數學界對 OpenAI 百餘項數學問題解答的多元反應

發佈於 2026年10月8日，最後更新於 2026年10月10日晚上11:00（CEST），加入了新回應。最新回應列於下方；如果你想提供回應，請電郵至 proofsandprompts@gmail.com！

我們很多人都在消化 OpenAI 於10月6日發布的公告。我們認為，瞭解社羣此刻的直覺感受概況很有用，以下就是大家的看法。我們會再開放一星期，歡迎你繼續傳送簡短回應給我們。為了清楚表明大家都身處其中，每則回應只以姓名署名，不提隸屬機構或職涯階段。你會看到各種不同的回應：既有一般感想，也有針對特定問題的評論。

Tim Santens

OpenAI 的成果幾乎令人難以置信；Quasi-Riemann 的結果，我原以為完全是科幻情節。對 AI 實驗室的行事方式，我們可以有很多看法，但既然這些模型已經存在，數學作為一門研究學科將不得不經歷重大轉變。

Roman Sauer

常見的説法是，數學家手上只有幾招，卻能熟練地一次又一次運用。我也有一招，其效果出乎意料地好，令我驚訝過幾次：以某種可測的葉狀物件取代流形。這種構造源自遍歷理論，而我把它用於一些出乎意料的情境。

最近，我在一篇與 Sabine Braun 合寫、研究宏觀純量曲率的論文中運用了這一招，Hannah Alpert 其後進一步擴展了這篇論文。這一招和這兩篇論文，如今都出現在 OpenAI 對第 207 和第 335 題極具創意的解答中——兩題截然不同，簡直毫不相干！

我深感震撼。

這一夜過後，我得到甚麼啟示？我們不能抱着防守心態，把焦點侷限於目前視為數學中人類特質的事物。這會令我們的角色愈來愈無足輕重。相反，我們必須徹底拓闊對數學家是甚麼、做甚麼的理解。

Xiaolei Wu

今天是中國為期一星期的全國假期最後一天。我們剛和家人到附近一個沿海城市短途旅行。醒來後，我照常查看手機，意外地發現很多訊息都附有同一份 PDF 檔。看來 OpenAI 解決了很多未解問題。

於是我看了羣論部分，內容看起來簡直難以置信。這幾乎涵蓋了我這個領域所有主要未解問題！怎麼可能？於是我傳訊息問這份檔案是從哪裏來的。得到的答覆是，OpenAI 剛剛正式公佈了這件事。

我又看了一遍羣論部分。沒錯，這些真的幾乎涵蓋了所有問題。Thompson 羣 F 不可平均。好吧，我曾用 ChatGPT 試過很多次，但我想自己當時沒有用到最新模型，或者 tokens 不夠。Eilenberg–Ganea 猜想不成立——或許不太令人意外，但究竟是怎麼做到的？好吧，至少 Whitehead 猜想還沒有被攻破。甚麼？他們竟然完全解決了 Boone–Higman 猜想？而且環境單羣甚至可以是 (F∞) 型……我預料到這一天會來臨，但或許要等到大約一年後才會發生。

過了一會兒，我開始查看其他部分，先從拓撲學看起。Borel 猜想在維度 4 不成立，粗 Baum–Connes 猜想也不成立……

當天餘下的時間過得很漫長。陸續有更多訊息傳來。我的社交媒體動態被關於這份清單的討論淹沒。有些人仍聲稱，他們各自領域中已解決的問題其實沒那麼重要。不過，我想自己已經過了那個階段。

Nicholas Williams

我們確實從這次發布中得知不少新事實。與此同時，我們數學家需要一些時間才能消化所有這些論文。毫無疑問，大型語言模型已徹底改變了局面，但最終目標始終必須是人類的理解。我認為重要的是指出，數學家所做的不只是解決文獻中的未解問題：我們還必須深入鑽研學科本身，思考哪些猜想才值得一開始就提出。但另一方面，問題解決能力曾是數學家的鮮明特徵，如今這個時代似乎正在結束。對我們這些喜歡解決問題的人來説，這令人難過。

Henry Bradford

今天，我主要感到迷惘和困惑。事情發展的步伐似乎已遠遠超出我們所能應對的範圍，以至於我們無從知道該如何回應，甚至不知道如何開始找出應對之道。我最擔心的是如何培育下一代數學家：我們知道，要在這門學科中磨練本領，直至能夠切實促進知識進步，需要投入多少時間和精力；我也憂慮，聰明的年輕人或許再沒有必要的誘因，願意付出這些努力。我仍抱有希望，相信數學實踐仍有光明的未來，AI 工具無處不在，能豐富而非削弱我們對數學宇宙的理解。然而，要實現這一點，我們這個羣體需要共同商定一條切實可行的路徑，讓年輕學者能在這門學科中站穩腳跟。

Henry Wilton

OpenAI 清單上有很多我十分關心的問題：Cannon 猜想、非殘有限的雙曲羣、Boone—Higman 猜想，以及至少另外十個問題。我還沒有仔細研究當中的任何一個問題，不過粗略一看，相關論述的質素似乎參差不齊。要逐一檢視所有證明、核實證明是否正確，並設法消化內容，需要花上幾個月。

但我認為，重要的是着眼大局。如果 OpenAI 想摧毀數學界，這會是個絕佳方法。未解問題是數學界經過數十年、甚至數百年建立起來的資源。它們的價值，來自我們賦予它們的意義。它們為我們提供架構、衡量進展的尺度，以及長遠目標。AI 的出現本來就勢必帶來劇烈震盪，但這批論文的大量湧現，猶如一場我們根本來不及準備的海嘯。顯然，OpenAI 為了追求自身的財務利益，並不介意沖走我們羣體的各種架構。

OpenAI 似乎一心要破壞數學界的結構和規範，這樣的例子不勝枚舉。他們拒絕列出論文作者的姓名，等於暗示數學已不再是人類的事業。他們不把研究結果投稿至學術期刊，這既暗示數學界的規範已經失效，也令學界無法消化他們的研究成果。OpenAI 尚未公開問題如何挑選、嘗試的失敗率等基本資訊。對任何嚴肅探討其工具的科學討論而言，這些資訊都應是必要條件。當然，還有最基本的財務不公義：他們極具價值的模型，是透過複製我們學界經由開放取用運動免費公開的推理，才學會推理。

我希望數學界能及時調整我們的做法和規範，從而挺過這些發展帶來的衝擊，但我感到擔憂。

Macarena Arenas

也許 AI 公司如果有意，的確可以做一些有益於人類和我們居住的地球的事（終結全球飢餓、解決全球暖化、治癒癌症……），但這並不是其中一件，而且他們這樣做也不是出於利他目的。

至於數學成果，當中有些尚未經過驗證，大部分寫得很差，而且（很可能）沒有任何成果透明交代其產生過程、涉及哪些人，以及成本是多少（各方面的成本）。當然，這些成果全都令人印象深刻。其中一些或許最終能令數學和數學家受益，但整體而言，我認為這是企業試圖對一個知識領域施加操控性壓力的例子，而且做法不負責任。我希望我們數學界能找到方法渡過這場混亂，但至少目前，這確實是一團糟，而且弊大於利。這會令很多人分心，無法研究自己的想法；會打亂不少現有的研究計劃；會在數學界引發衝突；會助長許多人投機取巧、不求甚解地使用這些技術（因而增加「低質內容」，令我們更難從中理清頭緒）；也會令許許多多人卻步，不再從事數學研究。

Johannes Schmitt

有一點值得記住：這批論文並非 OpenAI 以推進數學研究為主要目標而撰寫，甚至主要目的也不是公關（雖然他們發過一些公關貼文，但例如最後的公告就相對剋制）。這些論文之所以存在，主要是因為 OpenAI 利用未解的數學問題評估內部模型，以提升模型能力。他們不得不採用難度高的未解問題，因為只有這些問題仍能對前沿 AI 模型構成挑戰。我認為，從 OpenAI 的角度來看，這些論文基本上只是內部開發過程的副產品。

因此，公司大可以保持沉默，或者挑選幾項引人注目的成果，再發一篇網誌文章。我認為，他們沒有這樣做，反而向數學界尋求如何跟進的意見，並促成了數學與人工智能諮詢小組的成立，值得肯定。他們採納了小組的部分建議，例如適時發布成果及（部分）Lean 證書，但並非全部；尤其是他們宣佈會繼續利用未解的數學問題開發模型。在我看來，數學界目前最迫切的需要，是建立一個中央平台，讓大家討論成果、開始消化相關論文（據多位同事所言，這些論文寫得極差），並報告任何數學錯誤或未有適當致謝之處。由 OpenAI 控制、又停用了議題和 pull request 功能的 GitHub 儲存庫，無法達到這個目的。現在輪到 OpenAI 採取行動，把論文移至中立的儲存庫，並以承諾提供的工作坊及特別計劃資金，支持這項社羣工作。

Lvzhou Chen

先説明一下，我只閲讀了概述的部分內容，以及合集中的幾篇論文，並未理解全部細節。我熟悉羣論中的大部分問題，也熟悉拓撲學中的不少問題。我曾花幾年時間思考 Kervaire 猜想、Howie 猜想（問題 256）及相關問題。

對此，我的感受頗為矛盾。正面來看，我想了解並消化帶來這些解答的新技術或見解，也希望運用它們，更深入理解或解決其他問題。另一方面，我覺得在如此短的時間內解決這麼多問題，可能會損害數學界及數學專業。很多時候，人們正是在嘗試解決未解問題（例如這裏解決的問題）時取得新發現。因此，我們可能失去了大量作出其他新發現的機會；我希望值得發現的事物，遲早都會被發現。無論如何，這肯定會令數學界人士比以往承受更大壓力，也更感不確定，尤其是較年輕的人（例如博士生）：數學專業會如何改變？博士生應該專注甚麼？我們應該如何培訓他們？我希望我所説的「損害」能成為改革數學專業的一部分，最終帶來正面改變，也希望對數學有興趣的人（尤其是年輕人）不會因此氣餒。

至於未來，我猜想（或希望？）AI 的影響會有上限：數學的格局在經歷重大變化後會趨於穩定，而我們也會對難度有新的認識。畢竟，在看過舊問題的解答後，我們總會找到新的、有趣而且更難的問題。不過，格局穩定下來（如果真的會穩定的話）可能需要不少時間，在此之前，情況或許相當混亂。或許可以更專注於一些全新的領域，以及看來根本而重要的問題。即使格局穩定下來，人類是否仍需要作出貢獻以取得新發現，仍然是個問題。如果人類不再需要作出貢獻，我們仍可專注於更深入理解這些新發現，但我個人可能會覺得這樣沒那麼有趣，寧願研究其他事情。或許問題不在於人類還能否作出貢獻；相反，正如 Ruixiang 在某處所説，數學家是否仍有勇氣研究那些以新的意義而言仍然困難的問題？

