https://www.daniellitt.com/Writing on mathematics, research, and adjacent subjects.https://www.daniellitt.com/blog/2026/8/11/the-end-of-mathematicshttps://www.daniellitt.com/blog/2026/8/11/the-end-of-mathematicsTue, 11 Aug 2026 22:49:34 GMTI'm currently returning to Toronto from a summit on the future of mathematics, at OpenAI. Sebastian Bubeck asked me to talk a bit about the future we'd all like to avoid, where humans are mathematically disempowered. Jacob Tsimerman advi…https://www.daniellitt.com/blog/2026/5/12/problemsilikecomhttps://www.daniellitt.com/blog/2026/5/12/problemsilikecomTue, 12 May 2026 15:42:14 GMTAnnouncing a new project: problemsilike.com , a website collecting open problems that I, personally, like, with comments on their context, difficulty, and interest. The goal is to track progress on mathematical questions that I think are…https://www.daniellitt.com/blog/2026/2/20/mathematics-in-the-library-of-babelhttps://www.daniellitt.com/blog/2026/2/20/mathematics-in-the-library-of-babelSat, 21 Feb 2026 18:29:10 GMTMathematics isn't only about saying true things. It's about asking the right questions, being confused, stumbling about, getting distracted, being wrong, recognizing when you're wrong, being stuck. Mostly being stuck. It's about clinging…https://www.daniellitt.com/blog/2025/7/17/arxiv-in-troublehttps://www.daniellitt.com/blog/2025/7/17/arxiv-in-troubleThu, 17 Jul 2025 21:21:04 GMTSince June 15, 2025 there have been 8 papers posted to arXiv with the phrase “Hodge conjecture” in the title or abstract. By my count 6 of them (75%!) are LLM-generated nonsense, replete with hallucinated references; as you might imagine…https://www.daniellitt.com/blog/2024/5/30/a-short-story-about-prompt-engineeringhttps://www.daniellitt.com/blog/2024/5/30/a-short-story-about-prompt-engineeringThu, 30 May 2024 16:04:20 GMTThe idea for this (very short, and somewhat silly) story popped into my head a couple of weeks ago. Please enjoy. Toast , or, The Golems “It’s just not so simple as that,” the toaster replied. “You say you want your toast ‘medium brown,’…https://www.daniellitt.com/blog/2022/9/20/an-nserc-proposalhttps://www.daniellitt.com/blog/2022/9/20/an-nserc-proposalTue, 20 Sep 2022 17:17:28 GMTI try to make a habit of posting my grant proposals here after the application period has passed, both because I hope people might find them to be useful models, and because doing so is a good opportunity for a brief postmortem. If you’d…https://www.daniellitt.com/blog/2022/9/19/tiling-puzzle-solutionhttps://www.daniellitt.com/blog/2022/9/19/tiling-puzzle-solutionMon, 19 Sep 2022 16:44:17 GMTMy last post was a little tiling puzzle: you can read it here . In this post I want to quickly give the solution. Represent a red tile by and a blue tile by ; and think of the square in coordinate as the monomialhttps://www.daniellitt.com/blog/2022/8/5/a-tiling-puzzlehttps://www.daniellitt.com/blog/2022/8/5/a-tiling-puzzleFri, 05 Aug 2022 21:49:49 GMTHere are four magic triominoes: Each is made out of three squares, two red and one blue or two blue and one red, alternating in color. These squares have the…https://www.daniellitt.com/blog/2022/2/1/tensor-powers-of-faithful-representationshttps://www.daniellitt.com/blog/2022/2/1/tensor-powers-of-faithful-representationsThu, 03 Feb 2022 19:24:47 GMTLet be a finite group and a faithful representation, with a finite-dimensional complex vector space. The following is well-known: Theorem 1. Let be any finite-dimensional irreducible represen…https://www.daniellitt.com/blog/2020/12/26/the-geometry-of-the-sylow-theoremshttps://www.daniellitt.com/blog/2020/12/26/the-geometry-of-the-sylow-theoremsFri, 01 Jan 2021 17:51:37 GMTToday I want to explain some algebro-geometric manifestations of the Sylow theorems. This blog post is an expansion of a Twitter Thread