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<!DOCTYPE html><html lang="zh-Hans"><head hexo-theme="https://github.com/volantis-x/hexo-theme-volantis/#6.0.3"><meta name="generator" content="Hexo 8.1.2"><meta name="Volantis" content="6.0.3"><meta charset="utf-8"><meta name="robots" content="index,follow,max-image-preview:large"><link rel="canonical" href="https://blog.mhuig.top/notes/person/4"><meta http-equiv="x-dns-prefetch-control" content="on"><link rel="dns-prefetch" href="https://static.mhuig.top"><link rel="preconnect" href="https://static.mhuig.top" crossorigin=""><meta name="renderer" content="webkit"><meta name="force-rendering" content="webkit"><meta http-equiv="X-UA-Compatible" content="IE=Edge,chrome=1"><meta name="HandheldFriendly" content="True"><meta name="mobile-web-app-capable" content="yes"><meta name="viewport" content="width=device-width,initial-scale=1,maximum-scale=5"><meta content="black-translucent" name="apple-mobile-web-app-status-bar-style"><meta content="telephone=no" 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type="application/xml" title="MHuiG Blog Site Map" href="https://blog.mhuig.top/sitemap.xml"><link rel="author" href="https://mhuig.top"><meta name="author" content="MHuiG"><meta name="creator" content="MHuiG"><link rel="archives" href="https://blog.mhuig.top/archives/"><link rel="preload" href="/css/style.css" as="style"><link rel="preload" href="https://static.mhuig.top/npm/volantis-static@0.0.1660614606622/media/fonts/VarelaRound/VarelaRound-Regular.ttf" as="font" type="font/ttf" crossorigin="anonymous"><link rel="preload" href="https://static.mhuig.top/npm/volantis-static@0.0.1660614606622/media/fonts/VarelaRound/VarelaRound-Regular.ttf" as="font" type="font/ttf" crossorigin="anonymous"><link rel="alternate" href="/atom.xml" title="Magicland" type="application/atom+xml"><link rel="alternate" href="/rss2.xml" title="Magicland" type="application/rss+xml"><title>History of mathematics: Borcherds理查德·尤恩·博赫兹的生平、学术贡献与思想历程 - Magicland</title><meta name="keywords" content="历史,Person, History,人物,博赫兹,魔群月光猜想,顶点代数,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc="" name="description" content="本文记述了菲尔兹奖得主、数学家理查德·博赫兹的生平与主要成就。他证明了著名的魔群月光猜想，并为此创立了顶点代数与广义Kac-Moody代数等核心理论。文章也涵盖其早年经历、学术生涯的挑战与个人特质，以及近期致力于量子场论数学基础研究的工作 - MHuiG - Magicland"><meta property="og:type" content="website"><meta property="og:title" content="Magicland"><meta property="og:url" content="https://blog.mhuig.top/notes/person/4"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="本文记述了菲尔兹奖得主、数学家理查德·博赫兹的生平与主要成就。他证明了著名的魔群月光猜想，并为此创立了顶点代数与广义Kac-Moody代数等核心理论。文章也涵盖其早年经历、学术生涯的挑战与个人特质，以及近期致力于量子场论数学基础研究的工作"><meta property="og:locale"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" content="2026-03-31T09:07:00.000Z"><meta property="article:modified_time" content="2026-05-03T09:10:00.000Z"><meta property="article:author" content="MHuiG"><meta property="article:tag" content="History"><meta 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itemscope="" itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="Borcherds理查德·尤恩·博赫兹的生平、学术贡献与思想历程"><meta itemprop="description" content="本文记述了菲尔兹奖得主、数学家理查德·博赫兹的生平与主要成就。他证明了著名的魔群月光猜想，并为此创立了顶点代数与广义Kac-Moody代数等核心理论。文章也涵盖其早年经历、学术生涯的挑战与个人特质，以及近期致力于量子场论数学基础研究的工作"></span><span hidden=""><meta itemprop="image" content="/lib/favicon/android-chrome-192x192.png"></span><div class="article-meta" id="top"><span hidden="" itemprop="name headline"></span></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p> <span class="p logo center large">Borcherds</span></p><br><h1 hidden="">Borcherds</h1><div class="story post-story"><h2 id="生平"><a href="#生平" class="headerlink" title="生平"></a>生平</h2><p>理查德・尤恩・博赫兹（Richard Ewen Borcherds，1959- ）年出生于南非的开普敦（Cape Town）。他的父亲彼得・霍华德・博赫兹（Peter Howard Borcherds）早年学习电气工程，后将兴趣转向数学物理，并在 Cape Town 大学担任讲师；母亲是玛格丽特・伊丽莎白・格林菲尔德（Margaret Elizabeth Greenfield）。</p><p>Borcherds 家里共有四个孩子，受家庭浓厚的学术氛围影响，他的三个兄弟中有两位后来成了数学教师，其中 Michael Borcherds 是知名数学教学软件 GeoGebra 的首席开发者。</p><p>在 Borcherds 大约一岁时，全家搬迁至英国，父亲后来成为了 Birmingham 大学的物理讲师。</p><p>Borcherds 就读于 Birmingham 的国王爱德华学校（King Edward's School）。年少时的他展现出了非凡的逻辑天赋：他是一名出色的国际象棋棋手，14 岁时便斩获了当地 Midlands 地区 21 岁以下级别的象棋冠军。</p><p>此外，他还在国际数学奥林匹克（IMO）中代表英国先后斩获过银牌和金牌。</p><p>在孩童时期，他就接触到了大量前沿的数学内容，例如数学家考克斯特（Harold Scott MacDonald Coxeter，1907-2003）关于多面体（polyhedrons）的论文，以及 Cundy 和 Rollett 合著的经典读物《数学模型》（<em>Mathematical Models</em>）。</p><p>大学阶段，Borcherds 进入了 Cambridge 大学三一学院（Trinity College）学习。</p><p>在获得学士学位后，他继续留校攻读博士学位，师从康威（John Horton Conway，1937-2000）。</p><p>尽管后来大放异彩，但起初他在学术道路上充满了自我怀疑。他曾坦言，读博期间自己并没有取得多大进展，大部分时间都在为了保住职位而苦苦挣扎。</p><p>看到同龄人（例如 1986 年 Fields 奖得主唐纳森（Simon Kirwan Donaldson，1957- ））极其成功，他一度觉得自己显然没有作为研究型数学家的潜质，甚至好几次萌生了退学的念头。