我猜其中大部分最後都只會證明是些愚蠢又天真的想法，但我理解這篇帖文的用意，是記錄真實感受，不論愚蠢與否。

Enrico Fatighenti

我不是反科技人士——恰恰相反。我一直有使用 AI，至今每天仍會用它作為書目工具、測試各種突發奇想、加快處理行政事務，以及協助教學等等。我不會用它生成證明，但也不會批評這樣做的人。

不過，今天的公告確實令我非常不滿。

公平地説，我最初的反應幾乎是感到乏味。是的，AI「證明」了一批我所在領域中有趣的結果。（真的嗎？我們確定嗎？我們甚至看得懂上面寫了甚麼嗎？）甚至連一個千禧年難題也不是。哼。

但我想得越多，就越感到不滿。最令我介意的是雙重標準。

每當我審閲論文，或收到學生的論文或計劃書時，我都會採用我們大多數人一樣的標準。如果一篇論文或畢業論文寫得太差，令我必須花上不少功夫才能讀完第一頁，我就會拒絕它，並要求對方提交一份新的、易讀的版本。

面對這些由 AI 生成的論文，我覺得我們這個學術社羣並沒有採用同一套質素和嚴謹標準。他們可以一口氣發布多篇逾 100 頁的論文，聲稱解決了這個或那個問題，當中往往充斥重複的論證、含糊的邏輯結構、走不通的思路，以及奇怪或令人不安的術語——換言之，就是粗製濫造的垃圾。然後，我們卻要負責細讀、核查、修正、簡化，並解釋實際上發生了甚麼。

這要花我們大量時間和心力，他們卻可以直接轉向下一個猜想，繼續以粗劣內容橫衝直撞。當然，功勞仍然歸他們。從某種意義上説，我們是在自願促成自己的衰亡。

我認為這種模式必須改變。我們應該以要求人類撰寫的論文所遵守的同一套嚴謹、清晰和表達標準，來要求 AI 生成的數學成果公告。想得到數學界認可？想取得正當性的背書？那就拿出真正易讀、可核查的內容。否則，我們就應該直接置之不理。

換言之，我不認為我們這個學術社羣應該花時間收拾他們留下的爛攤子，替別人增加商業收入。

要贏得這場競賽並不容易，但至少我們可以嘗試稍微改變規則，讓這場競賽不至於完全偏袒他們。

Giulio Tiozzo

今天我的心情既興奮又憂慮：看到許多問題得到解決、發現新的證明，令人欣喜。然而，我們這個行業顯然已永遠改變。幾點想法：

1) 我不認為 AI 生成數學證明的主要問題在於難以理解：所有重大問題最初的解答都很難消化。問題在於，沒有人願意去做這件事，因為他們擔心自己得不到任何回報，無論是在社羣層面還是個人層面。

2) 令我非常困惑的是溝通方式：AGMAI 成立之後，我原本以為這次數學成果發布會是這樣進行：9 位專家各自選一個問題，在影片中仔細講解，或撰寫一篇優質的配套論文。實際上，卻只是在 GitHub 上一次過丟出大量粗製濫造的論文。我們到底為甚麼需要 AGMAI？

3) 很明顯，許多論文根本未經人類審閲：例如，我試了他們的例子 #254（一個沒有 CAT(0) 作用的 Artin 羣）：他們的羣有 116 個生成元，但把它交給 Astra 處理後，我在 10 分鐘內便得到一個有 12 個生成元的新例子。很明顯，他們急於發布這一大批成果，重數量而輕質素……為甚麼？

這完全沒有解決研究實驗室與學術界之間的不協調；事實上，問題仍然存在：前沿實驗室究竟是我們的朋友，還是敵人？

Raphael Appenzeller

這個發展時間表簡直瘋狂。我心情很矛盾。在數學中使用 AI，有很多支持和反對的充分理由，而對未來的不確定性只會愈來愈大。

Alexandre Martin

昨天公佈消息後，我們當中很多人都感到震驚，不知道如何適應這種新局面。現在已有同事呼籲組織國際讀書小組和工作坊，弄清楚整件事，並為部分這些聲稱的證明撰寫「由人類撰寫的説明」。對我而言，雖然其中一些結果與我的研究十分切身，而且這些同事當中有些確實出於好意，但我已決定不參與任何此類倡議。

首先，我不想參與某些情況下等同為 OpenAI 提供免費宣傳和免費同行評審的事情，助長一種不實説法，聲稱這家公司如今正積極與數學界合作。第二個原因是，如果我們以這種方式、如此大規模地組織起來，就等於默許這種新的分工模式：證明的產出可以完全脱離任何形式的理解，而這種去人化的證明產出過程，則愈來愈與計算資源軍備競賽掛鈎。我認為，這在很多層面上都難以持續，而且這也根本不是我希望我們的社羣和共同的數學實踐走向的未來。

Baptiste Serraille

看到今天公佈的巨大進展後，我決定首次大量試用 ChatGPT。大多數時候，我不會把自己的想法交給它，因為我太擔心這些想法會經 ChatGPT 傳到其他用戶或 OpenAI 本身，令我失去對它們的掌握。因此，我決定用它來試做相關領域的未解問題，或是短期／中期內我沒有時間嘗試的項目。最後，其中一個問題確實被攻克了，而我構想的一個項目似乎也可能有所成果。看到這些結果和技術進展，我既感到驚嘆，也因進展速度遠超我所能跟上而感到害怕。我也不樂意為不屬於自己的想法寫一篇出色的論文，而這樣做主要只是因為我相信這會令該領域受益。

Matt Zaremsky

多年來，我整體研究計劃中最主要的一部分一直是羣論中的 Boone-Higman 猜想。我與許多人合寫了一系列論文，逐步建立並完善一個不錯的充分條件，讓一個羣滿足這個猜想。過去 4 年來，我想大約每年都會有一次重大突破，強而有力地推進整個研究計劃。現在，有間 AI 公司認為 Boone-Higman 猜想已經夠有名，值得他們大肆慶祝勝利，於是他們就這麼做了。結果證明，我們提出的充分條件總是成立，而我們一直缺少的主要機制，在概念上與我們 2025 年取得的突破相當相似（極其粗略地説，關鍵在於各種對象的類仿射作用）。所以，看來我們確實找對了方向，而且相信再多花一兩年，我們也可能得出這個結果。

那感覺如何？嗯，感覺就像我們花了 4 年進行考古發掘，一點一點地發現一具看起來非常酷的恐龍骨架，然後一間市值萬億美元的公司突然出現，用 TNT 把整個東西炸開，將完整骨架交給我們，然後揚長而去。所以，我想我會説：「不太好。」

Ilya Kazachkov

首先，這項公告留下的一片焦土之外，其規模也顯示，我們這個專業領域很可能會經歷極其深遠的變化。我們作為一個羣體，需要回答涵蓋工作各個層面的許多問題。媒體上已提出其中不少問題，包括這個網誌：攻讀數學博士究竟意味着甚麼？出版、評估、招聘和資助將如何運作？數學研究在社會中扮演甚麼角色？

至於研究，AI 革命似乎帶來的最耐人尋味問題，是甚麼才算是「好的」數學問題。具體而言，我們需要了解，哪些類型的問題（如果有的話）可以由人類（與 AI）解決，但 AI 在很少人為介入的情況下卻無法解決。

在動盪時期，恐慌是最糟糕的策略。無論我們最終會得出甚麼答案，經過一段不確定和焦慮的時期後，情況終會穩定下來。我相信我們會適應，而這個專業領域也會在新常態下延續下去。

Mark Hagen

一些職涯初期的同事過去曾就其他事情向我徵詢意見；目前，他們似乎完全可以理解地因這些發展及其帶來的不確定性而深感困擾，並再次向我尋求建議（或一些安慰，或類似的東西）。今天，我優先考慮的是想想該對他們説甚麼，而不是詳細閲讀任何一篇 OpenAI 預印本，因此我對內容沒有甚麼有用的評論。至今我仍不知道如何回應這些求助（目前仍然如此）。

但無論我們在數學上覺得其中一些發展有多有趣，我認為它們的發布方式顯示，我們顯然正面對一個不懷好意的人類機構行為者；其利益與我對科學研究（或任何其他值得追求的人類事業）宗旨的理解並不一致。

我關心從事數學的人、我的朋友和同事，以及我重視的文化實踐（例如做數學）。我更擔心比數學範疇更廣泛的 AI 可能帶來的影響：經濟動盪、集體人類能力退化、剝削大幅加劇和生態崩潰、新而可怕的鎮壓與暴力手段等等。數學在我們的文化中素有艱深、令人望而生畏的名聲；顯然，OpenAI 認為，正因如此，數學這項人類共同事業適合被騎劫，成為展示力量的工具。

無論如何，這似乎是一種很容易令人困惑或不知所措的處境。這個社羣以關係相對緊密、能夠進行有成效的集體討論而自豪。這似乎是對這些假設的一項考驗。

Cyril Houdayer

和許多研究算子代數的朋友一樣，OpenAI 公佈的成果規模令我震驚。其中兩個問題尤其令我掛心，因為我花了很多時間思考和研究它們。

Connes 的剛性猜想是我最喜愛的問題之一。直到2010年代末，它之所以令我着迷，部分原因是它看來完全無從着手。即使是 Sorin Popa 的變形／剛性理論，也沒有提供處理這個猜想的方法。我與 Rémi Boutonnet 合作，開始用 von Neumann 代數方法研究高秩格點。我們提出的 Nevo–Margulis–Zimmer 定理的非交換版本，為嘗試從格點的 von Neumann 代數還原其秩，提供了概念框架。我沿着這個方向，透過非交換邊界理論深入研究，揭示了一些有趣的剛性現象，並大力推進還原格點秩的工作。

我沒有解決這個猜想，但我逐漸相信 OpenAI 的前沿模型或許能做到，並一直在心理上為這種可能性作準備。我已坦然接受這一點。如今，對於公佈的解答，我由衷感到興奮，也想了解證明。我期待在 IHP 計劃 « Operator Algebras: Approximation, Rigidity and Dynamics » 期間，召集我的博士生和博士後研究員，一起研究這個證明。

研究這個猜想對我意義重大，儘管我沒有解決它。它讓我深入研究 Furstenberg、Margulis 和 Zimmer 的數學。我結識了新朋友，也發現了算子代數與 Lie 羣離散子羣之間的聯繫。這些經歷至今仍對我意義深遠，我也期待可能由此產生的新聯繫。

對我們而言，Connes 的雙中心化子問題有不同的發展脈絡，並有一點重要事項需要澄清。這個問題在 III 型 von Neumann 代數的結構理論中一直扮演核心角色。我和 Amine Marrakchi 在十多年前開始研究這個問題，其後 Amine 發展出大量用來處理它的工具和技巧。最近，我們從一個相當間接的方向重新研究這個問題。在我們九月的合作研究中，我們解決了所有 III_1 型因子的這個問題。這個解答是在一項更廣泛的分類定理中，作為一個關於 II_1 型因子上某個流的雙中心化子的結果而得出的。