I wrote a few months ago, which you can find unrolled here . The Sylow Theorems and some examples Let…https://www.daniellitt.com/blog/2020/6/10/some-low-hanging-fruithttps://www.daniellitt.com/blog/2020/6/10/some-low-hanging-fruitWed, 10 Jun 2020 15:42:08 GMTToday is the #ShutdownSTEM strike , a day when non-black members of the academy strike in solidarity with the black lives matter movement, and take action to work to improve STEM academia. To quote their website: I wanted to briefly shar…https://www.daniellitt.com/blog/2020/5/2/scissors-integrationhttps://www.daniellitt.com/blog/2020/5/2/scissors-integrationSat, 02 May 2020 23:53:20 GMTOn Twitter Sarah Griffith made the following wonderful observation: Maybe this doesn't seem strange to anyone else, but I just realized I find it very strange that if you integrate a polynomial over Q between rational bounds you always g…https://www.daniellitt.com/blog/2020/4/28/grant-materialshttps://www.daniellitt.com/blog/2020/4/28/grant-materialsTue, 28 Apr 2020 19:40:11 GMTI was recently funded by the NSF Algebra and Number Theory grant DMS-2001196. Since other people may find it useful, I figured I would post my proposal here . I don’t claim at all that this is a model proposal, and there are a lot of cha…https://www.daniellitt.com/blog/2020/4/23/department-tea-and-revised-office-hours-tomorrowhttps://www.daniellitt.com/blog/2020/4/23/department-tea-and-revised-office-hours-tomorrowThu, 23 Apr 2020 16:02:57 GMTA historical April 2020 announcement about virtual UGA department teas and office hours.https://www.daniellitt.com/blog/2020/4/20/wagon-lessons-learnedhttps://www.daniellitt.com/blog/2020/4/20/wagon-lessons-learnedWed, 22 Apr 2020 16:00:31 GMTThis past weekend was the first-ever WAGON conference (Western Algebraic Geometry ONline), which was meant as the online iteration of the Western Algebraic Geometry Symposium (WAGS). Here “Western” refers to the western United States (wh…https://www.daniellitt.com/blog/2020/4/15/wagonhttps://www.daniellitt.com/blog/2020/4/15/wagonWed, 15 Apr 2020 17:41:18 GMTFor those of you who haven’t yet, you should register for WAGON ! Poster below — it’s going to be great.https://www.daniellitt.com/blog/2020/3/26/reflections-on-agonize-and-online-conferenceshttps://www.daniellitt.com/blog/2020/3/26/reflections-on-agonize-and-online-conferencesThu, 26 Mar 2020 18:44:18 GMTVideos are now up for AGONIZE. I’ve committed to organizing another one of these — stay tuned for more info. Since this was one of the first math conferences of our new era, I wanted to write a brief reflection on what worked and what (i…https://www.daniellitt.com/blog/2020/3/19/office-hourshttps://www.daniellitt.com/blog/2020/3/19/office-hoursThu, 19 Mar 2020 23:34:53 GMTA March 2020 announcement of temporary virtual office hours for the UGA arithmetic geometry community.https://www.daniellitt.com/blog/2020/3/19/agonizehttps://www.daniellitt.com/blog/2020/3/19/agonizeThu, 19 Mar 2020 18:07:31 GMTThis post announces the first ever AGONIZE conference ( A rithmetic G eometry is ON line I n Z oom, E veryone)! The conference will take place Wednesday, March 25, from 12pm-5pm EST. For more info and to register, see the conference webs…https://www.daniellitt.com/blog/2019/7/30/geometricity-and-galois-actions-on-fundamental-groupshttps://www.daniellitt.com/blog/2019/7/30/geometricity-and-galois-actions-on-fundamental-groupsTue, 30 Jul 2019 18:23:03 GMTA couple months ago I went to a fantastic conference on Arithmetic Topology at PIMS . The video from my talk is now up: enjoy ! I spend the first few minutes of the talk joking around about this cartoon:https://www.daniellitt.com/blog/2019/7/15/im-a-numberphilehttps://www.daniellitt.com/blog/2019/7/15/im-a-numberphileTue, 16 Jul 2019 16:02:00 GMTThe popular YouTube channel Numberphile has just