</p><p>然而，他最终克服了这些困难，于 1985 年凭借关于 Leech 格（Leech lattice，一种给出 24 维空间中球体极其密集堆积的模型）的优秀论文顺利取得博士学位。</p><p>毕业后，他的学术生涯主要在 Cambridge 大学和 Berkeley 之间交替，并于 1993 年正式被任命为 Berkeley 的数学教授，至今仍保留该职位。</p><p>在科研探索中，Borcherds 经历过许多有趣的插曲。1986 年他首创了极具前瞻性的 “顶点代数（vertex algebras）” 概念，但由于理论过于超前，起初几乎无人问津。</p><p>他回忆说，自己做学术报告时经常没人来听。直到有一次，由于海报出现了排版错误，把讲座标题错拼成了流体力学中的 “涡旋代数（vortex algebras）”，意外吸引了一大批流体物理学家。当然，当这群听众发现是个错别字后，立刻对他的纯数学理论失去了兴趣。</p><p>他最著名的成就是在 1989 年证明了著名的 “魔群月光猜想（monstrous moonshine conjecture）”。为了这个猜想，他苦思冥想了八年之久。</p><p>灵感降临的时刻非常具有戏剧性：当时他在旅行，乘坐的大巴车因山体滑坡被困。正是在这趟长达 24 小时、令人极其烦躁的旅途中，他百无聊赖地在脑海里做着计算，突然灵光一闪，找到了让所有理论拼图完美契合的关键思路。</p><p>凭借这些突破性的工作，Borcherds 于 1992 年获得了伦敦数学学会的初级 Whitehead 奖（Junior Whitehead Prize$）和欧洲数学学会奖；1994 年当选为皇家学会院士。1998 年，在柏林举行的国际数学家大会上，他登上了数学界的最高荣誉殿堂 —— 荣获 Fields 奖（和高尔斯（William Timothy Gowers，1963-）、孔采维奇（Maxim Lvovich Kontsevich，1964-）、麦克姆伦（Curtis Tracy McMullen，1958-）一起分享）。</p><p>对于伴随荣誉而来的名声，他表现得极其淡然与通透：<br>“在获奖前我觉得它极其重要，获奖后却觉得它毫无意义”。</p><p>相反，纯数学的发现能带给他极大的快乐：“当我证明了月光猜想时，我简直高兴得上了天（over the moon，语带双关）。如果得出了好结果，我会一连高兴好几天。有时候我想，这或许就是服用某些药物时的感觉吧，虽然我没亲自测试过这个理论。”</p><p>在个人生活方面，Borcherds 与拓扑学家乌苏拉・格里奇（Ursula Gritsch$）结为夫妻，并育有两个女儿。</p><p>他曾坦言自己可能带有阿斯伯格综合征（Asperger syndrome）的某些特质。</p><p>近年来，Borcherds 将科研精力投入到为量子场论建立严谨的数学基础中。同时他在社交媒体网站上开设了广受欢迎的个人频道，向大众免费普及高等数学和理论物理知识。</p></div><div class="story post-story"><h2 id="贡献"><a href="#贡献" class="headerlink" title="贡献"></a>贡献</h2><h3 id="一、证明魔群月光猜想（monstrous-moonshine-conjecture）"><a href="#一、证明魔群月光猜想（monstrous-moonshine-conjecture）" class="headerlink" title="一、证明魔群月光猜想（monstrous moonshine conjecture）"></a>一、证明魔群月光猜想（monstrous moonshine conjecture）</h3><p>这是让 Borcherds 赢得 Fields 奖的巅峰之作。1979 年，John Horton Conway 和 Simon Norton 提出的 “月光猜想” 揭示了有限单群与非紧 Riemann 面之间极度非平凡的函数同构关系。</p><p>在数论中，模群<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.561ex" xmlns="http://www.w3.org/2000/svg" width="7.255ex" height="2.253ex" role="img" focusable="false" viewBox="0 -748 3206.6 996"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="SL_2(\mathbb{Z})"><g data-mml-node="mi" data-latex="S"><path data-c="1D446" d="M133 157C133 181 136 201 141 217 141 227 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300 605 300 578 313 565 340 565 368 565 397 593 397 621M267 444C214 444 169 404 147 369 120 326 107 299 107 287 107 278 112 274 122 274 127 274 130 275 133 276 138 284 141 291 144 296 176 375 216 414 264 414 282 414 291 400 291 373 291 358 289 341 284 323L192-46C178-104 138-175 75-175 66-175 57-174 48-171 73-161 85-143 85-118 85-92 71-79 44-79 12-79-13-108-13-140-13-184 30-205 77-205 120-205 160-190 197-160 232-131 254-94 265-51L356 311C359 324 361 337 361 349 361 404 322 444 267 444Z"></path></g></g></g></svg></mjx-container> - 函数，其傅里叶展开（设<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.023ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 452 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="q = e^{2\pi\sqrt{-1}\tau}"><g data-mml-node="mi" data-latex="q"><path data-c="1D45E" d="M372 377C352 420 321 442 280 442 215 442 158 409 109 342 63 280 40 216 40 149 40 62 90-13 173-13 209-13 245 4 282 38L241-126C238-141 229-150 215-153 210-154 197-155 174-155 168-156 164-156 162-156 153-157 148-165 148-179L148-183C151-190 157-194 165-194 184-194 244-191 263-191 282-191 345-194 364-194 378-194 385-186 385-170 385-160 375-155 356-155 338-155 313-156 313-144 313-140 314-133 317-123L452 427C452 436 447 441 438 441 420 441 382 391 372 377M340 373C349 352 354 338 354 329 354 328 353 322 351 313L327 216C308 144 299 107 298 105 279 70 224 16 176 16 137 16 117 46 117 105 117 151 148 263 162 300 181 347 225 412 280 412 307 412 327 399 340 373Z"></path></g></g></g></svg><mjx-break size="4"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="9.569ex" height="2.857ex" role="img" focusable="false" viewBox="0 -1012.8 4229.6 1262.8"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="q = e^{2\pi\sqrt{-1}\tau}"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367 80 367C64 367 56 359 56 344 56 329 64 321 80 321L698 321C714 321 722 329 722 344 722 356 711 367 698 367M698 179 80 179C64 179 56 171 56 156 56 141 64 133 80 133L698 133C714 133 722 141 722 156 722 169 711 179 698 179Z"></path></g><g data-mml-node="msup" data-latex="e^{2\pi\sqrt{-1}\tau}" transform="translate(1055.8,0)"><g data-mml-node="mi" data-latex="e"><path data-c="1D452" d="M124 129C124 153 129 186 139 227L188 227C253 227 303 235 339 250 372 264 394 284 405 309 412 326 415 342 415 355 415 410 363 442 307 442 268 442 229 432 190 412 113 372 46 281 46 171 46 69 105-11 204-11 257-11 304 2 345 27 379 48 404 69 420 90 427 99 430 106 430 109 430 120 425 126 414 126 409 126 404 122 398 114 365 70 324 42 277 30 246 22 223 18 206 18 149 18 124 72 124 129M375 355C375 289 311 256 182 256L147 256C166 322 194 366 232 387 262 404 287 413 307 413 343 413 375 391 375 355Z"></path></g><g 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fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -817)"><g data-mml-node="math" data-latex="