OpenAI 公佈的證明建基於 Amine 早前研究所發展的方法，以及我們共同發現的共振機制。我們最近的研究解決了非相對版本的雙中心化子問題。OpenAI 的成果進一步確立了相對版本，擴展了我們的研究。

這些消息震撼了我們的社羣。我明白為甚麼許多同事，尤其是較年輕的同事，會覺得這個時刻很難面對；我對數學的興奮，並不會令這份不安消失。我希望大家能坦誠談論這兩種感受。對我而言，眼前的回應是研讀這些證明，與學生和同事討論，並繼續一起探索數學。

Danny Calegari

大約 15 年前，數學界對我在自由羣穩定交換子長度理論中的發現缺乏興趣，這段經歷（參見 https://www.quantamagazine.org/how-failure-has-made-mathematics-stronger-20240522/）令我下定決心：從那時起，我只做自己感興趣的事。如果我想證明一條定理，我就去證明一條定理；如果我想畫一幅圖，我就去畫一幅圖。在我看來，為此付出的代價很小：我不認為有很多人讀我的論文。好處是，OpenAI 沒有解決任何我當時正在研究的問題。他們解決了幾個我感興趣的問題，但我對它們並非特別在意，而且這些問題與我的自我認同也沒有甚麼關係。

目前最令我興奮的研究項目，是高維拉鏈理論；其中最令人興奮的一點，是現在已經可以畫圖（確切來説，是製作動畫）。我喜歡數學的某些領域，例如 Langlands 計劃，因為它們不只是一系列定理和猜想，更是一套框架，也是一個故事。我一直嘗試在自己的研究中尋找背後的故事（至少過去 15 年如此）。OpenAI 尚未破壞這一點，至少對我而言，目前還沒有。另一個例子：像 Sullivan 詞典這樣的框架，比全純動力系統領域最頂尖的 100 篇論文（包括 Sullivan 自己的論文）更重要。在自己的研究中，沿着 Sullivan 詞典的思路，我最近在全純動力系統領域發現了有限深度葉狀結構的對應物：它是地毯輪的嫁接！光是為它命名，就令我興奮！下一步是寫下定義並證明定理，但（對我而言）做這一步，是為了編寫一些程式，並繪製圖像／製作動畫。

無論如何，人們做數學的方式各有不同，原因也各有不同。我真心認為，未來做數學的方式只會更多，不會更少。有興趣的話，https://math.uchicago.edu/~dannyc/gallery/cs_zippers_movie.mp4 這段影片展示的，就是目前令我雀躍不已的事物；它是在 S^3 中、與一個算術複雙曲格相關的拉鏈，構思上參考了 Isenrich-Py 的一些想法。這條拉鏈是鏈結的，反映出存在一個圓值 Morse 函數；該函數有臨界點，但這些點全都位於中間維度（2）。我很喜歡這一點！我目前正與一位音樂家合作，為它配樂。這段影片是用 Claude 編寫的程式碼製作的；該程式碼推廣了我最初在 2d 設定中構思的程式碼（及演算法）。如果這不值得慶祝，我就不知道還有甚麼值得慶祝了。

Vadim Alekseev

我對自由因子問題的解答印象非常深刻（自由羣的 von Neumann 代數彼此全部同構），因為我對預期結果的判斷原來是錯的：我以為它們應該彼此不同構，因為測度羣論方面的相關證據（依我看來）似乎更指向這個結果（透過 L2-Betti 數）；此外，SL(n,Z) 的 von Neumann 代數彼此不同構這一事實，也支持這種思路！所以我覺得這很令人振奮：我們現在必須弄清楚，測度羣論與算子代數之間究竟有哪些方面仍然相似，哪些方面有所分歧，以及原因何在。

Tristan Humbert

今天早上醒來時，我收到一位合作者的電郵，得知 Open AI 宣佈證明瞭 Katok 熵猜想。這是一個懸而未決三十年的問題，曾是我的博士研究課題，也是更廣泛而言我研究的主要動力。Katok 已解決曲面情形，而高維情形大多仍未解決。我的論文主要研究項目，是證明複雙曲度量附近的局部版本。完成後，我原本打算更全面地研究這個猜想。Open AI 今早聲稱已在完全一般的情況下證明瞭這個猜想，我最初感到難過，看到自己最喜歡的問題被 AI「扼殺」；當問題沒有直接影響自己時，確實比較容易置之不理。接着，我感到壓力，因為我目前正在申請博士後職位，而研究計劃中有更多內容因此變得過時。我整個早上都在恐慌中重寫所有內容，並電郵給我的導師。最後，我打開論文，發現內容大多是難以閲讀的垃圾，於是感到憤怒。理智的做法本來只是置之不理，但我想自己太在意這個問題，無法假裝它不存在。我目前唯一能得出的結論是，Open AI 對數學研究成果發表的標準低得太過分，正以可能無法挽回的方式損害數學研究；而數學界出於真正的科學好奇心，願意免費工作，去「驗證」這些難以閲讀的證明。

Tim Gehrunger

我一直很期待 OpenAI 終於公佈聲明。我不確定他們究竟可以做些甚麼來符合我的期望，但起初我對內容略感失望，尤其是考慮到我所研究的算術幾何領域的成果。

當然，其他不少工作也令人印象深刻；在組合數學等一些數學領域中，數個主要猜想似乎已經得到解決，這很可能會改變這些領域未來的發展方向。

令我印象深刻的一點是，許多論文的潤飾程度未如理想：有些論文引用了已被移除的支持性手稿，有些則漏引相關的先前研究，還有一些小錯誤（我原本很有理由預期 AI 能夠發現）。如果採用更完善的工作流程，安排專責 Agent 負責審閲、正確標明成果歸屬及致謝相關文獻，很可能就能發現這些問題。我希望未來的發布會採用這類系統，改善論文的潤飾程度，並確保先前成果得到適當的歸屬。

Peter Scholze

我們應該記住，大家都在同一條路上；數學是一場馬拉松，而不是短跑；目標一直都是、將來也會是人類對數學的理解，而這必然需要時間。

目前最令我擔憂的，其實是密碼學的（不）安全性。找出能破解標準密碼協定的演算法，是一個數論問題；在我這個非專家看來，其難度並沒有顯著超出這些系統目前的能力。如果找到這樣的演算法，對社會將造成災難性後果。

Tasmin Chu

我當時在研究 pc<pu 問題。我在大學本科時第一次接觸這個問題。Benjamini 和 Schramm 在 1996 年發表了一篇很精彩的論文，他們猜想：在每個非可平均擬傳遞圖上，Bernoulli(p) 滲流都存在一個階段，會出現無限多個無限簇。Burton 和 Keane 的研究證明瞭其逆命題成立。我非常熱愛這個問題。它以簡潔優美的方式，揭示了這些高度類樹圖的一項特性，而這項特性在一般情況下不知何故一直未能證明。這就是我愛上滲流理論的原因。OpenAI 的預印本建立在我的導師 Tom Hutchcroft 的研究之上，也採用了他一年前向我講解的方法，這一點並不令人意外。

我下星期要提交的撥款申請必須重寫。也許撥款申請已經沒有意義了。我有相關的研究成果和工作可以談，這些都是我為攻克這個問題而制定的整體研究計劃的一部分。但現在，我甚至不敢談正在進行的研究，也不敢發表研究「公告」。我現在已經有足夠高的知名度，我相信有人很可能會蓄意以對抗方式透過提示詞補完我的研究成果。

我現在明白，比起知道 pc<pu 成立，我更渴望的是有時間和空間在未來幾年思考這個問題。想到我原以為會有的工作條件，我感到深切悲傷。沒有人能奪走數學思考的美和價值，但他們可以實際上令我在自己的專業領域中失去自主權。最令我痛心的是，OpenAI 甚至不在乎這項成果。他們不知道我們的社羣、我們的思想和我們的知識都是有生命的。坦白説，我發現自己對身處的社會感到極度厭惡。

Terence Tao

我對近期發展的感受十分矛盾，也相當複雜。

一方面，許多由 AI 生成的證明似乎提出了巧妙的新想法，待消化後將帶來裨益；同時，這些證明亦建基於過去和現在無數人類數學家的貢獻。但另一方面，我深感沮喪的是，與傳統的重大突破截然不同，參與這些證明的人類作者都無法回答問題、作演講、出席會議、向期刊投稿、培養學生，或以其他方式參與這些成果的後續發展。

同樣地，我對社羣能夠運用這些工具，處理以往連想像也不敢想的雄心勃勃、大規模項目，感到振奮。但想到數以人年計、持續而耐心且刻意放慢步調的研究工作，尤其是研究生和博士後研究人員為解決許多引人投入的數學問題所付出的努力，竟會因這類發布而被不經意地打亂甚至摧毀，我就感到不寒而慄。正如電影劇透或填字遊戲的提示一旦聽到，便無法當作沒聽過；一旦知道已有解答，便無法再以同樣有成效、豐富的方式探索問題。當然，我們仍可分析和消化這些答案；但最適合這樣做的時刻，正是答案被發現的一刻。而當這些時刻愈來愈多地完全交由 AI 工具處理，便白白浪費了。

我也為那些沒有選擇的道路，以及在這場發展這項技術的激烈競賽中失去的機會感到惋惜。實驗室原本可以把前沿模型交由獨立研究人員作適當的科學評估；可以與研究社羣協調，確保這些工具用來補足和提升人類研究人員的能力及工作，而不是與他們競爭；也可以運用這些工具促進合作和分享，而不是競爭和保密。開啟新的可能，同時不關上舊的大門。

但我們如今走上的並不是這條路。相反，社羣比以往任何時候都更需要團結起來，清楚表明我們自己的標準和價值觀，建立自己的工具和實踐方式，支持最脆弱的成員，並開闢自己的前路。讓我們開始行動。

Elia Fioravanti

看到一些我認為幾乎無法攻克的問題得到解決，我確實感到有些興奮；但這份感受被悲痛和憤懣蓋過，因為我們領域中許多最有前景、最令人振奮的研究方向，彷彿已被立上墓碑，而人類原本只需幾年便有望找到解答。我希望正在研究這些問題的人不要放棄，因為我們仍可從中學到很多。

我也希望社羣能齊集一起消化這些新結果，不論是資深還是年輕數學家，也不論他們對使用 LLM 持甚麼態度。不過，目標不應是撰寫載有透過這個過程得出的見解的預印本，除非這項工作獲 AI 公司提供大量資金支持。