released a video featuring yours truly — filming this with Brady Haran was a real pleasure! A couple of quick mathematical comments: in the video, I compute the Dehn invariant of the cube…https://www.daniellitt.com/blog/2019/2/23/arithmetic-and-representations-of-fundamental-groupshttps://www.daniellitt.com/blog/2019/2/23/arithmetic-and-representations-of-fundamental-groupsSat, 23 Feb 2019 21:12:47 GMTI gave a talk at the IPAM conference “ Braids, Resolvent Degree, and Hilbert’s 13th Problem ” this past Thursday. The conference was really inspiring — I enjoyed all the talks, but I’d in particular like to recommend Jesse Wolfson’s thre…https://www.daniellitt.com/blog/2019/2/12/holding-the-p-adics-in-the-palm-of-your-handhttps://www.daniellitt.com/blog/2019/2/12/holding-the-p-adics-in-the-palm-of-your-handTue, 12 Feb 2019 23:12:41 GMTA few years ago I made a picture of the 3-adic numbers in answer to a MathOverflow question (using the free 3d modeling and rendering software Blender , which I highly recommend despite its steep learning curve). Recently I discovered th…https://www.daniellitt.com/blog/2019/2/6/what-ive-been-thinking-about-recentlyhttps://www.daniellitt.com/blog/2019/2/6/what-ive-been-thinking-about-recentlyWed, 06 Feb 2019 17:00:55 GMTI wrote a MathOverflow answer just now which also happens to be a pretty good summary of some of the open questions I’ve been thinking about recently. Take a look: How does Frobenius act on the fundamental group of the projective line mi…https://www.daniellitt.com/blog/2018/10/30/more-tate-curves-more-problemshttps://www.daniellitt.com/blog/2018/10/30/more-tate-curves-more-problemsTue, 30 Oct 2018 16:03:55 GMTI’m giving an expository lecture on the Tate curve and non-archimedean geometry next Tuesday, November 6, as one of the preparatory lectures for Peter Scholze’s Chow Lectures in Leipzig . I thought the problem set for the lectures might…https://www.daniellitt.com/blog/2018/9/30/12-minutes-of-monodromyhttps://www.daniellitt.com/blog/2018/9/30/12-minutes-of-monodromySun, 30 Sep 2018 23:41:44 GMTHere’s a video of a (12 minute!) talk I gave on Friday about some of my recent work, mostly from this paper of mine . The talk is aimed at a general mathematical audience — please let me know if you have any questions! The video has rece…https://www.daniellitt.com/blog/2018/9/27/a-quick-comment-on-rhhttps://www.daniellitt.com/blog/2018/9/27/a-quick-comment-on-rhThu, 27 Sep 2018 04:19:15 GMTI recently commented for this article in Gizmodo on Atiyah’s supposed proof of the Riemann Hypothesis. In the article I assert there is an internal inconsistency (in fact there are several) in his paper. This post is meant to direct peop…https://www.daniellitt.com/blog/2018/7/14/the-boston-markin-conjecture-for-three-manifoldshttps://www.daniellitt.com/blog/2018/7/14/the-boston-markin-conjecture-for-three-manifoldsSat, 14 Jul 2018 20:18:44 GMTA couple of months ago I was in Oberwolfach, where Tomer Schlank told me about the following conjecture: Conjecture. ( Boston-Markin ) Let be a finite group. Then there exists a Galois extension of with Galois group…https://www.daniellitt.com/blog/2018/3/4/guest-post-ordinals-and-hydras-by-brian-lawrencehttps://www.daniellitt.com/blog/2018/3/4/guest-post-ordinals-and-hydras-by-brian-lawrenceSun, 04 Mar 2018 20:32:10 GMTA few days ago, the hydra game came up in discussion with Brian Lawrence; we'd both run into a somewhat mystifying analysis of the game via ordinals. Brian and I came up with a much more down-to-earth analysis (which experts will recogni…https://www.daniellitt.com/blog/2017/11/5/donate-to-mathoverflowhttps://www.daniellitt.com/blog/2017/11/5/donate-to-mathoverflowSun, 05 Nov 2017 21:28:22 GMTI just noticed that MathOverflow is soliciting donations at this link , or click here to donate directly. If you haven't