j(\tau) = q^{-1} + 744 + 196884q + 21493760q^2 + \cdots
"><g data-mml-node="mtable" data-latex="
j(\tau) = q^{-1} + 744 + 196884q + 21493760q^2 + \cdots
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d="M594 16C594 5 600 0 613 0L736 3C759 4 840 0 859 0 874 0 882 8 882 24 882 34 872 39 851 39 810 39 790 43 790 52 790 55 792 62 795 75L927 602C932 623 942 636 955 641 961 643 979 644 1008 644 1033 644 1044 645 1044 668 1044 678 1034 683 1013 683L882 683C853 683 853 681 840 662L484 108 408 657C404 682 402 683 373 683L237 683C215 683 204 682 204 660 204 649 215 644 236 644 256 644 297 647 297 631 297 629 296 623 293 613L167 110C158 72 135 49 100 42 75 39 42 43 42 16 42 5 48 0 60 0 78 0 141 3 159 3 178 3 242 0 261 0 276 0 283 8 283 24 283 33 276 38 261 39 219 39 198 52 198 78 198 82 199 89 202 100L332 621 415 26C418 9 424 0 434 0 442 0 450 7 459 20L848 627 712 82C706 60 696 47 681 42 675 40 656 39 625 39 604 39 594 37 594 16Z"></path></g></g></g></svg></mjx-container> 的前几个不可约表示维度依次是</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:47.983ex"><svg style="vertical-align:-.566ex;min-width:47.983ex" 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r_1 = 1,\quad r_2 = 196883,\quad r_3 = 21296876
"><g data-mml-node="mtable" data-latex="
r_1 = 1,\quad r_2 = 196883,\quad r_3 = 21296876
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transform="translate(889,0)"></path></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>数学家 McKay 敏锐地发现，比如，</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:33.025ex"><svg style="vertical-align:-2.036ex;min-width:33.025ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.204ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1400)"><g data-mml-node="math" data-latex="
\begin{aligned}
196884 &= r_1 + r_2 \\
21296876 &= r_1 + r_2 + r_3
\end{aligned}
"><g data-mml-node="mtable" data-latex="
\begin{aligned}
196884 &= r_1 + r_2 \\
21296876 &= r_1 + r_2 + r_3
\end{aligned}
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110C158 72 135 49 100 42 75 39 42 43 42 16 42 5 48 0 60 0 78 0 141 3 159 3 178 3 242 0 261 0 276 0 283 8 283 24 283 33 276 38 261 39 219 39 198 52 198 78 198 82 199 89 202 100L332 621 415 26C418 9 424 0 434 0 442 0 450 7 459 20L848 627 712 82C706 60 696 47 681 42 675 40 656 39 625 39 604 39 594 37 594 16Z"></path></g></g></g></svg></mjx-container> ，其特征标给出的 McKay-Thompson 级数</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:32.713ex"><svg style="vertical-align:-1.948ex;min-width:32.713ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.027ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1361)"><g data-mml-node="math" data-latex="
T_g(\tau) = \sum_{n\ge -1} Tr(g|_{V_n^\circ}) q^n
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232 125 232 120 232 116 230 114 228 109 223 52 8 52-8 52-17 57-22 67-22 72-22 79-17 88-6L133 47C168 1 224-22 300-22 366-22 424 5 476 58 528 111 554 170 554 236 554 283 538 323 505 355 490 368 470 379 445 388 422 393 400 399 378 405L312 423C279 432 254 469 254 509 254 552 271 589 306 621 341 653 380 669 423 669 515 669 561 619 561 520 561 502 557 482 557 465L557 462C560 455 565 451 573 451 582 451 588 459 592 474L645 691C645 700 640 705 630 705 625 705 618 700 609 689L566 637C538 682 491 705 424 705 361 705 304 681 254 634 202 586 176 531 176 468 176 396 223 339 282 323L387 296C441 281 475 262 475 195 475 149 457 108 422 72 387 36 347 17 302 17 204 17 133 60 133 157Z"></path></g><g data-mml-node="msub" data-latex="L_2" transform="translate(645,0)"><g data-mml-node="mi" data-latex="L"><path data-c="1D43F" d="M520 667C520 678 513 683 500 683L354 680 223 683C208 684 200 676 200 659 200 652 203 647 209 646 218 645 226 644 231 644 262 643 279 641 284 640 289 639 292 636 292 631 292 629 291 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给出了严密的证明。他没有局限于传统的有限群论，而是跨界利用了玻色弦理论中的无鬼定理（no-ghost theorem）来计算 Lie 代数同调（Lie algebra homology）。