Nick James

The issue I see with the current path (seb recently gave a talk at Princeton talking about oai progress, also have spoken to anthropic and other smaller groups/cos doing this) is that a lot of problems being tackled are very mechanical/"next step" reasoning. Hence a lot of stuff is being done in algebra/representation theory, low level analysis stuff, and combinatorics. So you'll have some clever tricks, but there are no real macroscopic structures that are being considered. Especially in low dimensional topology/differential topology which I've played the most around with in my math trajectory, there are so many areas where things can go wrong since you are using so much overlapping machinery. Heegaard Floer homology is a classic example of this -- will AI be able to see a lot of the underlying functional analysis underneath the surface of geometric calculations? There are just many considerations one makes when you aren't in the "problems solving" mode that a lot of these models operate in. Just some reflections......

has anyone tried just asking 5.5 to think about what open problems it wants to solve for 20 hours? i honestly think it might be the year of the harness. ridiculous abstract multi-agent prompt loops just kind of work now…

really pisses me off how much i yelled about how this becoming a major issue at openai and nothing happened ai labs don’t use much water, we could’ve wrote one blog or thrown a green leaf info icon at the bottom of gpt just so avoidable

@oprydai This is completely false. There are amazing classics professors who are smarter than most math or physics or computer science researchers I have ever met. I think having intuition of a field that is unbelievably deep and being multifaceted is what matters.

you degree of intelligence is directly proportional to how good you are at higher abstraction maths. let that sink in.

@dwarkesh_sp @reinerpope This is awesome, I love this format! Please make more episodes like this in the future :)

Did a very different format with @reinerpope – a blackboard lecture where he walks through how frontier LLMs are trained and served. It's shocking how much you can deduce about what the labs are doing from a handful of equations, public API prices, and some chalk. It’s a bit technical, but I encourage you to hang in there - it’s really worth it. There are less than a handful of people who understand the full stack of AI, from chip design to model architecture, as well as Reiner. It was a real delight to learn from him. Recommend watching this one on YouTube so you can see the chalkboard. 0:00:00 – How batch size affects token cost and speed 0:31:59 – How MoE models are laid out across GPU racks 0:47:02 – How pipeline parallelism spreads model layers across racks 1:03:27 – Why Ilya said, “As we now know, pipelining is not wise.” 1:18:49 – Because of RL, models may be 100x over-trained beyond Chinchilla-optimal 1:32:52 – Deducing long context memory costs from API pricing 2:03:52 – Convergent evolution between neural nets and cryptography

Some reflections on imperfections, Picasso, and design. njames.xyz/blog-post.html…

@mathladyhazel The first math book I bought (in middle school!) was “Introduction to Number Theory” by George E. Andrews. I love Dover! Its proof of Fermat’s little theorem bugged me growing up, I didn’t like the proof. Burnside’s lemma and fields/rings made it click like magic later on :)

These Dover math book covers are so beautiful.

One thing I’ve noticed among college students is that startups sometimes aren’t taken seriously purely because of the vocabulary and register used in conversation. Simply swapping “startup” for “company” tends to flip a switch, so that the idea is treated with far more respect and treated seriously. The same holds for “engineer” or “entrepreneur” instead of “builder,” “founder,” or “hacker” (the last of which is especially seen as cringe).

Science is about to get absolutely nuked. Unless we get extremely strict about providing and opening up code and data and documenting lab experiments rigorously, a torrent of credible-looking but fraudulent papers is upon us.

To solve hard open math problems, we need AI models to train and self-improve indefinitely without more external data. Humans can self-improve, so AI should as well if it imitates humans. So we let AI also conjecture, prove, and also be self-guided with some tastes.

@ns123abc @elonmusk This is the introductory chapter of any general relativity textbook worth its salt. Many models are far past this being an achievement as well.

Grok 4.3 "write an academic paper on general relativity" It produced a 5-page LaTeX paper: >Einstein field equations >Schwarzschild metric >tensor notation and Christoffel symbols >starlight deflection diagram >Newtonian vs GR comparison table >8 properly formatted citations >author: "Grok" xAI team cooked fr

A cool website illustrating migration patterns, urbanization, poverty, and other metrics across the globe: clicme.liser.lu/dashboard/?lat…

@deedydas It’s also frustrating that this sort of result is being championed on X, when amazing papers are largely ignored online and only passed around by academics. A great paper shown by a friend today: arxiv.org/pdf/2604.08473 Positive mass theorem extended to arbitrary dimension…

@deedydas This is not what the paper claims (which already is slightly dubious). It claims all elementary functions can be expressed in this way by such an operator, but requires the constant 1 too. Many functions are not elementary.