donated yet, please consider it -- the site is an important place for discussion, and I know that I benefitted from…https://www.daniellitt.com/blog/2017/11/2/a-non-prorepresentable-deformation-functorhttps://www.daniellitt.com/blog/2017/11/2/a-non-prorepresentable-deformation-functorFri, 03 Nov 2017 01:50:51 GMTI've been teaching a topics course on deformation theory this semester; so far we've covered quite a few interesting things -- the Tian-Todorov theorem and Green-Lazarsfeld's generic vanishing theorem, for example. I just wanted to use t…https://www.daniellitt.com/blog/2017/9/3/hire-mehttps://www.daniellitt.com/blog/2017/9/3/hire-meTue, 05 Sep 2017 15:00:00 GMTI'm currently on the market for tenure-track jobs in mathematics. The application process is somewhat formal -- I think it's hard, in a research statement, to get across my passion and excitement for mathematics (though hopefully this co…https://www.daniellitt.com/blog/2017/7/18/the-parity-of-zero-the-primality-of-two-and-other-mysterieshttps://www.daniellitt.com/blog/2017/7/18/the-parity-of-zero-the-primality-of-two-and-other-mysteriesWed, 19 Jul 2017 16:01:00 GMTFrom time to time, I try to speak or write about mathematics for general (non-mathematical) audiences. If you've done this, you know it's pretty hard -- in large part because it's hard to know what people know, despite my best attempts t…https://www.daniellitt.com/blog/2017/5/11/villani-for-parliamenthttps://www.daniellitt.com/blog/2017/5/11/villani-for-parliamentThu, 11 May 2017 17:21:03 GMTThe (relatively new) French party En Marche! (from which the winning presidential candidate, Macron, hails) has nominated spider-brooch wearing mathematician Cedric Villani as a parliamentary candidate! For the full list of candidates, s…https://www.daniellitt.com/blog/2017/5/9/uniformization-over-finite-fieldshttps://www.daniellitt.com/blog/2017/5/9/uniformization-over-finite-fieldsTue, 09 May 2017 20:04:19 GMTI just asked this question on MathOverflow. It's basically idle curiosity, but I've now been idly curious about this for several years, so I figured I might as well ask it publicly. Please let me know if you have any thoughts! In short…https://www.daniellitt.com/blog/2017/4/26/constructive-criticismhttps://www.daniellitt.com/blog/2017/4/26/constructive-criticismWed, 26 Apr 2017 18:16:27 GMTHere is a classical example of a non-constructive proof. Thm 1. There exist irrational numbers such that is rational. Proof. If is rational, then we may take . Otherwise, we may tak…https://www.daniellitt.com/blog/2017/3/18/the-lost-worldhttps://www.daniellitt.com/blog/2017/3/18/the-lost-worldMon, 20 Mar 2017 14:00:00 GMTI. Opening the Airlock While I was in Arizona last week (running a study group for this year's Arizona Winter School on perfectoid spaces), I took the opportunity to visit another world, 50 minutes outside of Tucson. Dreamt up by a group…https://www.daniellitt.com/blog/2017/2/22/sawin-on-severis-conjecturehttps://www.daniellitt.com/blog/2017/2/22/sawin-on-severis-conjectureThu, 23 Feb 2017 14:54:00 GMTOne of my favorite questions is: for which is the moduli space of -pointed genus curves unirational/uniruled? Will Sawin has just posted a beautiful paper on the ArXiv answering…https://www.daniellitt.com/blog/2017/2/13/mumford-at-the-methttps://www.daniellitt.com/blog/2017/2/13/mumford-at-the-metMon, 13 Feb 2017 14:39:28 GMTI was at the Met yesterday, where by chance I ran across the Concinnitas Project . (Pictures below the fold.)https://www.daniellitt.com/blog/2017/1/30/cs-lewis-on-commutative-algebrahttps://www.daniellitt.com/blog/2017/1/30/cs-lewis-on-commutative-algebraMon, 30 Jan 2017 14:54:13 GMTRead the whole essay here . Sorry for the pause in blogging -- I should start up again