</p><p>他证明了作用在特定的 “魔群 Lie 代数” 上的恒等式恰好能导出<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.667ex" xmlns="http://www.w3.org/2000/svg" width="5.188ex" height="2.36ex" role="img" focusable="false" viewBox="0 -748 2293.3 1043"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="T_g(\tau)"><g data-mml-node="msub" data-latex="T_g"><g data-mml-node="mi" data-latex="T"><path data-c="1D447" d="M344 631C344 628 343 621 340 611L208 83C204 68 200 59 197 55 188 44 154 39 94 39 63 39 49 42 49 16 49 5 56 0 69 0 120 0 192 4 235 3L317 2C331 2 386 0 403 0 420 0 428 8 428 24 428 34 416 39 391 39 339 39 309 41 300 46 297 48 295 52 295 58L430 603C433 617 436 626 439 629 443 635 467 638 511 638 566 638 604 634 623 625 642 616 652 595 652 561 652 543 649 517 644 483 642 475 641 469 641 464 641 453 646 447 657 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data-mml-node="mo" data-latex="(" transform="translate(1004.3,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="mi" data-latex="\tau" transform="translate(1393.3,0)"><path data-c="1D70F" d="M468 431 192 431C139 431 91 397 47 330 34 310 27 297 27 292 30 284 31 279 43 279 50 279 56 283 62 292 94 341 134 366 183 366L260 366 165 56C158 34 155 21 155 18 155-2 166-12 187-12 198-12 208-9 214-3 227 9 228 12 232 33L297 366 459 366C494 366 511 378 511 403 511 426 493 431 468 431Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(1904.3,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></svg></mjx-container> 必须满足的一组被称为 “可复制方程（replicable identities）” 的递推关系，从而彻底解决了这一长达十余年的代数悬案。</p><h3 id="二、引入顶点代数（vertex-algebras）"><a href="#二、引入顶点代数（vertex-algebras）" class="headerlink" title="二、引入顶点代数（vertex algebras）"></a>二、引入顶点代数（vertex algebras）</h3><p>为了给物理学家和代数学家（Frenkel、Lepowsky 和 Meurman）构造的月光模<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="2.723ex" height="1.595ex" role="img" focusable="false" viewBox="0 -683 1203.7 705"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="V^\circ"><g data-mml-node="msup" data-latex="V^\circ"><g data-mml-node="mi" data-latex="V"><path data-c="1D449" d="M671 680C652 680 592 683 573 683 558 683 550 675 550 660 550 650 557 645 570 644 598 643 612 633 612 616 612 607 607 595 598 580L300 107 234 619C234 636 255 644 298 644 318 644 327 651 327 668 327 678 321 683 309 683 287 683 209 680 187 680 167 680 99 683 79 683 64 683 56 675 56 660 56 649 66 644 85 644 102 644 114 642 122 640 138 635 137 633 140 614L218 4C221-13 229-22 242-22 255-22 265-15 273-2L629 564C652 600 674 623 696 632 711 639 730 643 752 644 763 645 768 652 769 667 770 678 764 683 752 683 737 683 686 680 671 680Z"></path></g><g data-mml-node="mo" transform="translate(862.4,363) scale(0.707)" data-latex="circ"><path data-c="2218" d="M356 250C356 300 347 321 312 356 283 385 247 400 206 400 124 400 56 332 56 250 56 168 124 100 206 100 288 100 356 168 356 250M309 250C309 208 303 200 279 177 259 157 235 147 206 147 149 147 103 193 103 250 103 307 149 353 206 353 263 353 309 307 309 250Z"></path></g></g></g></g></svg></mjx-container> 提供严格的代数操作空间，Borcherds 在 1986 年以极高的代数品味首创了 <strong>顶点代数（vertex algebras）</strong> 理论。</p><p>这本质上是对二维共形场论（conformal field theory，CFT）中手征代数（chiral algebras）的严格数学公理化。</p><p>在传统的结合代数中，乘法是一个固定的二元映射。但在顶点代数中，受弦理论中 “状态 - 场对应（state-field correspondence）” 的启发，向量空间<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.742ex" height="1.595ex" role="img" focusable="false" viewBox="0 -683 770 705"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="V"><g data-mml-node="mi" data-latex="V"><path data-c="1D449" d="M671 680C652 680 592 683 573 683 558 683 550 675 550 660 550 650 557 645 570 644 598 643 612 633 612 616 612 607 607 595 598 580L300 107 234 619C234 636 255 644 298 644 318 644 327 651 327 668 327 678 321 683 309 683 287 683 209 680 187 680 167 680 99 683 79 683 64 683 56 675 56 660 56 649 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58 311 20 345-11 392-11 461-11 482 70 498 144M341 374C350 353 355 339 355 330 355 326 354 321 353 314L304 122C301 111 294 99 285 87 248 41 212 18 177 18 138 18 118 48 118 107 118 131 124 167 136 215 157 300 187 357 224 388 244 405 263 413 282 413 309 413 329 400 341 374Z"></path></g></g></g></svg></mjx-container> （状态）都对应一个以形式变量<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.057ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 467 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="z"><g data-mml-node="mi" data-latex="z"><path data-c="1D467" d="M162 92 145 90C193 137 230 171 255 194L351 283C374 304 398 330 421 359 452 397 467 420 467 428 467 437 462 442 452 442 445 442 439 438 434 429 409 388 386 368 364 368 342 368 326 386 317 398 293 427 269 442 246 442 220 442 193 430 165 404 137 378 123 352 123 326 123 316 128 311 139 311 146 311 152 316 156 325 167 354 194 368 235 368 249 368 269 363 296 354 320 345 343 340 364 339 308 284 265 244 234 217L145 134C130 119 111 99 90 74 59 35 43 12 43 3 43-6 48-11 59-11 65-11 71-6 78 4 103 43 130 63 158 63 174 63 192 51 212 26 232 1 255-11 278-11 312-11 346 7 382 42 418 77 436 111 436 144 436 153 431 158 420 158 413 158 407 153 