116 | Guth taught her the beauty of mathematics. | 古斯教她数学的美. |
| 117 | Wang later said he showed her how to think systematically about research and, most importantly, how to develop a sense of mathematical taste. | 王后表示,他教她如何系统地思考研究,最重要的是,如何培养数学味觉. |
| 118 | This means knowing which problems are truly deep and which are just hard, and it became her guide. | 这意味着知道哪些问题是真正深刻的,哪些是困难的,它成为了她的向导。 |
| 119 | After MIT, her career took off quickly. | 在麻省理工学院毕业后,她的职业生涯迅速起飞. |
| 120 | She held a postdoc at the Institute for Advanced Study in Princeton from 2019 to 2021, then became an assistant professor at UCLA in 2021. | 她于2019年至2021年在普林斯顿高等研究院担任博士后,后于2021年成为加州大学洛杉矶分校的助理教授. |
| 121 | By 2023, she joined NYU's Courant Institute, and by 2025 to 26, she held professorships at both NYU and in France. | 到2023年,她加入了纽约大学的库兰特学院,到2025年到26年,她同时在纽约大学和法国担任教授职务. |
| 122 | Her research sits at the intersection of Fourier analysis, geometric measure theory, and incidence geometry. | 她的研究坐落在傅里叶分析,几何测量理论,和事件几何的交汇处. |
| 123 | Harmonic analysis is the process of breaking complex signals into simple waves using the Fourier transform. | 谐波分析(Harmonic analysis)是利用傅里叶变换将复杂信号破解成简单波的过程. |
| 124 | For example, it's like taking the mixed sounds of an orchestra and separating them into the individual notes each instrument plays. | 例如,它就像把一个管弦乐团的混合声音分解成每个乐器演奏的单个音符. |
| 125 | Since waves are key to sound, images, and quantum physics, harmonic analysis is a foundation for understanding the physical world. | 由于波是声音,图像,和量子物理的关键,谐波分析是理解物理世界的基础. |
| 126 | The Fourier transform she studies is the same math used in practical applications such as image compression and MRI scans. | 她研究的傅里叶变换是图像压缩和核磁共振扫描等实际应用中使用的相同数学. |
| 127 | Geometric measure theory, in contrast, studies how size, shape, and dimension behave for very irregular or jagged sets of points, such as fractals. | 相形之下,几何测量理论研究了大小,形状,和维度对于非常不规则或被划断的几组点,如分形的动作. |
| 128 | Wang focused on problems involving tubes, curves, and dimensions because she thinks very spatially. | 王专注于涉及管子,曲线和维度的问题,因为她认为非常空间化. |
| 129 | The Kakeya problem is right at the intersection of these fields. | 卡凯亚问题恰好位于这些领域交汇处. |
| 130 | The way needles or tubes are arranged in geometric measure theory models, how high frequency waves overlap and interact in harmonic analysis. | 针或管在几何测量理论模型中的排列方式,高频波在谐波分析中如何重叠和相互作用. |
| 131 | No mathematician works completely alone, and the story of the 3D Kakeya proof is truly a partnership. | 没有数学家完全单独工作,3D卡基亚证明的故事就是真正的伙伴关系. |
| 132 | Joshua Zoll was that partner. | 约书亚・佐尔就是那个搭档 |
| 133 | Born in 1986, Zoll earned his bachelor's degree from the California Institute of Technology in 2008, followed by a PhD from UCLA in 2013. | 佐尔出生于1986年,2008年在加州理工学院获得学士学位,后于2013年获得加州大学洛杉矶分校博士学位. |
| 134 | His research interests are very similar to Wang's, focusing on harmonic analysis, combinatorics, incidence geometry, and Kakeya type problems. | 他的研究兴趣与王氏非常相似,注重谐音分析,梳理,事件几何,以及卡凯亚类型问题. |
| 135 | Great proofs are almost never sudden flashes of inspiration. | 伟大的证据几乎从来不是 突然的灵感闪烁。 |
| 136 | They are built on the work of many earlier mathematicians. | 它们建立在许多早期数学家的作品之上. |
| 137 | Before the 2025 proof, experts spent decades chipping away at Kakeya step by step. | 在2025年的证明之前,专家们花了几十年的时间一步一步地在卡基亚大开杀戒。 |
| 138 | In 1971, Roy Davies successfully solved the two dimensional case of the Kakeya problem. | 1971年,罗伊·戴维斯成功解决了卡基亚问题的二维案件. |
| 139 | However, the 3D case remained a stubborn fortress. | 然而,3D案仍然是一个固执的堡垒. |
| 140 | In 1995, Tom Wolfe achieved a major milestone by proving a lower bound, showing that a 3D Kakeya set must have dimension at least 2.5. | 1995年,汤姆·沃尔夫(Tom Wolfe)通过证明下限,表明3D卡基亚集必须至少有2.5个维度,从而实现了一个重要的里程碑. |
| 141 | In 2014, Nets Katz and Terence Tao developed a strategic roadmap, identifying that any counterexample to the Kakeya conjecture would have to possess a pathological property they called stickiness. | 2014年,Nets Katz和Terence Tao制定了战略路线图,确定卡凯亚猜想的任何反例都必须拥有他们所称的粘性病理属性. |