This Polish theoretical physicist just proved you can recreate all math functions from JUST one operation. E(a, b) = e^a - ln(b) Every single operation: +, -, x, / , trig, log, as you can see below. Extremely mathematically elegant.

@KiaGhods @karpathy After writing an entire autograd engine in rust over the weekend: "It was surprisingly some of the most fun I've had in a long time." lmfao never change @KiaGhods

happy to share something small I made recently: micrograd-rust -- rewrote @karpathy's scalar autograd engine in Rust and published it as a Python package; same api, 1-2 OOMs faster, scales to 1M+ params where Python stack overflows. `pip install micrograd-rust` github.com/kiaghods/micro… had a lot of fun working on!

I was told I should make this a while ago, but I've also finished creating a rough version of a bookshelf for my personal website. njames.xyz/bookshelf.html Hopefully you find something good on it! I'm planning on changing out some entries every once and a while.

Writing more blogs about literature, since lots of people in tech have to read more genres outside of entrepreneurship self-help and research to be honest: njames.xyz/blog-post.html…

It’s important to note how much human interaction went into orchestrating this “ecosystem.” The mathematical community has to be very careful of overstating the consequences and importance of such results. Otherwise, it’s just bad science. It’s very unclear how such models can solve diverse sets of problems with vastly different toolsets and frameworks. I’d love to see how models perform on solving problems in low dimensional topology, arithmetic geometry, higher homotopy theory, etc. Let’s raise the bar beyond classical combinatorics!

Three weeks ago I shared that Claude had shocked Prof. Donald Knuth by finding an odd-m construction for his open Hamiltonian decomposition problem in about an hour of guided exploration. Prof. Knuth titled the paper Claude’s Cycles. The story didn't end there. The updated paper shows the story got much bigger. For the base case m=3, there are exactly 11,502 Hamiltonian cycles. Of those, 996 generalize to all odd-m, and Prof. Knuth shows there are exactly 760 valid “Claude-like” decompositions in that family. The even case, which Claude couldn’t finish, was then cracked by Dr. Ho Boon Suan using GPT-5.4 Pro to produce a 14-page proof for all even m≥8, with computational checks up to m=2000. Soon after, Dr. Keston Aquino-Michaels used GPT + Claude together to find simpler constructions for both odd and even m, by using the multi-agent workflow. Dr. Kim Morrison also formalized Knuth’s proof of Claude’s odd-case construction in Lean. So yes: the problem now appears fully resolved in the updated paper’s ecosystem of human + AI + proof assistant work! We went from one AI solving one problem to a full mathematical ecosystem (multiple AI systems, multiple humans, formal verification) running in parallel on a problem that stumped experts for weeks. We are living in very interesting times indeed. Paper (updated): www-cs-faculty.stanford.edu/~knuth/papers/…

There has always been a big discussion surrounding "agency" and whether people are "agentic." One thing I've been thinking about recently is how bravery fits into that picture. Stories going back thousands of years talk about bravery. Take Homer for example; in Book VI of the Iliad, Diomedes says to Glaucos: ἀτὰρ μὲν νῦν γε πολὺ προβέβηκας ἁπάντων σῷ θάρσει, ὅ τ᾽ ἐμὸν δολιχόσκιον ἔγχος ἔμεινας (Il VI 125-126). This roughly translates to, "but now you have exceeded all others in your courage, given that you faced my far-shadowing spear." Another translation of θάρσει (from θάρσος) is "boldness." I think a fun way of thinking about it is potentially "moxie." Point is, this is something people have thought about since literature was a thing. Why don't people talk about this term more often in business? Huge decisions are made every day. But it gets made so boring when words like "risk tolerance" are used instead. I think there is an overemphasis on productivity, shipping, and formulaic terminology. It's helpful, sure, but it's a lot less inspiring, isn't it?