402 143 387 100 344 63 288 63 264 63 186 92 162 92Z"></path></g></g></g></svg></mjx-container> 为参数的 Laurent 级数，即 “顶点算子（vertex operator）”：<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.793ex" xmlns="http://www.w3.org/2000/svg" width="3.589ex" height="1.793ex" role="img" focusable="false" viewBox="0 -442 1586.4 792.7"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="a_{(n)}"><g data-mml-node="msub" data-latex="a_{(n)}"><g data-mml-node="mi" data-latex="a"><path data-c="1D44E" d="M498 144C498 153 493 158 482 158 474 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data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(6)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="36" d="M383 504C416 504 432 521 432 555 432 627 378 666 304 666 221 666 155 627 106 548 63 480 42 403 42 316 42 189 65 100 112 47 152 1 198-22 251-22 312-22 362 1 401 47 438 91 457 144 457 205 457 266 439 318 403 361 365 407 316 431 257 431 205 431 165 402 138 346L138 352C138 465 166 561 226 605 252 624 279 633 306 633 342 633 368 623 385 602 351 602 334 583 334 553 334 525 355 504 383 504M344 340C355 317 361 272 361 206 361 141 356 98 345 76 325 35 294 14 251 14 222 14 200 24 184 44 171 60 162 74 158 85 146 116 140 163 140 227 140 255 144 282 151 308 164 355 201 399 256 399 295 399 325 379 344 340Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>这相当于用一整族的无穷双线性运算取代了单一的乘法。</p><p>Borcherds 提取了物理中算子乘积展开（OPE）的精髓，引入了核心的 “局部性公理（locality axiom）”：对于任意两个场算子，必然存在足够大的整数<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.993ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 881 683"><g stroke="currentColor" 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(z-w)^{-N}[Y(a,z),Y(b,w)] = 0
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viewBox="1278 -823 1 1146"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:7" transform="translate(0,675)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(7)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="37" d="M475 604C482 613 485 626 485 644L243 644C174 644 135 648 128 657 125 660 122 667 120 676L89 676 55 464 88 464C98 520 106 550 112 555 115 558 146 560 205 560L401 560 295 410C214 295 174 171 174 36 174-3 190-22 223-22 256-22 272-3 272 36L272 87C272 239 296 349 343 416Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>这在代数上完美刻画了物理量子场论中 “类空分离测量相互独立” 的微观因果律。如今，顶点算子代数（VOA）已成为几何表示论的核心语言，在推进几何 Langlands 纲领（geometric Langlands program）的过程中扮演了基石角色。</p><h3 id="三、创立广义Kac-Moody代数（Generalized-Kac-Moody-algebras）"><a href="#三、创立广义Kac-Moody代数（Generalized-Kac-Moody-algebras）" class="headerlink" title="三、创立广义Kac-Moody代数（Generalized Kac-Moody algebras）"></a>三、创立广义 Kac-Moody 代数（Generalized Kac-Moody algebras）</h3><p>经典半单 Lie 代数以及无限维的 Kac-Moody 代数，完全可以由其广义 Cartan 矩阵（generalized Cartan matrix）<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 750 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A = (a_{ij})"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g></g></g></svg><mjx-break size="4"></mjx-break><svg style="vertical-align:-.667ex" xmlns="http://www.w3.org/2000/svg" width="6.744ex" height="2.364ex" role="img" focusable="false" viewBox="0 -750 2981.1 1045"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A = (a_{ij})"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367 80 367C64 367 56 359 56 344 56 329 64 321 80 321L698 321C714 321 722 329 722 344 722 356 711 367 698 367M698 179 80 179C64 179 56 171 56 156 56 141 64 133 80 133L698 133C714 133 722 141 722 156 722 169 711 179 698 179Z"></path></g><g data-mml-node="mo" data-latex="(" transform="translate(1055.8,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 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data-latex="a_{ii} = 2"><g data-mml-node="msub" data-latex="a_{i i}"><g data-mml-node="mi" data-latex="a"><path data-c="1D44E" d="M498 144C498 153 493 158 482 158 474 158 468 151 465 137 445 58 421 18 394 18 377 18 368 32 368 60 368 73 372 97 381 132L438 357C443 376 445 387 445 392 445 412 434 422 412 422 391 422 377 410 370 387 349 424 319 442 281 442 216 442 159 409 109 343 63 281 40 217 40 150 40 63 91-11 175-11 218-11 260 12 300 58 311 20 345-11 392-11 461-11 482 70 498 144M341 374C350 353 355 339 355 330 355 326 354 321 353 314L304 122C301 111 294 99 285 87 248 41 212 18 177 18 138 18 118 48 118 107 118 131 124 167 136 215 157 300 187 357 224 388 244 405 263 413 282 413 309 413 329 400 341 374Z"></path></g><g data-mml-node="TeXAtom" transform="translate(562,-150) scale(0.707)" data-latex="{i i}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="i"><path data-c="1D456" d="M284 621C284 648 271 661 244 661 216 661 188 633 188 605 188 578 202 565 229 565 257 565 