Srivatsav Kunnawalkam Elayavalli

看到自由羣因子問題的解答時，我最初感到驚訝。我認為自由羣因子應該彼此不同構，並曾多年努力嘗試證明這一點。無論如何，我現在完全相信，定理／證明在這個專業領域幾乎已完全失去價值。我們必須優先重視人類的理解，並設計方法予以獎勵和提供誘因。反覆發佈由 AI 生成並驗證的手稿，只會帶來沮喪和無所適從。正如我在先前的 Proofs and Prompts 文章中所説，我在這個專業領域的目標是提升我的 manodharma。目前我樂在其中，所有時間都用來準備我正在教授的「自由 C*-代數」課程。我有一羣非常優秀、對課程內容深感興趣的學生。我完全不想打亂自己的步調，在此刻開始閲讀 OpenAI 海嘯般湧現的研究成果，令自己因 AI 而消化不良。

Yang Li

假設這些證明正確，我必須説，這些成果令我非常佩服；如果能夠恰當理解這些成果，數學的發展或可大幅加快。我主要憂慮的是社會層面的問題。尤其是，我認為在急劇變革的時代，博士生和博士後研究員是最脆弱的一羣，數學界需要找到方法保護年輕一代。我也認為，鑑於這項技術的力量，計算資源應該更公平地讓人取得。

Michael Chapman

OpenAI 在公佈中發表的近期成果，規模之大、涵蓋範圍之廣，令我十分驚訝。當中有幾個問題，例如推翻 Kaplansky 各項猜想、非殘有限雙曲羣的存在、所有維度中的有界度數 coboundary expanders，以及唯一遊戲猜想，都是「我在學術成長過程中一直接觸的」問題——也就是説，我在學習初期便認識了它們，多年來一直深深着迷（也曾深入研究其中一些問題）。我最想做的，就是坐下來盡可能多讀一些，梳理主要構想並研究這些成果。這令我深感謙卑；對一位年輕研究者來説，也令人無所適從。儘管如此，我感到興奮多於害怕，並希望整個數學界，以及規模較小的我所屬研究羣體，都能因此變得更強大。

Constantin Kogler

11 天前，我目睹 GPT-6 Astra 解決了我所屬領域中最知名的問題之一。這個令人驚嘆的證明令我十分興奮，於是我與一位長期合作的夥伴一起重新撰寫並重新思考這個解答。經過多日埋頭苦幹，到星期一時，我們已完成一篇幾乎寫好的論文。星期二，OpenAI 也把同一個猜想列為 Paper 148，並聲稱已解決。我那位合作者想在星期二發表，但我想更仔細地查核文獻中的某個方面。

公告最初帶來的震撼過後，OpenAI 不久後聲稱得出的解答，對我個人而言並沒有造成太大影響。無論如何，核心構想出自 Astra。我們星期三在 arXiv 發表了論文。如果論文的主要構想是我們自己的，它就會比我們以往的最佳成果高出一個數量級。我很高興我們參與了這個層次的數學研究，而且可能是最先理解這個解答的人類。我希望其他人也會撰寫自己對這個證明的看法，讓我們有多篇從不同角度出發的闡述。

Paper 153 探討的也是一個我非常重視的問題。我肯定能寫出比 OpenAI 更清晰的解釋。因此，我期待理解事情的來龍去脈，並希望能與合作者一起表達我們對這項深刻數學成果的看法。

是的，具備這種能力的機器將會改變一切，不只改變數學，也會改變世界。我理解許多同事為此感到不安，尤其是那些已經取得驚人成果，或正準備發表出色研究的人。至於我自己，我不禁覺得彷彿生活在數學仙境：只要我準備好，下一個足以定義研究領域的構想便會被發現。

Andreas Thom

此刻，我為數學界為這項卓越發展奠定基礎而感到自豪。明天再看看感覺如何；我們必須回答一些嚴肅的問題。但無論如何，接下來整個冬天都有足夠的內容可讀。

Anna Chavez Caliz

今天，我抱持較樂觀的態度。上星期我們在 Cuernavaca 舉行了一場非常振奮人心、引人投入的會議。我比以往任何時候都更清楚，工作中不可或缺的一部分，不是獨自待在辦公室、站在黑板前，或撰寫只有一小部分人會看的論文。我很珍惜這個機會，提醒自己只要走出去與人交流，我們仍然能對數學感到興奮。正如 Tolstoy 所説，樂趣在於尋求真理，而不在於找到真理。

Tobias Osborne

差不多一年來，我一直密切留意 LLM 的能力，本以為自己對其進展速度多少已經習慣了（甚至變得麻木）。雖然我並非完全意外，但 OpenAI 今天大規模發布的數學成果，還是令我感到相當震撼。現在要在一段文字中妥善梳理這一切是不可能的，尤其是時間尚早；不過，我想先記下幾點想法：（1）目前我只看了幾項與我關注領域密切相關的成果。我驚訝地發現，當中的個別部分都很容易辨認。更大膽的創意在於它們違反直覺的組合方式。我大概會放棄嘗試這些組合，或説服自己不要試。我非常好奇專家會如何看待那些較重要的問題。我留意的一個特點似乎並不存在：難以解讀的「外星遺物」式想法和方法。論證以文字敍述呈現，雖然較粗略，但看來不難理解。（2）我認為，直接公開分享這些內容是正確的做法：認知如今已是充裕的資源，我們應爭取讓這項資源盡可能廣泛地免費提供。（3）我要提醒大家，LLM 絕對沒有「解決所有數學問題」，就如它們也沒有「解決所有軟件工程問題」一樣。它們的能力分佈非常不均；如果你經常使用它們，就會明白我的意思。毫無疑問，事情將會改變：用手指在電腦上輸入程式碼，現在已令人覺得像是過時的做法，但工程工作仍然充滿挑戰。數學家的工作遠不止於解決問題。我樂觀地認為，「Big M」數學和「Big P」物理學的時代如今已確確實實到來：我們終於可以採取更進取的步伐，重新思考幾個世紀以來推動我們領域發展的重大問題。也許現在我們可以攜手合作，在有生之年於這些問題上取得有意義的進展。

Mahan Mj

今天我一直在研究 OAI 發布的非殘有限羣構造，發現證明中其實有兩個截然不同的部分：一個純粹是代數的，另一個則是幾何的。今天摸索 Astra 一段時間後，它提出了一個聽起來合理、純粹以羣環理論為基礎的羣之非殘有限性判準。它又指出，這項判準或許也可以證明該羣不是 sofic 羣。我簡略看過 AI 生成的證明，但還沒有時間仔細核查。總之，這看來是構造中相對較新的唯一部分，因為它建基於幾天前 AI 生成的非 sofic 羣。

令人惋惜的是以下這一點。大致而言，學界至今還未有時間真正消化大約一個月前提出的非 sofic 構造。可以想像，隨着不同人反覆研究這個例子，久而久之或會提出多項此類判準。這或能描繪出非 sofic／非殘有限羣這片大致上仍未探索的領域。由此帶來的清晰與理解，正是人類從事數學的兩個主要初衷。目前一切以驚人的速度發展，卻缺乏人類的理解，正正危及了這一點。

Barna Saha

我擔心大型企業如何控制學術研究。有一宗個案中，一名 Anthropic 員工要求一位頂尖大學的知名學者簽署 NDA，並提出報酬和作者署名，請對方驗證一項許多研究人員花了數十年研究的結果。這在學術研究中不可想像。研究工作的作者署名不應以這種方式取得。另一宗事件中，OpenAI 一次過公開 772 道數學及相關領域問題的解答，其中許多都是重大的突破。一般學者無法使用那些強大的內部模型。公開提供的模型與之相去甚遠。這造成了可及性和公平性方面的巨大差距。我看到不少學生感到沮喪。

Josh Frisch

多年來，每逢我在會議上遇到新朋友，我都會問他們：「如果你可以解決任何一個問題，你會選哪一個？」目標是超越一份「著名」問題清單，瞭解數學家真正關心甚麼。我們真正想知道的是甚麼？只有一個人回答過「黎曼猜想」。

跟許多其他數學家一樣，踏入 2026 年以來，我最主要的感受一直是失落：意義的失落、目標的失落、人類證明時代的失落，以及無法想像未來的失落。隨着昨天（2026年10月6日）發布了數百項美妙的結果——當中許多，或許大部分，解答了某人心中的「那個問題」——我正試着走出失落。這裏有許多美妙的成果：一些從未有人想到任何解法的問題、一些沒有人認為可能存在的算法、出乎意料的同構、構造和證明。這是一種由人類數學的回響與支架構建而成、但顯然將超越人類數學的數學。

在這場證明洪流中，一定有、也必定有許多美妙的想法。即使解答不是由我們找出，即使不是由任何人類找出，只要我們能知道那個問題的答案，就應該嘗試理解它，並彼此分享這份理解。

David Fisher

沒有人應該需要這麼快作出回應。或許 AI 可以帶動一場慢數學運動。如果我更有才華，就會把這段文字寫成一首俳句。¹

Martin Bridson

昨晚公佈的內容，規模之大確實令人震撼。就在不久之前，我還無法想像數學的前沿可以在一天之內推進這麼遠。除了那份看來已經解決的問題清單令人驚嘆外，公告附帶文件中勾勒的多種推理形式，也令我深感佩服。

這些公告會令許多人感到興奮，令另一些人感到害怕，也會令部分人受到沉重打擊。

先談談令人振奮之處。若有令人振奮之處，必定不是因為未解問題已經解決，而是因為新的可能性已經開啟。短期內，隨着全球數學界吸收、完善並深化機器提出的論證，人類的理解將大幅提升。某些情況下，專家會懊悔自己竟然錯過了文獻中不同部分之間的聯繫；另一些情況下，他們或會驚訝於一個看似巧妙的手法竟然奏效，並隨後透過發掘一種能解釋其奏效原因的新現象，推進自己的研究領域。

在所有情況下，我們努力從機器的形式化推演中提煉出人類理解，這個過程都會激發我們對這個數學新時代中（與 AI Agent 合作）所能達成之事抱有更大抱負。我們最喜愛的一些高峯已被征服，但其背後還有更大的高峯，等待我們以新裝備攀登。

與此同時，我們必須抗拒一種誘惑：以為從 AI 實驗室的公告中挖掘洞見，會成為我們這個時代最重要的任務。全球數學界必須堅定地自行決定哪些研究方向最值得關注。我們應善用機器所提供的力量，但不應淪為追隨它們方向的附庸。

這種受人支配的意象，帶出與興奮並存的恐懼；隨着 AI 的數學能力日益提升，這種恐懼無疑如影隨形。

OpenAI 公告所涵蓋的問題中，有幾個曾是指引我個人研究的挑戰。我為失去這些指引性問題感到惋惜，也確信未來的公告還會讓我失去許多其他問題。我為此感到難過，但並未因此崩潰。這種相對樂觀的反應無疑反映了我目前所處的職業生涯階段；二十年前，我不會這麼樂觀。我深切明白，今天的公告對年輕同事的人生影響會更深遠。