soon.https://www.daniellitt.com/blog/2016/12/13/graph-theory-and-mathfraksl2https://www.daniellitt.com/blog/2016/12/13/graph-theory-and-mathfraksl2Thu, 15 Dec 2016 14:18:00 GMTHow many unlabeled graphs are there with vertices and edges? (More below the fold.)https://www.daniellitt.com/blog/2016/11/28/jie-liu-on-projective-spacehttps://www.daniellitt.com/blog/2016/11/28/jie-liu-on-projective-spaceMon, 28 Nov 2016 21:01:59 GMTExciting news! Jie Liu has proven a conjecture of mine -- thereby resolving an old conjecture of Sommese. I'll briefly explain his result in this post.https://www.daniellitt.com/blog/2016/11/14/whats-wrong-with-the-worldhttps://www.daniellitt.com/blog/2016/11/14/whats-wrong-with-the-worldMon, 14 Nov 2016 20:18:04 GMTIn 1910, G.K. Chesterton (one of my favorite authors -- I highly recommend The Man Who Was Thursday and the Father Brown mysteries) diagnosed all the world's problems. You can read his famous diagnosis, What's Wrong With the World , at P…https://www.daniellitt.com/blog/2016/10/19/biosphereshttps://www.daniellitt.com/blog/2016/10/19/biospheresWed, 19 Oct 2016 13:09:00 GMTThis is Biosphere 1: And this is Biosphere 2:https://www.daniellitt.com/blog/2016/10/11/we-already-live-in-rationaliahttps://www.daniellitt.com/blog/2016/10/11/we-already-live-in-rationaliaWed, 12 Oct 2016 13:20:00 GMTIn June 2016, Neil deGrasse Tyson proposed (not entirely seriously, as this series of tweets should make obvious) the creation of a new country: Rationalia, governed by the dictum " All policy shall be based on the weight of evidence ."…https://www.daniellitt.com/blog/2016/10/6/a-minimal-proof-of-the-fundamental-theorem-of-algebrahttps://www.daniellitt.com/blog/2016/10/6/a-minimal-proof-of-the-fundamental-theorem-of-algebraFri, 07 Oct 2016 14:27:00 GMTWhen I was in graduate school, I came up with what I think is a nice proof of the fundamental theorem of algebra. At the time, I wrote it up here somewhat formally; I thought it might make a nice blog post, since the formal write-up obsc…https://www.daniellitt.com/blog/2016/10/3/-lemmashttps://www.daniellitt.com/blog/2016/10/3/-lemmasTue, 04 Oct 2016 03:06:37 GMTA dilemma is a difficult choice between two alternatives. I recently learned that there is a word for a choice between three alternatives: trilemma. But what if I have a hard choice between four options? I was curious, so I did some goog…https://www.daniellitt.com/blog/2016/9/28/are-shimura-varieties-kpi-1shttps://www.daniellitt.com/blog/2016/9/28/are-shimura-varieties-kpi-1sWed, 28 Sep 2016 19:07:03 GMTLet be the moduli space of principally polarized Abelian varieties of dimension . The complex-analytic space (stack) associated to is a ; that is, its only non-vanishing homotopy group…https://www.daniellitt.com/blog/2016/9/25/weapons-of-math-destructionhttps://www.daniellitt.com/blog/2016/9/25/weapons-of-math-destructionMon, 26 Sep 2016 03:15:57 GMTI've just finished reading Cathy O'Neil's book Weapons of Math Destruction , which I highly recommend. (One notable feature of the book is that the skull and cross-bones on the cover is the second known example of mathematical piracy .)…https://www.daniellitt.com/blog/2016/9/19/the-typographical-equivalent-of-a-knife-fighthttps://www.daniellitt.com/blog/2016/9/19/the-typographical-equivalent-of-a-knife-fightMon, 19 Sep 2016 15:21:47 GMTI've been thinking recently about typefaces -- the four to eight readers of this blog may have noticed that the font used in the body text of these posts has changed. I've also been thinking about best practices for mathematical typesett…https://www.daniellitt.com/blog/2016/9/1/imaginary-worldshttps://www.daniellitt.com/blog/2016/9/1/imaginary-worldsFri, 16 Sep 2016 16:59:37 GMTOne of my favorite books growing up was Dougal Dixon 's Man after man: an anthropology of the future , which imagines the development and speciation of humanity in the far future -- under