284 593 284 621M259 138C237 59 205 19 164 19 151 19 144 28 144 47 144 64 173 150 232 306 240 329 244 347 244 360 244 409 210 445 161 445 118 445 84 420 59 369 39 328 29 301 29 288 29 279 34 275 45 275 58 275 60 281 64 295 87 375 118 415 158 415 171 415 178 406 178 387 178 373 175 357 168 338 145 275 121 208 101 155 86 115 78 88 78 74 78 25 114-11 162-11 205-11 239 14 264 64 283 103 293 130 293 145 293 154 288 159 277 159 274 159 259 150 259 138Z"></path></g><g data-mml-node="mi" data-latex="i" transform="translate(345,0)"><path data-c="1D456" d="M284 621C284 648 271 661 244 661 216 661 188 633 188 605 188 578 202 565 229 565 257 565 284 593 284 621M259 138C237 59 205 19 164 19 151 19 144 28 144 47 144 64 173 150 232 306 240 329 244 347 244 360 244 409 210 445 161 445 118 445 84 420 59 369 39 328 29 301 29 288 29 279 34 275 45 275 58 275 60 281 64 295 87 375 118 415 158 415 171 415 178 406 178 387 178 373 175 357 168 338 145 275 121 208 101 155 86 115 78 88 78 74 78 25 114-11 162-11 205-11 239 14 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306 627 352 556 352 470 352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0 450 180 417 180C409 129 402 100 396 91 391 86 361 84 306 84L139 84 236 179C304 243 390 312 419 365 439 400 449 435 449 470 449 588 357 666 237 666Z"></path></g></g></g></svg></mjx-container> ，这对应于根系中长度平方为正的 “实单根（real simple roots）”。</p><p>Borcherds 极其大胆地放宽了这一限制，允许对角线元素<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="2.488ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1099.9 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="a_{ii} < 0"><g data-mml-node="msub" data-latex="a_{i i}"><g data-mml-node="mi" data-latex="a"><path data-c="1D44E" d="M498 144C498 153 493 158 482 158 474 158 468 151 465 137 445 58 421 18 394 18 377 18 368 32 368 60 368 73 372 97 381 132L438 357C443 376 445 387 445 392 445 412 434 422 412 422 391 422 377 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="a_{ii} < 0"><g data-mml-node="mo" data-latex="<"><path data-c="3C" d="M666-45C683-52 701-39 701-23 701-13 696-6 687-2L153 250 687 502C696 506 701 513 701 522 701 539 693 547 677 547 673 547 669 546 666 545L92 273C82 268 77 261 77 250 77 239 82 232 92 227Z"></path></g><g data-mml-node="mn" data-latex="0" transform="translate(1055.8,0)"><path data-c="30" d="M249-22C390-22 460 92 460 320 460 473 428 575 365 625 330 652 291 666 250 666 109 666 39 551 39 320 39 136 88-22 249-22M361 524C368 489 371 425 371 332 371 240 367 172 360 128 347 48 310 8 249 8 226 8 203 17 182 34 155 57 139 104 132 176 129 201 128 253 128 332 128 419 131 480 136 513 145 568 163 603 191 618 213 630 232 636 249 636 314 636 350 583 361 524Z"></path></g></g></g></svg></mjx-container> 。这意味着他在 Lie 代数系统中合法化了范数平方为零或负数的 “虚单根（imaginary simple roots）” 的存在，由此创立的全新代数结构现称为 <strong>Borcherds Lie 代数</strong> 。</p><p>即使引入了看似破坏结构的虚根，这类代数依然保留了经典 Lie 代数最优美的性质 —— <strong>Weyl 分母公式（Weyl denominator formula）</strong> 依然成立，只需在公式中加入由虚根贡献的修正项。</p><p>Borcherds 将月光顶点代数与一个由洛伦兹格分次的系统结合，成功构造出了 “魔群 Lie 代数（Monster Lie algebra）”。</p><p>在这个广义代数中，对应根的重数恰好等于<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.464ex" xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.959ex" role="img" focusable="false" viewBox="0 -661 412 866"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="j"><g data-mml-node="mi" data-latex="j"><path data-c="1D457" d="M397 621C397 648 383 661 356 661 328 661 300 633 300 605 300 578 313 565 340 565 368 565 397 593 397 621M267 444C214 444 169 404 147 369 120 326 107 299 107 287 107 278 112 274 122 274 127 274 130 275 133 276 138 284 141 291 144 296 176 375 216 414 264 414 282 414 291 400 291 373 291 358 289 341 284 323L192-46C178-104 138-175 75-175 66-175 57-174 48-171 73-161 85-143 85-118 85-92 71-79 44-79 12-79-13-108-13-140-13-184 30-205 77-205 120-205 160-190 197-160 232-131 254-94 265-51L356 311C359 324 361 337 361 349 361 404 322 444 267 444Z"></path></g></g></g></svg></mjx-container> - 函数的系数，其广义分母公式也正是彻底攻克月光猜想的终极代数方程。</p><h3 id="四、发现Borcherds积与奇异Theta提升（Borcherds-products-and-singular-theta-lift）"><a href="#四、发现Borcherds积与奇异Theta提升（Borcherds-products-and-singular-theta-lift）" class="headerlink" title="四、发现Borcherds积与奇异Theta提升（Borcherds products and singular theta lift）"></a>四、发现 Borcherds 积与奇异 Theta 提升（Borcherds products and singular theta lift）</h3><p>这是 Borcherds 在代数几何和自守形式理论中最深刻的贡献。在研究魔群 Lie 代数的分母公式时，Borcherds 意外发现无穷乘积与高维自守形式之间存在普遍的内在联系。