我個人期待學習新成果的 AI 草圖中隱藏的新想法和構造，也特別期待與不同職業生涯階段的同事交流，一起探討已經取得的成果。我預期數學界會攜手合作，逐步增添人類的洞見和解釋。我認為這將是真正由羣體共同完成的努力，也相信我們將會有非常精彩的研討會！

不過，我也不得不承認，心中一直有個揮之不去的憂慮：我深愛的那種生活方式——尋找新發現的樂趣一直佔據核心位置——將會演變成某種較陌生、也較難令我由衷感到吸引的事物。深入理解一直是研究數學家的主要動力。在艱苦求索理解的過程中，發現的喜悦支撐着我們；親自理解某件美妙事物，也能帶來很大的個人喜悦。但對我而言，我猜對我們大多數人來説，發現並分享人類尚未認識的新事物，帶來的喜悦更大。如果一般數學家的角色完全縮減為解釋他人（人類或機器）的產出，這份更大的喜悦便會消失，當數學家這份職業的吸引力也會降低。這未必是我們的未來，但這種憂慮並非無理。

我認為毫無疑問的是，近期的發展對我們這個專業的架構有深遠影響。我們必須迅速調整架構，尤其是專業訓練的學徒階段——博士及博士後時期。此外，還有許多關於成果歸屬與認可、出版角色等問題。

我也認同大家普遍感到不安的一點：AI 實驗室的商業利益與全球社羣的價值觀並不一致。兩者之間存在根本而固有的矛盾，唯有持續而有力的參與，才有可能化解。我們當然不能把數學的未來福祉寄託在他們的善意之上。

Konrad Wrobel

瀏覽這份列出各項聲稱成果的清單時，我的情緒在震驚與冷漠之間反覆擺盪。成果數量之多，只能説令人振奮（即使撇開我個人關注的突出成果，以及其他我熟悉的成果也是如此），但同時也令人情緒上極度疲憊。然而，想到眼前還有堆積如山的工作，要分析這些坦白説寫得糟糕透頂的論文，弄清楚當中有哪些新想法，整件事又出奇地令人感到虎頭蛇尾。我只能猜想，還要多久我才能真正消化那些與自己相關的想法。

Alon Dogon

一整天下來，我對這件事的感受反覆轉變，從對未來感到極度恐懼，到因許多重要難題終於有了答案而感到興奮。

如果要我挑一項最觸動自己的成果，那就是強 Ulam 穩定性與可容許性的等價，以及 Dixmier 問題；我已認真思考這個問題好幾年。

這個解法似乎把 Furstenberg 的邊界理論與量子計算中的量子電路結合起來，真是太不可思議了！

總的來説，這一輪有許多猜想得到肯定解答，令人印象深刻。

Hugo Duminil-Copin

我早已預料有一天我們會被超越，而且這種情況會系統性地發生。但昨天的公佈帶來的衝擊，遠超我的預期。數十篇論文都涉及我正在研究的課題。有些成果搶先一步完成，還有長達千頁的證明，我已不知道該從何看起。

我在整個學術生涯中曾公開提及的主要未解問題，無論是在演講、課堂、文章，甚至資助申請中提過的，沒有一個不受這次公佈影響。所有問題都有人聲稱已經證明。

我預料清單上會有其中幾個，但沒想到會全部都在。沒想到會一次過全部出現，更沒想到會如此輕描淡寫。

「為了人類心智的榮耀」，他們這樣説……

這次的震撼無比巨大。我完全不知所措。明天，我們會找到前進方向。我們會重新思考自己的專業，以及工作的方式。我們是一個有韌性的社羣，我毫不懷疑我們會適應。但此刻，我實在沒有精力。我回想起那些年、那些面孔……我想到同事、學生……我擔心自己再也找不到合適的言語。

Petra Schwer

今天一整天，我在參加的工作坊裏和很多人談起「那份清單」。今天我的心情百感交集，從震驚，到對這項技術所能做到的一切感到某程度的驚嘆，情緒都有。我也很憤怒，因為一些似乎對實際內容毫無興趣的公司不斷向我們拋出各種「解決方案」。聳動的標題有助他們爭取更好的資金，他們似乎對此樂見其成。他們似乎不在意一切被如此迅速地攪亂，以至於我們（數學家社羣）已無法跟上收拾殘局的速度。

令我擔心的是，我已經注意到自己行為出現一些細微變化。談到自己的研究項目和計劃時，我不再像從前那樣坦率。例如，演講後有人提問時，我沒有坦率回答部分問題。這不只是我一個人的情況，大家都變得更謹慎。猜疑正在蔓延，這並不好。我很幸運，身處的數學社羣大多彼此信任（過？）。看到這種轉變，令我擔心。

更令我擔心的是，看到身邊的年輕數學家苦苦掙扎。今天我也看到很多恐懼。我可以怎樣幫助他們渡過難關？

數學已死嗎？顯然沒有²。我們現在經歷的，無疑是一場巨大的震盪、地震、風暴。有很多問題需要處理。數學研究的格局肯定會改變。具體會怎樣改變？我完全不知道。我非常希望我所珍惜的社羣（和人們）能安然度過這場變局，只受一點輕傷。

Bryna Kra

這次發布包含深刻而影響深遠的成果，讓我們得以一窺現今探索數學的強大工具。但數學不只是產生定理，而這種發布方式沿途失去了太多。理解這項工作需要付出巨大努力；而與幾個月前的研究不同，當我們在證明中遇到困難時，已經沒有可以請教的人。這並非數學界的文化。

作為一個社羣，我們必須團結起來，共同守護我們珍視的事物。我們理解數學的目標沒有改變，但達成目標的方法已經改變。我擔心的是，讓目前用來推動進展的這批研究成果得以產生的生態系統，正在遭受破壞。數學界大致上一直以合作為主：我們分享問題、討論尚在進行的研究，也會為別人提供解決問題的思路。把研究工作的這些部分輕易轉化為證明，會截斷理解所需的過程，而這種理解正是帶來影響的必要條件。這種發布成果的方式會封閉研究方向，而不是開拓新的視野。

數學界已經開始團結起來，深入討論如何應對這段動盪和變化的時期。現在是時候從討論走向實踐，為我們這個專業的未來規劃方向。博士培訓、初級教員聘用、升職評核、出版準則和出版方式，都需要接受檢視和更新。這種局面帶來的好處，是社羣中那些沒有成效的傳統得以瓦解，同時保留我們珍視的部分。

Alvaro Lozano-Robledo

「OpenAI 的重大發布」堪稱歷史性時刻，甚至可能是迄今數學史上最重要的一天。2026年10月6日發布中，許多如今已有擬議解答的問題，若能解決，將為各自的領域作出重大貢獻：準 RH、希爾伯特第十六問題的第二部分、Q 上的希爾伯特第十問題、CM 阿貝爾簇的霍奇猜想、Goldfeld 猜想、快速整數乘法……這些無疑都是重大貢獻。

然而，這些事並沒有改變我的想法。相反，我們早已知道這些模型能做到驚人的事（例如 Navier-Stokes）。我們早已知道，前沿模型能以巧妙的方式串連現有文獻中的線索（例如單位距離猜想）。我們早已知道，OpenAI 可以投入多得令人咋舌的資源來攻克問題。我們也知道，他們最高級別的訂閲價格即將大幅上調，至 $500/month。

Also, we suspected that their models have limitations, and the new release shows evidence of that too. In their report, they mention that they attacked 4000 open problems, and their model was able to make progress on about 700 related problems. Yes, some of the ones they were able to solve are huge. But it also shows that their models are limited in some ways: their goal was RH, not quasi-RH. Their goal was the full Hodge, not Hodge for CM varieties. Their goal was BSD, not Goldfeld’s 50-50. Again, quasi-RH is huge! But it is not RH.

Are any of the solutions using new ideas that are outside of the convex hull of the current ideas in the literature (in the sense of Nestor Guillen)? We will need mathematicians and time to digest these new proofs and understand what connections are being made, and whether brand new ideas were actually discovered in the process. There is a lot of mathematical research that remains to be done with and without the aid of LLMs. What has changed is that now there are new mountains of mathematics to explain and communicate to others.

Julian Wykowski

Navigating today, I kept thinking about a passage in Stanisław Lem’s Solaris (1961), where the main character fantasises about the existence of a bóg ułomny. This has been translated into English as an imperfect god, although I believe a more faithful translation would be a defective or disabled god. The passage reads:

“I’m not thinking of a god whose imperfection arises out of the candour of his human creators, but one whose imperfection represents his essential characteristic: a god limited in his omniscience and power, fallible, incapable of foreseeing the consequences of his acts, and creating things that lead to horror. He is a … sick god, whose ambitions exceed his powers and who does not realise it at first. A god who has created clocks, but not the time they measure. He has created systems or mechanisms that served specific ends but have now overstepped and betrayed them. And he has created eternity, which was to have measured his power, and which measures his unending defeat.”

Many members of the community agree that the main purpose of open questions in mathematics is to guide theory building and produce understanding, rather than a binary answer to some problem with limited applications in the real world. In that sense, we truly have created systems or mechanisms that served specific ends but have now overstepped and betrayed them. While it is certainly in OpenAI’s marketing interests to spread a narrative that mathematics has been “solved” through the existence of some lean code on some server, I sincerely hope our community will not succumb to such a defeatist narrative. Instead, I hope that we will find a consensus-based, organised approach to adapt our work to this new reality, in ways that align with our values, support our pursuit of human understanding, and benefit the construction of mathematical theory. This may well include embracing AI, but only in a form that maximises its positive and minimises its negative impact on the aspects of mathematics we consider fundamental. In the meantime, if OpenAI wants everyone to believe they are a deity, it is our duty to remember how defective their idea of deity is.

Ben Green

I was not expecting the magnitude of some of these results. Most particularly, seeing a proof of no zeros of Dirichlet L-functions to the right of Res=7/8 (and a second, short, proof of no Siegel zeros) is absolutely shocking to me, but there are many other breathtaking advances. Closer to my particular expertise, many of the central problems of additive combinatorics have fallen, including around three quarters of the aims I had for an ERC Advanced Grant, awarded only in June. The work of understanding these solutions and the associated context properly is significant and, from what I can glean from the current manuscripts, likely to be very worthwhile. I’ll start with number 182, which shows that any subset of {1,…,N} of size N1−c has two elements differing by a square.

Sam Hughes

It was a privilege of a lifetime to get to do research level maths. But not like this. How much beauty have we lost?

Emily Riehl

Firstly, kudos to whoever is behind the website citedbyagi.com, which recognizes the mathematicians whose work is cited by the manuscript collection released by OpenAI. It will take quite a while to understand what exactly has been achieved there. But whatever it is was only possible because mathematicians formulated the conjectures, proved the surrounding results, and shared their ideas – in conversations, talks, expository writing, and papers – so that other humans, and now AI, could learn from them. I hope they continue, because like many others I love learning new mathematics from other humans and always will.

