Daniel Litt ✓
@littmath · Toronto, Ontario · joined 12 Aug 2010
Assistant professor (of mathematics) at the University of Toronto, visiting Harvard Fall 2026. Forever confused. Algebraic geometry, number theory, etc. He/him.
62 930Followers
932Following
32 770Posts total
519.4KViews on collected posts
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@littmath it's all an excuse to think about Alice and Bob. who are they, where do they run, why do they keep picking balls from urns as the world around them fades away?
570 views · 10 likes
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Apologies for the long self-indulgent post; hopefully it's a bit useful to see how, other than one-shotting things, the models can be used to do mathematics, and how at least some professionals are thinking about this.
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So there's a mathematical question I've been thinking about on and off for about 10 years. For the first 5 or so years I made very little progress, but learned a bit about the general area (while working on other things). In the last 5 years I've made some modest progress with
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SITUATION DETECTED: andrew wiles IPOs for a trillion dollars
29.8K views · 480 likes
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I don't know about the chess analogy or the views of people with money, but I do think there's pretty broad agreement that furthering human knowledge is an inherently good thing.
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The problem with is that maths has, at least for the last 100 years or so, justified its existence (i.e. funding) through the idea that furthering human knowledge is an inherently good thing. And I just don't think the people with money believe that anymore...
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This is right. People are (understandably) concerned about negative second-order effects of these solutions, especially to problems that have been central to a field for some time and guided past work.
But, don't you believe in mathematics? There will always be more to learn.
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"math is dead because a computer solved a problem"
Every serious mathematician knows that a solution to a famous problem immediately generates dozens or hundreds more problems ... that's how it's always worked https://t.co/meVr73pUTC
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@littmath Most liked post about non-abelian cohomology 🏅
Congrats on finishing!
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You can read the paper here: https://t.co/QFtNNK5Aql
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This project has been in the works for some time. One fun aspect of it is that this paper is the first one I've written in part after AI tools exceeded some invisible threshold of usefulness for math research tasks. I'll talk a bit about this some other time! https://t.co/DaABZOy
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This is really an excuse to develop non-abelian analogues of the structures on cohomology Katz exploits; for example, a non-abelian analogue of the Hodge index theorem. https://t.co/8P8t0tsDPx
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There is an associated (non-linear!) differential equation--see e.g. below--controlling this action. We prove an analogue of Katz's theorem for these equations: the arithmetic of this differential equation controls the action of \pi_1(S,s) on representations of \pi_1(X_s). https:
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This paper with Josh instead studies connections between topology and arithmetic mediated not by cohomology, but by what we (and others) refer to as "non-abelian cohomology" -- spaces parametrizing representations of the fundamental group of a variety.
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The players are as follows. Let f: X-->S be a smooth projective morphism of complex varieties, and fix s in S. The topological object we will study is the natural (outer) action of \pi_1(S,s) on \pi_1(X_s), and the induced action of \pi_1(S,s) on representations of \pi_1(X_s).
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These are the ODEs satisfied by so-called "period integrals": loosely speaking, functions obtained by integrating polynomially varying families of algebraic functions.
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Katz's proof relies on the structures on and compatibilities between different kinds of cohomology theories of algebraic varieties; for example, the Hodge index theorem, and various compatibilities between de Rham cohomology in positive characteristic and characteristic zero.
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In general the p-curvature conjecture is open, but it is known (due to work of Katz) for certain ODEs, called Picard-Fuchs equations or Gauss-Manin connections, which come from the cohomology of algebraic varieties. https://t.co/wyFADOnFhd
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The starting point for this paper is a different connection between arithmetic, geometry, and topology--the Grothendieck-Katz p-curvature conjecture. It predicts that the algebraicity of solutions to a linear ODE is controlled by the arithmetic of the ODE.
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One of the basic themes of 20th century mathematics is that the topology of the set of complex solutions to a system of polynomial equations is controlled by the arithmetic of the polynomials.
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For example, the Weil conjectures (proved by Dwork, Grothendieck, and Deligne) show that the Betti numbers of a (smooth projective) variety defined by polynomials with integer coefficients are controlled by the number of solutions to those polynomials over finite fields.
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This control is mediated in most cases by an invariant of a variety called its cohomology, which has both topological and arithmetic incarnations. Loosely speaking, a large part of the theory of motives is about compatibilities between these incarnations.
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The basic objects of study here are algebraic varieties--shapes defined as the set of solutions to a system of polynomial equations--and polynomial maps between them. https://t.co/Zf97ynVrdC
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New paper with Josh Lam, about which I'm really excited! I want to try to briefly explain what the point is in this thread. https://t.co/c98Inw9Mgu
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เทียบกับบัญชีขนาดเดียวกัน
8 โพสต์จาก 90 วันที่ผ่านมา เทียบกับช่วง 10K–100K ผู้ติดตาม แสดงในวงกว้าง แต่มีผู้ชมตอบสนองน้อย.
ยอดดูมัธยฐาน26 770บัญชีนี้895ค่ามัธยฐานของ 10K–100K
การเข้าถึง, %42.54%บัญชีนี้3.22%ค่ามัธยฐานของ 10K–100K
การมีส่วนร่วม, %1.41%บัญชีนี้1.98%ค่ามัธยฐานของ 10K–100K
| ตัวชี้วัด | บัญชีนี้ | ค่ามัธยฐานของ 10K–100K | อัตราส่วน |
|---|---|---|---|
| ยอดดูมัธยฐานต่อโพสต์ | 26 770 | 895 | 29.9× |
| การเข้าถึง (ยอดดู ÷ ผู้ติดตาม) | 42.54% | 3.22% | 13.2× |
| อัตราการมีส่วนร่วม | 1.41% | 1.98% | 0.71× |
บัญชีอื่นในช่วงนี้ → เปรียบเทียบกับบัญชีอื่น → ค่าอ้างอิงเหล่านี้คำนวณอย่างไร →
Growth & engagement
How the posts we collected actually performed: views and reaction rate post by post, what the audience did with them, and where the follower count goes.
Views per post
2.7K14 Jan
2.6K
2.5K
3.9K
3.6K
10.4K15 Jan
31.5K2 Sep
23.7K
40.4K3 Sep
20.8K
29.8K4 Sep
47.7K5 Sep
7.1K
570
Last 14 collected posts, oldest on the left. The scale is logarithmic: one post can outrun the rest a hundred times over.
Engagement rate per post
1.20%14 Jan
1.32%
1.77%
1.03%
1.35%
0.34%15 Jan
0.33%2 Sep
1.23%
0.24%3 Sep
1.26%
1.68%4 Sep
1.56%5 Sep
1.57%
1.93%
Reactions — likes, reposts, replies and quotes — divided by views. Median for 10K–100K accounts is 1.98%.
What the audience does
Likes80.3%3 119 in total
Reposts3.4%133 in total
Replies3.4%132 in total
Quotes0.4%16 in total
Bookmarks12.5%486 in total
Share of every reaction we collected for this account. Replies mean argument, reposts mean endorsement, bookmarks mean the post was worth keeping.
The follower curve appears once this account has two daily snapshots — we take one a day, and this one is on its first.
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