</p><p>早年间，像 Dedekind<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.489ex" xmlns="http://www.w3.org/2000/svg" width="1.124ex" height="1.489ex" role="img" focusable="false" viewBox="0 -442 497 658"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\eta"><g data-mml-node="mi" data-latex="\eta"><path data-c="1D702" d="M379 442C318 442 268 415 229 362 222 407 187 442 136 442 95 442 65 409 44 344 34 311 29 292 29 286 29 277 34 272 45 272 50 272 53 273 56 275 61 284 64 291 65 298 83 374 106 412 133 412 151 412 160 398 160 371 160 358 155 330 144 289L87 61C84 48 78 22 78 17 78-3 89-13 111-13 130-13 144-3 151 17 152 22 158 47 169 90L191 179C203 228 212 265 219 290 222 297 230 311 243 331 279 385 323 412 376 412 409 412 425 390 425 347 425 332 422 312 415 286L303-169C301-177 300-182 300-185 300-206 311-216 332-216 354-216 369-202 376-173L488 274C493 295 496 315 496 332 496 404 451 442 379 442Z"></path></g></g></g></svg></mjx-container> 函数这样的无限乘积公式仅存在于一维情况（如在<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.561ex" xmlns="http://www.w3.org/2000/svg" width="7.255ex" height="2.253ex" role="img" focusable="false" viewBox="0 -748 3206.6 996"><g 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28 22 28 4 43 0 66 0L572 0C601 0 607 5 609 31L636 226C636 241 629 248 615 248 610 248 605 246 602 243 596 231 593 222 592 215 571 117 480 43 362 43L228 43 596 643C602 654 605 660 605 663 605 682 593 685 569 685M156 642C144 635 131 625 116 612L119 642M455 642 546 642 179 43 87 43M517 43C538 56 558 73 575 92L568 43Z"></path></g></g><g data-mml-node="mo" data-latex=")" transform="translate(2817.6,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></svg></mjx-container> 上）。</p><p>在当时的代数几何界，人们普遍认为高维自守形式不可能拥有如此整齐的纯乘积结构。</p><p>Borcherds 彻底打破了这一偏见，他建立了一种被称为 “奇异 Theta 提升（singular theta lift）” 的强大积分映射。</p><p>该映射将定义在<mjx-container class="MathJax" jax="SVG" 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data-mml-node="mo" data-latex="-"><path data-c="2212" d="M698 270 80 270C64 270 56 263 56 250 56 237 64 230 80 230L698 230C714 230 722 237 722 250 722 262 710 270 698 270Z"></path></g><g data-mml-node="mfrac" data-latex="\frac{2}{n}" transform="translate(1000.2,0)"><g data-mml-node="mn" transform="translate(255.4,394) scale(0.707)" data-latex="2"><path data-c="32" d="M237 666C186 666 143 648 106 612 69 576 50 534 50 483 50 449 75 424 106 424 136 424 161 450 161 480 161 513 137 536 105 536 102 536 100 536 98 535 117 584 161 627 224 627 306 627 352 556 352 470 352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0 450 180 417 180C409 129 402 100 396 91 391 86 361 84 306 84L139 84 236 179C304 243 390 312 419 365 439 400 449 435 449 470 449 588 357 666 237 666Z"></path></g><g data-mml-node="mi" transform="translate(220,-345) scale(0.707)" data-latex="n"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18 427 18 420 28 420 47 420 61 426 84 438 115 478 224 498 296 498 333 498 403 451 442 381 442 322 442 271 416 230 363 222 407 187 442 136 442 80 442 58 390 44 345 34 313 29 294 29 287 29 278 34 273 45 273 50 273 53 274 56 276 61 285 64 292 65 299 83 375 106 413 133 413 151 413 160 399 160 371 160 358 155 331 144 290L87 63C84 50 78 24 78 19 78-1 89-11 111-11 130-11 144-1 151 19 153 24 159 49 170 92L191 181 221 295C232 318 249 341 270 365 299 397 335 413 378 413 411 413 427 391 427 348 427 310 406 234 363 120 356 102 353 87 353 74 353 25 390-11 438-11 482-11 517 14 542 64 561 104 571 131 571 144 571 153 566 158 555 158 552 158 537 149 537 137Z"></path></g><rect width="624.3" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> 的 “弱全纯模形式（weakly holomorphic modular forms，即允许在无穷远尖点处有极点的主模形式）” 作为输入，提升为正交群<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.561ex" xmlns="http://www.w3.org/2000/svg" width="6.981ex" height="2.253ex" role="img" focusable="false" viewBox="0 -748 3085.7 996"><g stroke="currentColor" 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61 426 84 438 115 478 224 498 296 498 333 498 403 451 442 381 442 322 442 271 416 230 363 222 407 187 442 136 442 80 442 58 390 44 345 34 313 29 294 29 287 29 278 34 273 45 273 50 273 53 274 56 276 61 285 64 292 65 299 83 375 106 413 133 413 151 413 160 399 160 371 160 358 155 331 144 290L87 63C84 50 78 24 78 19 78-1 89-11 111-11 130-11 144-1 151 19 153 24 159 49 170 92L191 181 221 295C232 318 249 341 270 365 299 397 335 413 378 413 411 413 427 391 427 348 427 310 406 234 363 120 356 102 353 87 353 74 353 25 390-11 438-11 482-11 517 14 542 64 561 104 571 131 571 144 571 153 566 158 555 158 552 158 537 149 537 137Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(2696.7,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></svg></mjx-container> 对称空间上的亚纯自守形式（meromorphic automorphic forms）。</p><p>这些生成的高维自守形式不仅能在边界展开为极具美感的无限乘积（即 Borcherds 积），而且它们的零点和极点极其有规律地分布在 “Heegner 除子（Heegner divisors）” 上，且分布位置与重数完全由输入的低维模形式的 Fourier 展开主部（负次幂项）的系数精准决定。</p><p>这一 “极点发生器” 机制彻底颠覆了高维自守形式的研究。