the influence of both genetic engineering and apo…https://www.daniellitt.com/blog/2016/9/12/morita-theory-tannaka-duality-and-approximate-tannaka-dualityhttps://www.daniellitt.com/blog/2016/9/12/morita-theory-tannaka-duality-and-approximate-tannaka-dualityMon, 12 Sep 2016 17:12:40 GMTLet be a ring -- it is well-known that the category of (left) -modules does not determine . For example, the functor is an equivalenc…https://www.daniellitt.com/blog/2016/9/8/varieties-with-infinitely-generated-automorphism-grouphttps://www.daniellitt.com/blog/2016/9/8/varieties-with-infinitely-generated-automorphism-groupFri, 09 Sep 2016 02:47:11 GMTJohn Lesieutre has just sent me an exciting new preprint in which he constructs a smooth projective variety such that is discrete, and is not finitely generated. Whether or not such varieties exi…https://www.daniellitt.com/blog/2016/9/6/integral-househttps://www.daniellitt.com/blog/2016/9/6/integral-houseTue, 06 Sep 2016 19:01:30 GMTI'm teaching Calculus III at Columbia this semester, and am kind of amazed at the exploitative prices charged for Stewart's Calculus, 8th Edition, "Early Transcendentals" . (Not that anyone knows what "Early Transcendentals" are). On the…https://www.daniellitt.com/blog/2016/9/3/taaaghttps://www.daniellitt.com/blog/2016/9/3/taaagSat, 03 Sep 2016 18:55:07 GMTStill at UGA, at the very enjoyable but weirdly named conference TAAAG . The conference featured a very interesting talk by Padmavathi Srinivasan , as well as mini-courses by Michel Raibaut , Ben Williams , and Arnav Tripathy (whose webs…https://www.daniellitt.com/blog/2016/9/1/my-herohttps://www.daniellitt.com/blog/2016/9/1/my-heroFri, 02 Sep 2016 04:08:16 GMTThis is a picture of Alexander Grothendieck, taken near the end of his life: I took this picture of a page in a book that was just lying around in Oberwolfach…https://www.daniellitt.com/blog/2016/8/30/sl4mu2-and-a-mod-8-congruencehttps://www.daniellitt.com/blog/2016/8/30/sl4mu2-and-a-mod-8-congruenceThu, 01 Sep 2016 19:41:47 GMTThis is a continuation of a previous post . Recall that we wanted to prove the following claim: Claim. Let be a representation of such that no irreducible subrepresentation of descends to . Then i…https://www.daniellitt.com/blog/2016/8/31/more-tautological-classeshttps://www.daniellitt.com/blog/2016/8/31/more-tautological-classesThu, 01 Sep 2016 03:31:44 GMTStill at UGA -- I just saw a great talk by Jason van Zelm (a student of Nicola Pagani who apparently does not have a webpage), constructing non-tautological cycles on for ...https://www.daniellitt.com/blog/2016/8/29/krashen-the-partyhttps://www.daniellitt.com/blog/2016/8/29/krashen-the-partyTue, 30 Aug 2016 03:35:26 GMTI'm at UGA for the week, in between SWAG and TAAAG . Today Danny Krashen gave a great talk on this paper of Auel, First, and Williams. The paper is one of the latest in a long tradition of papers which construct counterexamples by making…https://www.daniellitt.com/blog/2016/8/28/families-of-curves-wantedhttps://www.daniellitt.com/blog/2016/8/28/families-of-curves-wantedMon, 29 Aug 2016 01:26:40 GMTAn interesting problem Let be a large positive integer. Recently I've been looking for a family of curves with the following properties: is flat and proper of relative dimension …https://www.daniellitt.com/blog/2016/8/28/swaghttps://www.daniellitt.com/blog/2016/8/28/swagSun, 28 Aug 2016 04:16:48 GMTI'm currently at SWAG , the awesomely named Summer Workshop in Algebraic Geometry at Georgia. Jonathan Wise gave a really amazing talk today about logarithmic geometry, which got me very excited about the possibility of using log geometr…https://www.daniellitt.com/blog/2016/8/25/starting-a-bloghttps://www.daniellitt.com/blog/2016/8/25/starting-a-blogThu, 25 Aug 2016 18:43:54 GMTI'm thinking of starting a blog. Let's see how this goes.