| 142 | Where tubes pointing in similar directions clump uncomfortably close together. | 指向类似方向的管子会挤合在一起 |
| 143 | Concurrently, Larry Guth provided another vital insight, showing that any 3D counterexample would have to be grainy, organized into small 3D chunks. | 与此同时,拉里·古斯提供了另一个至关重要的洞察力,表明任何3D反例都必须是谷状的,组织成小的3D块. |
| 144 | Wang and Zoll used the earlier bounds and translated the problem into families of thin tubes, which provided a basic picture of the rotating needles. | 王和佐尔使用早期的界限,将问题翻译为薄管的家族,这提供了旋转针的基本画面. |
| 145 | The first battlefield was the sticky sets, a victory they claimed in 2022 by proving tubes couldn't clump together pathologically based on their angles. | 第一个战场是粘着的套装,他们声称2022年的胜利是通过证明管子无法根据角度在病理上凝聚在一起. |
| 146 | The second battlefield involved dealing with what was left, Guth's grainy sets. | 第二个战场涉及处理剩下的东西 古斯的粮仓 |
| 147 | A grainy set is made up of tiny, chopped up sections of tubes, or grains, where a massive number of tubes overlap. | 一个谷状的装置由细小的,被切碎的管子部分组成,或者说由大量管子重叠的谷子组成. |
| 148 | Instead of looking at the long tubes, Wang and Zoll zoomed in and analyzed how these microscopic grains could intersect. | 与其看长管,王和佐尔放大 并分析了这些微粒是如何相交的。 |
| 149 | They created a step-by-step bootstrapping process that is said to be remarkable. | 他们创造了一个渐进的拖鞋过程,据说是惊人的。 |
| 150 | They started with Tom Wolf's earlier bound of 2.5 and used it in their grain overlap equations. | 他们从汤姆·沃尔夫早先的2.5捆绑开始,并在他们的谷物重叠方程中使用. |
| 151 | The way the grains fit together raised the bound a bit. | 谷物的结合方式 使束缚提高一点点。 |
| 152 | Then they used this new, higher number to repeat the process, raising the bound even more each time. | 然后他们用这个新的,更高的数字来重复这个过程,每次提高束缚. |
| 153 | Terrence Tao compared this process to a perpetual motion machine, because a small geometric improvement could be applied repeatedly until it reached the full 3D dimension. | 特伦斯 陶将这个过程比作一个永恒的运动机,因为一个小的几何改进可以反复应用,直到达到完整的3D维. |
| 154 | Now that the foundation is secure, the impact on harmonic analysis and partial differential equations is huge. | 现在基础是安全的,对口琴分析和部分微分方程的影响是巨大的. |
| 155 | Mathematicians can be confident that wave equations will remain stable, without worrying about Kakeya-type problems. | 数学家们可以相信波的方程会保持稳定,而不用担心卡凯亚型的问题. |
| 156 | Mathematicians often judge a theorem's importance by the new methods it creates, even if there's no immediate product. | 数学家们常常用它所创造的新方法来判断一个定理的重要性,即使没有直接的产品. |
| 157 | The grain analysis and projection techniques Wang and Zoll developed are now standard tools. | 所开发的谷物分析和投影技术王和佐尔现在是标准的工具. |
| 158 | Wang's recent work is already moving toward the Stein restriction conjecture. | 王刚的作品已经向施泰因限制猜想发展. |
| 159 | Pure math usually takes a long time to show its benefits. | 纯粹的数学通常需要很长的时间才能显示其好处. |
| 160 | It won't lead to a new microchip or engine right away. | 它不会马上导致一个新的微芯片或引擎. |
| 161 | But Kakeya problems are closely connected to Fourier analysis, wave equations, number theory, cryptography, and randomness theory. | 但卡凯亚问题与傅里叶分析,波浪方程,数字理论,密码学,随机性理论密切相关. |
| 162 | This proof will lead to the creation of MRI machines, but it also strengthens the theory behind technologies based on waves and signals. | 这种证明会导致核磁共振机的产生,但也加强了基于波和信号的技术背后的理论. |
| 163 | Mathematicians care about this work even without instant applications, because understanding the limits of wave interference helps engineers build more stable systems in the future. | 数学家即使没有即时应用也关心这项工作,因为了解波干扰的极限有助于工程师在未来建立更稳定的系统. |
| 164 | It's the same thing as inventing a new light bulb versus understanding how light itself works. | 这与发明新灯泡 和了解光本身如何工作是一样的 |
| 165 | After the February 2025 Kakeya breakthrough, she collected the Salem Prize, the Ostrowski Prize, and the ICCM Gold Medal of Mathematics. | 2025年2月卡基亚突破后,她获得了萨利姆奖,奥斯特罗夫斯基奖,以及ICCM数学金质奖章. |
| 166 | The Association for Women in Mathematics gave her the 2025 Sadowski Prize in Analysis, featuring her trailblazing ideas. | 数学界妇女协会授予她2025年萨多夫斯基分析奖,以她开创性的思想为主. |