</p><p>如今，它被广泛应用于显式构造 K3 曲面、Enriques 曲面以及志村簇（Shimura varieties）的模空间几何结构，成为了现代算术几何中无可替代的解析工具。</p></div></div><div class="footer"></div><div class="article-meta" id="bottom"><div class="new-meta-box"></div></div><div class="prev-next"><a class="prev" href="/notes/person/3"><p class="title"><i class="fa-solid fa-chevron-left" aria-hidden="true"></i>Conway英国数学家约翰·康威的多领域开创性贡献：从生命游戏、超现实数到有限单群</p><p class="content">约翰·霍顿·康威是一位极具创造力的英国数学家。他的贡献横跨多个领域，其发明的生命游戏普及了元胞自动机概念，创建的超现实数统一了数系与博弈论，并在有限单群领域发现了以他命名的Conway群</p></a><a class="next" href="/notes/person/5"><p class="title">Wall英国数学家C.T.C. Wall的学术生涯与在拓扑学及奇点理论中的奠基性贡献<i class="fa-solid fa-chevron-right" aria-hidden="true"></i></p><p class="content">本文介绍了英国数学家C.T.C. Wall的生平及其开创性工作。他统一了高维流形分类的手术理论，建立了著名的Wall群。其在四维流形稳定化、有限性障碍及奇点分类等领域的研究深刻塑造了现代拓扑学</p></a></div><div class="recommended-article"><div class="recommended-article-header"><i class="fa-solid fa-bookmark fa-fw" aria-hidden="true"></i> <span>推荐阅读</span></div><div class="recommended-article-group"> <a class="recommended-article-item" href="/notes/Zeta/48.html" title="魔群" rel="bookmark"><img src="https://static.mhuig.top/npm/imbox@0.0.15/c/64.webp" class="lazyload" data-srcset="https://static.mhuig.top/npm/imbox@0.0.15/c/64.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="魔群"> <span class="title">魔群</span></a> <a class="recommended-article-item" href="/notes/person/3.html" title="Conway英国数学家约翰·康威的多领域开创性贡献：从生命游戏、超现实数到有限单群" rel="bookmark"><img src="https://static.mhuig.top/npm/imbox@0.0.15/c/64.webp" class="lazyload" data-srcset="https://static.mhuig.top/npm/imbox@0.0.15/c/64.webp" 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class="title">Irving Segal欧文·西格尔：联结算子代数与宇宙学的数学物理学家</span></a></div></div></article><article class="post white-box shadow floatable blur" id="comments"><span hidden=""><meta itemprop="discussionUrl" content="/notes/person/4#comments"></span><p ct=""><i class="fa-duotone fa-comments"></i> 留言区</p><div id="layoutHelper-comments"></div></article></div><aside id="l_side" itemscope="" itemtype="http://schema.org/WPSideBar"><section class="widget text desktop mobile pjax"><header><a href="/notes/"><i class="fa-duotone fa-book fa-fw" aria-hidden="true"></i> <span class="name">Notes</span></a></header><div class="content"></div></section><section class="widget list group desktop mobile pjax"><header><i class="fa-duotone fa-p fa-fw" aria-hidden="true"></i> <span class="name">History of mathematics</span></header><div class="content"><ul class="list entry navigation"><li><a class="flat-box" title="/notes/person/" href="/notes/person/" active-action="action-notesperson"><div class="name"> Welcome</div></a></li><li><a class="flat-box" title="/notes/person/1" href="/notes/person/1" active-action="action-notesperson1"><div class="name"> Hilbert</div></a></li><li><a class="flat-box" title="/notes/person/2" href="/notes/person/2" active-action="action-notesperson2"><div class="name"> Thompson</div></a></li><li><a class="flat-box" title="/notes/person/3" href="/notes/person/3" active-action="action-notesperson3"><div class="name"> Conway</div></a></li><li><a class="flat-box" title="/notes/person/4" href="/notes/person/4" active-action="action-notesperson4"><div class="name"> Borcherds</div></a></li><li><a class="flat-box" title="/notes/person/5" href="/notes/person/5" active-action="action-notesperson5"><div class="name"> Wall</div></a></li><li><a class="flat-box" title="/notes/person/6" href="/notes/person/6" active-action="action-notesperson6"><div class="name"> Waldhausen</div></a></li><li><a class="flat-box" title="/notes/person/7" href="/notes/person/7" active-action="action-notesperson7"><div class="name"> Dehn</div></a></li><li><a class="flat-box" title="/notes/person/8" href="/notes/person/8" active-action="action-notesperson8"><div class="name"> Whitehead</div></a></li><li><a class="flat-box" title="/notes/person/9" href="/notes/person/9" active-action="action-notesperson9"><div class="name"> Iwaniec</div></a></li><li><a class="flat-box" title="/notes/person/10" href="/notes/person/10" active-action="action-notesperson10"><div class="name"> Sarnak</div></a></li><li><a class="flat-box" title="/notes/person/11" href="/notes/person/11" active-action="action-notesperson11"><div class="name"> Ramanujan</div></a></li><li><a class="flat-box" title="/notes/person/12" href="/notes/person/12" active-action="action-notesperson12"><div class="name"> Hardy</div></a></li><li><a class="flat-box" title="/notes/person/13" href="/notes/person/13" active-action="action-notesperson13"><div class="name"> Littlewood</div></a></li><li><a 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