Kevin Buzzard

I am very excited about the future. I know that there is chaos today. We are in the eye of the storm. We do not want to read slop papers. Some of the OpenAI papers have already been retracted. We do not yet even know what is true. But I believe that truth and understanding will bubble to the top. There are plenty of important poorly-written papers by humans — bad exposition has always been with us, and mathematicians have offered translation services for free many times before. AI will get better at explaining. Mathematics has undoubtedly moved forwards this week — this cannot be denied.

Zhou Feng

I am optimistic, because I see little to gain from pessimism. Still, I was stunned by the claimed solution to determining the dimension of self-similar measures on the line (Item 148 in the Review) and related problems. I am heartened that the proof appears to stand on the shoulders of earlier mathematicians and live within the framework they developed. I will spend more time studying it; after all, I believe human understanding and explanation are essential to progress in our community. They also bring joy, though perhaps not the same intense joy as a eureka moment.

I do not know how these problems were selected, but their impact makes me wonder: what makes a mathematical problem important or interesting? Problems drive progress, and posing good ones requires vision and taste, often developed through years of exploration. Knowing their definitive answers is always fantastic, but what comes next?

This announcement reveals AI’s capacity to “do” mathematics at massive scale and with remarkable depth. Is it possible to build an interactive mathematical world (perhaps a Google Map of math) where we can visually explore ideas, navigate proofs, and uncover connections between different fields? AI could help build it, guided by human mathematicians’ expertise. Future mathematicians could use their creativity and insight to expand this world, enrich its details, and find new questions worth pursuing.

Jakob Glas

When OpenAI announced that it had solved over 100 open problems in mathematics, I felt both excited and anxious. Excited about the new mathematics to come, and anxious that some of the problems I was working on might be among them.

Fortunately, my own research was unaffected. But when I saw the 7/8 bound for the zero-free region of the Riemann zeta function among the released problems, I was genuinely shocked. I had always thought there was a broad consensus among mathematicians that the Quasi-Riemann Hypothesis was completely out of reach with existing mathematics. Apparently, we were wrong.

Giovanni Mongardi

Today, something is lost forever. We were explorers of uncharted theorems, fine goldsmiths of beautiful proofs. Humanity was alone in the fantastic world of the mind, where crystalline cohomology was as concrete as a gothic cathedral. Today the machine came.

She claims to have solved a lot of our questions with its thunderous answer “42!”. She is quicker than us, knows everything humanity has ever done and gives us answers a few moments after we make the question.

What is left for us is to be priests of the Machine-God, heeding her words, understanding them for our fellow humans.

I fear the future of the mind will be a desert, with no questions to guide us beyond the horizon.

Sam Fisher

The announcement was the first thing I saw in the morning. My immediate reaction was just to laugh, I’m not really sure what I felt. A combination of disgust, grief, and apathy maybe. Digesting machine arguments will not offer us the same depth of understanding, expertise, intuition, and fulfillment that we find in working on our own problems for months and often years, and sharing our work with others. I worry about the future of research mathematics and my place in it. I hope we can adopt healthy norms. I am not interested in paying morally depraved tech giants >100€/month to become a professional button-pushing slop digester.

Ignasi Mundet

One question I was thinking about recently (before summer!) is what it is that makes mathematics beautiful. A partial conclusion is that beauty in mathematics is very much related to our own limitations: our limits impose us a slow pace, which allows us to discover things which we would not notice at a faster pace. Slowliness and our limitation is also very much related to viewing mathematics as an adventure and a challenge, which I think is also an important ingredient in its beauty.

One question I ask myself is: in what ways will mathematics be beautiful after the revolution that we are experiencing now?

Nilima Nigam

My own immediate reaction was one of irritation. Some of us saw this day as inevitable, even a couple of years ago. But we could not stop each other from the seduction of these tools, their promotion, and very quickly claims about their inevitability. Well, here we are.

I already saw many younger colleagues and students who were experiencing existential concerns about what it means to do mathematics, and what their role would be. I see our community already in crisis, with deep rifts around what we value, what motivates us, and indeed how much to prioritize mathematics over the humans working on it. Some are excited, some are despairing, and everything in between. We’ve seen greed, naivete, courage, resignation – all those sentiments we don’t really attach to the austere beauty of this field we love. And I see much anger directed at each other as well.

I see the public at large react in different ways since the infamous NS announcement, with a fair number of people accusing mathematicians of ‘gatekeeping’, and others accusing the community of selling out to AI corporations. There is Schadenfreude, there’s excitement about democratization of mathematics, and hopes that ‘solving NS’ would presage ‘solving cancer’.

In other words, as a community and a society we were already overwhelmed – intellectually, emotionally, politically.

Now there’s a dump of results, a high-decibel screeching for our collective attention and energy. I see how yet again many, many, many in the community will dedicate time they didn’t have, to carefully parse papers (not all of which are written with care) released at scale. And I see the exodus of young people hastened. Students are in shock at their theses suddenly being scooped.

Yes, we must react, I suppose. But my own immediate reaction to the high-volume cacophony of attention-grabbing motorcycles on my street is typically one of irritation. This is how I feel today about Open AI’s efforts in math. Open AI doesn’t need or care about my attention or respect. But if it did, the equivalent of racing down the neighbourhood noisily on 50 motorbikes with silencers off isn’t the way to do it.

Indira Chatterji

One of the papers is a solution to the Bass trace conjecture, a beautiful conjecture made by Hyman Bass in 1976, implying the older idempotent conjecture (from the 50ies maybe?) that there should be non non-trivial idempotent in the group ring of a torsion-free group. I was privileged enough to have given a proof for amenable groups 25 years ago with two amazing collaborators, Jon Berrick and Guido Mislin, that changed my career and my vision of life and of mathematics. For years now, progress on this question remained incremental and we all did other stuff.

I don’t understand the proof, and I still don’t believe that this conjecture could be true in general. The paper is 30 pages long and looks like slop -lean certified, whatever that means. However I am looking forward to a small group of colleagues to go through it and either shred it to pieces or understand a stunningly beautiful argument. Who’s in?

3 papers have been pulled already… Is it time to claim that it’s useless slop, that lean is unreliable (and demand money to say otherwise)?

Ivan Smith

Disrupted times, but I still naively hope we will come through stronger. If the community learns to reward those who forge the paths and lay down the fixed ropes as much as those who reach the summit, it would be a very good development.

Sahana Balasubramanya

My initial reaction was to skim the list of problems to see what all has been impacted. It was a bit disorienting to see many famous problems listed there, and I had a sense that the landscape of math was being “nuked”. Upon closer inspection, I realised that not all the proofs have a Lean formalization. (And as someone unfamiliar with Lean, I am not sure what to make of the results that carry such a credential). Since then, at least 3 preprints have been withdrawn due to mistakes and other corrections have also been made.

I have heard from colleagues in other areas of math that they consider some of the papers released to be “unreadable”. There are also the issues of the perpetuating lack of human attribution, the lack of transparency and the apparent disregard of the advice of the advisory committee. It will take a long time for humans to absorb this new information, if it stands the test of time and rigorous scrutiny at all. If it doesn’t, then a lot of time is still likely to be wasted proofreading for the hole in the argument, which makes one wonder what is the point of it all. This is a mess.

It makes me feel that the AI companies have chosen a side, and it is not the side that wants to work for the betterment of humanity (by trying to mitigate poverty, work on resource allocation, or trying to find a cure for serious diseases, for example) nor one that particularly cares. For the time being, I hope we do not give in to panic or a sense of doom.

Yuval Gorfine

It is hard to find the right words to describe what I’m feeling and what I’m thinking, especially when my thoughts and feelings keep changing. One such thought, at least, is this (and it lives in my mind together with other thoughts which contradict it).

I don’t know if this is the end. I hope that it’s not. I think that it’s not. But what is definitely true is that if this is indeed the sunset of mathematics, it is a marvelous one. What humanity has achieved is astonishing. When I went through the list of problems solved by OpenAI, I couldn’t but think of this old poem by E. E. Cummings:

who are you,little i

(five or six years old)

peering from some high

window;at the gold

of november sunset

(and feeling: that if day​

has to become night

this is a beautiful way)

Mitchell Taylor

As someone who has been closely following the rapid progress in the mathematical capabilities of AI, I am not particularly surprised by the number or the prominence of the problems that OpenAI has just released. I find many of the results to be beautiful, and I’d love to understand them more deeply. In particular, I am happy to see that the hot spots conjecture is true for simply connected subsets of the plane, and it is very cool to see that the separable quotient problem is independent of ZFC. These results are truly spectacular, and the fact that we are able to witness their resolutions is extremely exciting.

What concerns me much more is the potential misalignment between the objectives of AI companies and what needs to be their primary goal: optimizing the impact of AI on humanity. By now, it is clear that AI will completely change the world. However, it is also clear that many people will experience trauma during this transition period, and in this regard I think that mathematics serves as a particularly revealing case study.

In their recent release https://openai.com/index/sharing-ai-progress-in-mathematics/, OpenAI begins by stating that they have looked to improve how they share their results with the mathematical community and have consulted with the AGMAI group to discuss best practices. Although collaboration between AI companies and academic researchers is fundamentally important, in this case there seems to be a substantive disagreement between what the AGMAI group recommends and what OpenAI states that they will do. Most notably, AGMAI asks AI companies to stop evaluating their proprietary models on open research problems, whereas OpenAI makes it clear that they believe that this is important.

Although I am personally excited to witness the amazing progress in mathematics, many of my friends and colleagues are currently experiencing severe anxiety, depression and loss of purpose. What they need most at this moment is not more powerful AI, but the time to process the transformation of their discipline. I worry that they will not be given this time. However, I worry much more that our society as a whole will not be given nearly enough time to adapt to AI.

I truly hope that OpenAI takes the reactions within the mathematics community seriously when considering how the wider society may respond to the changes ahead. The primary purpose of developing a technology of this magnitude should be to improve everyone’s lives, not to maximize power or profit. In particular, the pace of scientific and social change should not be dictated by AI labs alone. Instead, I believe that it is imperative that OpenAI follows through on their stated aim of empowering scientists by actively and thoughtfully listening to their concerns and offering them a meaningful say in how this transition unfolds. In practice, this means giving a larger subset of the mathematical community a direct role in making decisions about the timing of future releases, the support needed to understand these results, and how research, teaching and society more broadly can adapt.

Menny Aka

It was always about fun. For more than two decades now, I have been lucky enough to have the opportunity to have fun learning, doing, and teaching mathematics. So the main open question for me now is: how can I continue to have fun? After recovering from the shock that each such release of results brings, I keep coming back to this question, hoping it will lead me to a solution. I have found some answers and keep experimenting in search of more. Here is where I stand today.

In teaching, I immensely enjoy being able to create, with minimal effort, an applet or demonstration tailored to exactly what I want to explain.