| 167 | At the same time, NYU named her a silver professor. | 同时,纽约大学也为她指定了一位银质教授. |
| 168 | This momentum points toward the highest honor in mathematics. | 这一势头表明了数学的最高荣誉。 |
| 169 | The Fields Medal is awarded every four years at the International Congress of Mathematicians. | 菲尔兹奖章每四年在国际数学家大会上颁发一次. |
| 170 | It's often called the Nobel of Math because it brings instant lasting recognition. | 它经常被称为"数学的诺贝尔",因为它带来即时的持久认可. |
| 171 | Recipients must still be under 40 on January 1 of the award year. | 获奖者在获奖年份的1月1日仍须在40岁以下. |
| 172 | The next ICM is scheduled for Philadelphia, July 23 to 30, 2026. | 接下来的ICM定于2026年7月23日至30日在费城举行. |
| 173 | Wang, born in 1991, easily meets the age eligibility requirement, and the timing in 2026 fits perfectly with the digestion of her proof. | 王,生于1991年,容易满足年龄资格要求,2026年的计时与她证明的消化完全吻合. |
| 174 | On Polymarket, a major prediction platform, the contract for who will win the 2026 Fields Medal | 在Polymarket,一个主要的预测平台, 合同谁将赢得2026菲尔兹奖章 |
| 175 | has shown Wang's chances staying between 80% and 87.5 in April 2026. | 显示王在2026年4月的停留机会在80%到87.5之间. |
| 176 | In betting terms, odds of 1 seventh show that many people see her as the clear favorite. | 从赌注上看,七分之一的概率 表明很多人认为她是明显最喜欢的 |
| 177 | After Wang, the odds drop sharply. | 王后,机率急剧下降. |
| 178 | Jacob Simerman, known for his work on the Andre Oort Conjecture, is next with a 56% to 75% chance, while Sam Raskin and Jack Thorne are around 48%. | 雅各布·西默曼(Jacob Simerman)以安德烈奥特猜想(英语:Andre Oort Controduction)的作品而闻名,他以56%到75%的概率成为下一个,而萨姆·拉斯金和杰克·索恩则约为48%. |
| 179 | Wang has spoken publicly about her relationship with exercise and her love of sports. | 王 has公开谈她与锻炼的关系和对体育的热爱. |
| 180 | She makes time to rest, refusing to force math when exhausted, and has engaged in table tennis, fencing, and even online taekwondo. | 她让时间休息,在疲惫时拒绝强迫数学,还从事桌球,栅栏,甚至在线跆拳道. |
| 181 | Talking about fencing, she said, fencing is an activity which requires different kinds of postures, and these postures must be accurate. | 她说,在谈论栅栏时,栅栏是一种需要不同姿势的活动,这些姿势必须准确。 |
| 182 | Therefore, after I have fenced, I tend to be obsessed with those postures that I have done badly. | 因此,在我被围起来之后,我往往沉迷于那些我做得不好的姿势。 |
| 183 | Importantly, she believes in the value of rest for serious thinking. | 重要的是,她相信休息对认真思考的价值。 |
| 184 | Her thoughts on procrastination and the value of starting early show that she is practical. | 她关于拖延的想法和开始早期的价值表明她是实用的. |
| 185 | Her advice to students is simple. | 她对学生的建议很简单. |
| 186 | Look for those with mental resilience. | 寻找那些有精神复原能力的人. |
| 187 | These details matter because they show how she manages the challenges of difficult work. | 这些细节很重要,因为它们显示了她如何应对困难工作的挑战。 |
| 188 | A mathematician like Wang doesn't stop after reaching one big goal. | 像王一样的数学家在达到一个大目标后不会停止. |
| 189 | She looks for the next challenge. | 她寻找下一个挑战。 |
| 190 | Recent reports and conference talks show Wang is already moving beyond Kakeya and working on the Stein restriction conjecture. | 最近的报告和会议会谈显示,王已经超越了卡基亚,致力于斯坦因限制猜想. |
| 191 | The restriction problem created by Elias Stein in the 1960s is the next big step. | Elias Stein在20世纪60年代产生的限制问题是下一个大步骤. |
| 192 | She is now working with Shukun Wu of Indiana University on a geometric approach to the restriction conjecture. | 她现在与印第安纳大学的Shukun Wu合作,研究限制猜想的几何方法. |
| 193 | Instead of relying exclusively on traditional harmonic analysis, Wang and her collaborators are now making new estimates of how often thin tubes in space cross using a purely geometric perspective. | 王氏与其完全依靠传统的谐波分析,不如现在与其合作者用纯几何视角对太空交叉中的薄管的频率作出新的估计. |
| 194 | More generally, her research continues to connect Fourier analysis, geometric measure theory, and incidence geometry. | 更一般地说,她的研究继续连接着傅里叶分析,几何测量理论,以及事件几何. |
| 195 | With four de-Kakeya and related restriction problems ahead, it's clear that Hong Wang's story is still unfolding. | 随着前面四个德甲及相关的限制问题,显然洪旺的故事仍在发展. |
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