I am at COLM'26 this week in San Francisco with my students as well as folks from @Rabdos_AI . We have three cool papers on timely and exciting topics; drop by if you are attending! > MathDuels: A Self-Play Benchmark That Grows arxiv.org/abs/2604.21916 Wednesday 11-1, Grand Ballroom #121 Leaderboard: mathduels.ai/ > Detecting Safety Violations Across Many Agent Traces arxiv.org/abs/2604.11806 Thursday 11-1, Grand Ballroom #129 > Do We Need Frontier Models to Verify Mathematical Proofs? arxiv.org/abs/2604.02450 Thursday 4:30-6:30, Imperial Ballroom #21

This is the first semester in my teaching career that I have been able to use LLMs to improve the learning experience. One way I’ve been doing this is simply feeding my static slides from each lecture into an agentic framework to create live demos in class. For my software analysis class (cis547.github.io/), I used Claude Code to demo different program correctness techniques, from random testing using AFL++ to theorem proving using Dafny. In the past, I spent hours just debugging the setup of these tools. Now Claude does it autonomously for me in *under 5 minutes*, giving me a standalone Docker container with a web UI. Check out this demo of an automated debugging technique called Delta Debugging that I covered this week: static.us.edusercontent.com/… As a bonus, you yourself learn subtle new things about the technical content. For me this happened with a concurrency fuzzing technique called PCT, where Claude showed how P(bug per run) is higher but rate of bugs per second is lower if you put threads to “sleep” versus simply preempt them. It is an even bigger privilege to be teaching in these times. LLMs can now do almost anything so it all comes down to instructor taste and judgment.

Huge congratulations to Adam, my recently graduated PhD advisee who worked on this newly minted model -- on the topic of safety which is becoming a big deal!

What do surjective / injective functions have to do with optimizing and verifying ML kernels? Read on! In math/logic foundations of CS, we learn early on about “surjective” and “injective” functions. Thereafter, most of us have only a hazy recollection of these words, let alone why we learnt them or if we will ever apply them. I was certainly in this camp. Until recently, when I worked on proving equivalence of tensor programs with my amazing PhD student Paul Biberstein (paulbiberstein.me/) and colleague Joe Devietti: arxiv.org/abs/2609.19611. Suppose you want to check if tensor programs P and Q are equivalent. This is decidable but intractable. Since 90% of time is spent in 10% of code, it is often the case that we can express P and Q as: P = F o R o G Q = F o S o G for programs R and S such that checking R == S becomes tractable. At first glance, it seems that one can always strip F and G. But this turns out to be true only if F is surjective and G is injective! Here’s why: If R == S, then it is easy to see why P == Q. But if R != S, there is an input X such that R(X) = Y and S(X) = Z and Y != Z. Lifting this counterexample to P and Q requires F to be surjective and G to be injective. F is surjective means for all B: exists A: F(A) = B. So one can plug in B = X from the above counterexample and apply the inverse of F to it. G is injective means X1 != X2 => G(X1) != G(X2). So one can plug in X1 = Y and X2 = Z in the above counterexample. Putting these two together, we see that there exists an input A such that P(A) != Q(A). In other words, one can “lift” the counterexample of R == S to a counterexample of P == Q provided F is surjective and G is injective. When I first memorized things like “injective == one-to-one” and “surjective == onto”, I wish I had seen a practical example. In the LLM era, we have an opportunity to tailor explaining abstract concepts to individuals’ methods of learning. My research group is building the formal machinery for optimizing and verifying ML kernels, and this paper is our first on this topic. It only scratches the surface of classic symbolic methods and math concepts one can bring to bear in this domain. A lot more to come in future months!

Congrats to Anthropic on shipping Claude Opus 5.5! Here's an especially noteworthy article by Boris Cherny on using this model to formally verify the Claude Agent SDK using Lean. Rabdos is an industry leader in creating high-quality Lean data; check out our benchmark at rabdos.ai/math-index/lean

📣 We're growing our team at Rabdos AI! We're looking for a 𝗥𝗲𝘀𝗲𝗮𝗿𝗰𝗵 𝗘𝗻𝗴𝗶𝗻𝗲𝗲𝗿 to lead our work on mathematical formalization in Lean. You'd turn informal math into formal proofs and build autoformalization pipelines alongside frontier AI models, working with some of the sharpest mathematicians and computer scientists I know. If you love Lean, rigorous math, and hard open problems, I'd love to hear from you. Please share with anyone who comes to mind! 👇

Are endpoint protection systems like CrowdStrike prepared for large-scale agentic workloads? My research group has been running 100's of codex agents in parallel on a machine this summer. The agents are not coordinating (or at least aren't supposed to): each works on an independent research task in a STEM domain such as math, SWE, materials, biochem, ... In a matter of days, they have started tripping conventional security software in ways that have alarmed IT folks and are very hard to debug. The only solution seems to be to isolate the machine from the network, but that cuts off the agents from even basic web search. This and recent incidents of agents running amok at frontier AI labs present a massive opportunity for research and startups in AI x security. How do we specify / enforce security policies for agents? How do we triage alarms? Seems especially difficult since good vs. bad behavior is superficially indistinguishable : agents exploring how to use a low-level systems tool could be misconstrued as trying to breach security; a sequence of such seemingly random tool calls would pose an even greater challenge. Feels like the early days of the wild west, and a harbinger of things to come ...

Every year I take an international trip, I measure in-flight internet connectivity as an indicator of global technological progress. 2026 represents a *huge* step up in this measure, entirely single-handedly due to @Starlink. They have essentially solved this problem from 0 to 1.