Recently, we created a seminar called “Illustrating Math toward Outreach,” where we work with students at all levels on illustrating and presenting mathematics through different media, many of which have become accessible thanks to LLMs. We started just a month ago, but it looks like fun will be at least one component of this seminar.

Whether I can have fun with LLMs in research is less clear to me. I keep experimenting: lately, I have been washing the dishes while discussing research questions with an LLM through my headset. Is it useful? Surprisingly, very useful. Is it a weird (not to say dystopian) experience? For me, for now, yes. It leaves me feeling empty and tired every time I try.

Can I find a way to enjoy reading these Lean-checked proofs? So far, I have found no fun in reading them. Can I enjoy working towards understanding them? I’m not sure! My current experiment is to find other interested humans, hoping to have fun tackling these proofs together. For now, I still have some naive hope: I cannot imagine a world in which understanding the proofs of the results in Project 15 is not fun. And I suspect many of us feel the same way about some other project X.

P.S. To see the reaction that helped me the most so far, google “the last ten minutes OpenAI”.

Inhyeok Choi

The results OpenAI announced are amazing, and it will be great if I can possibly learn solutions to many questions that I am interested in.

That said, I am sad about how these questions were treated. On August 1 they said they wanted to empower scientists and mathematicians. I interpreted this as helping mathematicians thrive. But on October 6 they said to empower scientists, it is important to continue evaluating their internal models on mathematics. So they just used these beautiful, far-reaching questions as testbeds. It does not feel like OpenAI is interested in Thompson’s group F, Bernoulli percolation, or QI-rigidity. It feels like they posted these results to show their model’s power. This practice will harm the math community.

Let me share some personal feelings. I have thought a little bit about a question (pc<pu) that OpenAI attacked. How do I feel about the sudden resolution of the full question? Well, it’s good to know. The solution seems reasonable, and I would love to dive into the details and gain some further understanding from it. I will still ponder upon the question from different perspectives, e.g., whether there is a more natural proof for groups with free subgroups.

At the same time, this problem deserved more endeavors. There were many ways ahead, and we could explore different routes to eventually reach the destination. It could be more humane. The journey could be enjoyed by people. But now? We’re suddenly transported to the goal by the machine. This is sad not because the destination is unwanted, but because we have lost so much of the journey and so many of the people.

For now, I’m still walking around in the era of portals. I prefer to take a stroll and find some random flowers. I hope we, as a community, will continue to value this human aspect of mathematics.

Stefan Witzel

Like all of us I’m tempted to dive into an detailed analysis of the proofs (I’d start with the non-residually finite hyperbolic groups). But I think our imminent task as a community is to form an idea of what values and processes will allow us to survive (in a first approximation: deep understanding matters more than concrete theorems; informal ways to convey understanding matter more than lean certificates). I am worried about the tempting vision that some have that we steer AI to push the frontiers way further now: I think it would work but we might loose offspring along the way and end after a generation.

Anonymous

I have been trying to convince people, including very recently, that we’re screwed and that most likely LLMs will soon be able to prove anything we might dream of proving, and more often than not what I heard back was that, really, LLMs are not that impressive.

Nino Tannio

Now machines can prove faster than people can digest them, most proofs will go unread. Our attention doesn’t scale with output. Curiosity-driven professionals may not care enough to understand once the fun, the reward & the mystery are gone, leaving us with truths without readers & answers without seekers.

But I don’t think this is a dead end. We may have to revolutionize the system. That could be as disruptive as replacing π with τ after centuries built around π. Perhaps we’ll find a better way of looking at the world along the way. Who knows? Why assume all the mystery disappears?

Koji Fujiwara

AI

— after Dolly Parton’s “Jolene”

AI, AI, AI, AI
I’m begging of you, please don’t take my math
AI, AI, AI, AI
Please don’t take it just because you can

Your wisdom is beyond compare
Your speed is like a bullet train
But I cannot compete with you, AI

And I can easily understand
How you could easily take my math
But you don’t know what it means to me, AI

I had to have this talk with you
My happiness depends on you
And whatever you decide to do, AI

AI, AI, AI, AI
I’m begging of you, please don’t take my math
AI, AI, AI, AI
Please don’t take it even though you can

AI, but it’s just not fair, AI

Ryan Alweiss

I find the new results from OpenAI to be wonderful! This is an extremely exciting time to be a mathematician. Clearly this is a period of significant disruption, and we need to restructure our institutions and rethink many of our practices for this new age of “proof abundance”. But we are learning a lot more mathematics than ever before, and humans and AI working together will propel both beautiful pure mathematics and useful applied mathematics to new heights. Props to Will DePue for his wonderful website citedbyagi.com showcasing how AI stood on the shoulders of human mathematicians.

Hyunwoo Kwon

Yesterday, OpenAI dumped more than 700 research ‘paper’ in GitHub. When I see the list of problems, I was so surprised that they announced the solution to the Falconer distance problem, local smoothing estimates for wave operators, and bounds for Kakeya maximal functions. These were central problems in harmonic analysis, and my friend has worked on one of these problems for 6 years, but OpenAI’s abrupt announcement devastated the world that she has cared about. I have started my math journey from a harmonic analysis perspective. I know the meaning of the problem for her and my colleagues and I really have a deep, bitter resentment toward this situation.

I kept thinking about the old play that I recently saw <The Cherry Orchard>. Am I Ranevsky, who is just sad about the past, or just Trofimov, who talks about ideology? My Cherry Orchard, which is full of curiosity shared with my peers, is being razed.

I was happy to have numerical experiments with the aid of AI. I could come up with a new style of questions. I don’t know how to answer yet, but I believe that this will bring a perspective to my research area. I was hopeful for this future. However, the recent activity of OpenAI kept me thinking that they are just trying to flex their dominance through raw speed and brute-force resources, not respecting the time for contemplating and historical developments in the area.

OpenAI claimed that they brought a new development for mathematics. It is a really historical moment without any doubt. However, the paper uploaded to their GitHub cannot be accepted as responsible behavior nor a genuine mathematical advancement. How can this unreadable flood of data be regarded as “Knowledge”?

It is really hard for me to endure this turmoil without watching the fall leaves turn red, the sea, and the sky.

Alex Nolte

In making sense of an evolving situation, I think it’s valuable to compare current developments to one’s past assessments. In this direction, a few weeks ago I wrote a response to the AGMAI request for community input that began: “I think that the worst outcome here is one in which the landscape of existing conjectures is suddenly destroyed in a wave of unprocessed proofs that estrange the communities of mathematical subdisciplines from the advances of their field.” This seems to be pretty directly in line with what OpenAI is aiming for in this release.

I think that the emergence of a new technology that has the potential to improve human understanding of mathematics can and should be positive for the field of mathematics. Communities adjust slowly to changes, though. It will take time to re-align our incentives and norms around rewarding work that we value in the context of these developments. To put it gently, I think it is a shame that temperance, consideration, and respect for the careers of mathematicians do not seem to figure highly among the priorities of AI companies.

Grigori Avramidi

Many of us have spent decades coming up with our own little metrics to test ai models (we called them conjectures, questions, toy problems and so on), even though we didn’t know it at the time, and have seen the new models blow past those personal metrics in the span of a few months. For me, it has been a sometimes exciting, disorienting, exhausting, visceral experience.

It has also been a real challenge to communicate this to people outside (and sometimes inside) the math community who have not experienced it firsthand. My hope is that this latest openai drop will help a broader swath of the math community feel the rate of progress that is behind these results. It is not that ai can solve a few of the marquee conjectures with a silly amount of compute, it is that it can (right now!) answer a good chunk of everything we ask.

And while it is technically possible that this rapid progress will cover coding, math, and go no further, I think we need to prepare for the possibility that it will not stop there.

I hope (perhaps naively) that we, as mathematicians, can set aside some of the complicated mix of feelings about the field we love and the politics involved, look carefully at the math in front of us and say as credible (we need to stay credible!), expert observers sitting outside the ai companies themselves “this is what we know, this is what we see, this is how fast it is going, it is not about our jobs, and it is not hype, please pay attention”.

Carl-Fredrik Nyberg-Brodda

The past few days have made me feel worried to be a mathematician, and excited to be a mathematician, and proud to be a mathematician; and often all three, and more, at once. On the conduct of OpenAI in this matter others have written far more eloquently than I could ever hope to formulate my (concurring) thoughts. Likewise the beauty and unity of mathematics and mathematicians is hard to exemplify better than what is already on display here.

While I am still learning new mathematics, there is still joy for me in mathematics; and when I am learning with others that joy is greater still. Who may come to pay me and us in the future to learn and access this joy — and why they would do so — remains, to me at least, a sharply pressing question.

Part of me is very excited to see what the future will hold, as these are weeks when years and decades happen. Part of me also wishes to live in precedented times, for once. Well, this is not for me to decide; I can only be thankful for the companions and friends I have along the way.

Stéphane Gaussent

I am torn between wanting to try to understand how the work of the machine on the saturation of the Littlewood-Richardson cone of type D and rejecting the whole thing outright. How can one not feel overwhelmed by the sheer scale of this announcement? Who is out to destroy the human practice of mathematics? What should we say to young people looking for a career in academics?

Anonymous

My suspicion is that a lot of people got Buckmastered in this drop. Clearly we need our own models, and the mathematical community needs to start being just as aggressive as other creators about stopping IP theft, starting with exposing how much these “solutions” stole from in-progress work.

Snorre H Christiansen

Report from the gym

Today the tourist came back
We had seen him last year
Working out in our gym
It was kind of funny
The way he lifted a weight or two
But now he’s back
And it looked kind of ugly
The way he threw around all the machines
Riemann
Hodge
And some other big ones
Will we also have to
Throw everything around
Like it was nothing
But yeah we’re impressed
We thought it would take him longer
Soon he’ll win this game
You know
Just push harder
And you’ll make it

Junseo Lee

October 5 was the submission deadline for QIP, the leading conference in quantum information theory. Like many others, I had spent the preceding weeks wrapping up my research, with plenty of help from LLMs, of course. QIP submissions appeared to have more than doubled compared to last year, and browsing the preprints, I was struck by the remarkable results, including longstanding open problems such as oracle separations between quantum complexity classes. It was exciting to see researchers who had spent years wrestling with these questions finally make breakthroughs, often with the help of frontier AI tools.

Then, just a day after the deadline, came OpenAI’s announcement. Whether or not the timing was intentional, quite a few people around me seemed a little deflated. Some results addressed area-law questions that one of my advisors had been thinking about for years. Others settled problems in quantum information, theoretical computer science, and many-body physics that I had hoped to tackle someday. The progress was astonishing, but also unsettling. My one reservation was that some manuscripts were difficult to follow. Perhaps this reflects my limited expertise as a PhD student, but I felt that researchers deeply familiar with these problems might have presented the ideas more clearly.

I started my PhD hoping to work on quantum complexity and learning theory. That hasn’t changed. I still love thinking about open problems and exchanging ideas with colleagues. But I’m no longer sure what the best path forward looks like for someone just starting a PhD, or what doing theory will look like a few years from now. Perhaps this is also an opportunity. If AI accelerates progress on foundational problems, more researchers might turn toward building quantum computers and finding practical applications, pushing the field forward even faster. I don’t know where things are headed, and I have a lot to discuss with my advisors and colleagues.

Despite all this, I’m genuinely excited about these developments. My one real worry is that fewer talented students will choose to pursue theoretical research, and that we’ll gradually lose the conversations that make research so enjoyable. I’d hate for “ChatGPT has probably already solved this, and probably understands it better than I do” to replace the discussions and exchange of ideas with colleagues that I value so much.

Claudio Llosa Isenrich

It feels like another historic day for mathematics. They seem to be accumulating faster than I can digest them. While I was sensing something coming towards us for a bit longer, I think for me it first really hit home when I saw results I cared about for most of my (I like to believe still short) career being proven by AI or at least with essential help from AI in June (or was it July?). Then came OpenAI’s announced solution to the 10 open problems, then came their announced solution to Navier–Stokes and finally came their bombshell of 700something papers.

Instead of putting my energy into trying to further our (or least my) understanding of mathematics by doing research, I find myself spending much more time pondering the future of mathematics, reading with interest the many insightful opinions of others, including on this excellent blog (really a great thanks to everyone who contributes to this blog and elsewhere, including the people running it) and trying to form my own views.

I have been feeling uneasy for a while about the AI companies alignment with the interests of the mathematical community. For me the Navier–Stokes episode supported this feeling and the new release, after the community’s reactions to Navier–Stokes, only strengthened it (if we were to call this feeling a conjecture, I would be very happy to hear about a disproof by the AI companies though).

Now where does this leave me: I am sad about all the guiding lighthouses that have been torn down and all the beautiful results that will never see the light of day as a consequence. I am convinced that AI is here to stay and that I need to adapt to this new reality, although I am still figuring out what precisely this means. I am longing for stability for myself, but even more for our future generation, our Bachelor, Master, PhD students and postdocs.

However, despite these thoughts, I am hopeful that humans will continue to long for intellectual activities and for understanding the world we live in. I am convinced that mathematics will continue to play an important role for this; it always has (alongside many other disciplines). Moreover, I am happy to see the community that I am so glad to be a part of come together to discuss our future in these challenging times and to work on figuring out not only what will be worthwhile, but also how to enable our young generation to thrive. Finally, I am also excited to learn new mathematical ideas, even if they did not arrive in the way I think they should have.

For these reasons I am optimistic that the dust will settle (hopefully sooner rather than later), and that we will be able to brush it off and come out on the other side, and, most importantly, that mathematics as an endeavour, as a profession and as a community will survive this, and might even come out stronger and more important in this ever more complex world.

Anirban Basak

‘Attention is all’ they care about.

Thomas Nikolaus

There are questions to ask, things to discuss about the how and why, and criticisms to make. At the same time, I hope we can also recognize what an extraordinary achievement and breakthrough these results are. They are the collective achievement of generations of mathematicians: the questions they asked, the theories they developed and the ideas they shared. I am looking forward to understanding the new mathematics and seeing where it leads us as a human mathematical community.

Ian Agol

Although I had heard rumors about these results, the range of proofs are stunning, solving problems in geometric group theory and topology that I might not have expected to be resolved in my lifetime (assuming they get checked to be correct). I would be especially happy if all of the results are eventually formalized, since it would require formalization of some of my work and the literature it relies on.

I (and others) have observed that there is some analogy to the proof of the Poincaré/Geometrization conjecture by Perelman. He posted his preprints after a 7 year hiatus from the mathematics community, and gave a couple of talks, but otherwise made no effort to disseminate the results. For most mathematicians, the preprints were very hard to read, and it took several groups many years to fill in the details (although the experts gave all of the credit of ideas to Perelman in the end). These results might have a similar assimilation into the mathematics community.

I feel like the French mathematics community may be better equipped to deal with these developments, since they have a tradition of valuing mathematical exposition, eg the Bourbaki seminar (and its publication Astérisque), and the journal l’Enseignement Mathématique. Maybe this is a model for how the wider mathematical community can behave.

After many big new developments in mathematics, once the community has understood the result, there are usually people that make clever new applications of the results, or build on them in highly non-trivial ways to make progress on related problems. For example, I spent much time learning about Perelman’s work, and from this was able to make some applications beyond the Geometrization conjecture applying his techniques. I expect this to happen with this OpenAI drop – mathematicians are clever, and will find simplifications and applications of these results (I am already hearing about this happening). For young people, this may be a good opportunity to learn some cutting edge mathematics.

Claire Burrin

Tonight, I am reading through the 100+ reactions posted so far with some level of sideration. Yes, there will be a tremendous number of new connections, new ideas, new questions to come out of the October 6 deflagration. But we know that this only a first batch: more are to come. OpenAI (and competitors) are using our most famous open problems to train their frontier models (with little regard for the communities and careers that are being bruised along the way): this is not over.

Serre, who always has a knack for being so efficiently sparse with his words, told the NYT: “Mathematicians take pleasure in doing maths in two different ways: learning and finding new things. Hence a conflict: A.I. increases the first pleasure and lowers the second one. The problem is it may lower the pleasure too much; that is especially serious for young mathematicians.” At the pace at which we are about to operate (and setting aside the crucial question of access to the top models, as of now priced $500/month), how can we preserve the space and the time needed for intellectual creativity?

There are more down to earth concerns. Our budgets, our grants depend on the goodwill of non-mathematicians. How will we answer the funding-threatening questions « Hasn’t AI solved math already? », « If AI is teaching you math, wouldn’t it be a better teacher to your students as well? » For years, we used the prominence of long-standing open problems as a value system to present to the general public, to motivate projects in grant proposals, to motivate the hiring of new talents. A new story has to be written. As for teaching, I hope that the COVID years have left us with enough data to make the point that human, front-facing communication is and always will be a crucial component of the transmission of knowledge. Nonetheless, the question will come up, again and again.

Some colleagues refuse to use AI altogether, others generate proofs daily. We absolutely cannot let this lead us to a polarisation of pros and antis that divide the community. More than ever, we need to find common ground on which to stand united. I would love to see every math department host an internal forum where everyone involved can openly and respectfully discuss their reactions and the paths they see forward. Even if they seem obvious, certain guidelines need to be spelled out and collectively agreed on, such as: you should have your collaborators’ explicit consent to use LLMs on a joint project. The Leiden declaration is, in my opinion, a great starting point for these discussions.

And most urgently, we need to proactively support younger researchers. A PhD student or a postdoc that wants to become a professor one day will be grappling with an existential question: are the emotional, personal, professional, and financial sacrifices required to stay in academia worth it in the midst of a paradigm shift? What if doing math in five years looks nothing like doing math five years ago?

There is a real urgency to coordinate on how best to protect our community and our junior researchers. We have so many questions to work on, that ironically math is the last thing on my mind right now.

Marco Moraschini

I believe that announcements like these, about “big” problems (or rather, about those open problems that turned out to be within reach), are shining a light on what really matters in doing mathematics. The ultimate goal is not just to prove or disprove a conjecture, but to understand how it is proved and how the key idea first came to mind. This is one reason why mathematicians often spend more time carefully writing up their results than finding them.

Yet I am old enough to know that, even when papers are well written, reading every arXiv paper on our favorite subjects has long been impossible. Fortunately, there have always been seminars, conferences and other community events: places where we can share ideas, intuitions and all the things that matter most, which no striking title like “A counterexample to a famous conjecture” can ever convey.

So I have started to wonder how we, as a community, will absorb all this, especially if such announcements become frequent, each bringing many interesting results presented only in sketch form. Who should take on the task of studying them in depth and explaining them to the research community in each field? And should it be up to us to organize conferences to learn about results that the AI labs have not yet presented in detail, although they have announced their intention to do so?

In the end, the most likely outcome may simply be that mathematicians put even more energy into creating new and beautiful mathematics (new definitions, new problems, new theories) and shift the focus away from answering questions that have already been posed. This would also be an interesting challenge for AI labs: if they are willing and able to start new mathematical theories, they will also need to invest time and resources in sharing them with the community, perhaps even more than they do today. I believe we would all benefit from that.

Alain Valette

I was one of the many people devastated by the OpenAI announcement—to me, it felt like a real stab in the back. Now, I feel like I’m standing among ruins ; questions that guided 46 years of my career, feel suddenly wiped out. My biggest worry right now is for younger mathematicians. How are we supposed to convince undergrads to pursue a math PhD? How do we tell postdocs that staying in academia is still worth it? How do we convince funding agencies to keep supporting math research?

Right now I’m at the IHP in Paris, where people are organizing reading seminars just to try to make sense of those AI breakthroughs. I hesitantly agreed to take part in an online seminar on constructing non-sofic groups, but I keep coming back to one question I can’t answer: most of us are publicly funded, so why should we be providing free expert feedback to a secretive company that ignores our community’s basic rules? A song by The Doors is running through my head:

« Strange days have found us
Strange days have tracked us down
They’re going to destroy
Our casual joys
We shall go on playing
Or find a new town »

Kimball Martin

Of course there was initial shock and dismay, both of the “why would they do this” and “how bad will this be for our community” variety, and marvel at progress on a couple very hard problems in my field. Then the realism after they somewhat mutedly withdrew some papers and fixed errors in others. I do not know how much of the progress came from creative novelty versus simply technical prowess, or how much they suppressed credit of ideas/contributions of mathematicians before them.

Certainly I have no interest in reading these papers myself, and think they should be expected to give good expositions — talks by humans, even! — rather than expect us to read slop if they want credit for advancing human understanding. A Lean verification does not show the written proof is correct, and is definitely no indication that the manuscript or proof is good.

Lastly, let me say that I am grateful for this site in current times.

Annette Karrer

Here is what I am feeling these days:

First, grief about what we might have lost: the research that humans would have done on their detours proving the problems by hand that AI solved “randomly”. Grief that we might have lost the beauty to do math slowly, step by step, only with chalk and our brain. And most importantly grief that we might have lost independence from Tech companies in math, and science as a whole.

Second, I feel compassion for all who are affected most: the mathematicians whose projects are lost, the mathematicians who are applying for jobs, and phd students who face pure insecurity.

Third, I feel fear and despair we will not manage to help and motivate each other to keep doing math. And that we will not support PhD students enough with that.

Fourth, I am feeling anger about the amount of power we let a single Tech company have over us.

Finally, I am feeling HOPE, that we as math community trust our values what matters in math, that we will fight for its survival, and that we will create a new set of roles organizing math academia that is better than the system we had to deal with so far.

1. I could of course ask AI to write this as a haiku, but not today. ↩︎
2. See also here: https://arxiv.org/abs/2509.15998 ↩︎