One trivial observation at a time | Everything Mathematics, Optimization, Machine Learning, and Artificial Intelligence

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TL;DR: This is a short summary of our paper Neural Concept Verifier: Scaling Prover-Verifier Games via Concept Encodings by Berkant Turan, Suhrab Asadulla, David Steinmann, Kristian Kersting, Wolfgang Stammer, and Sebastian Pokutta. The paper was accepted as a spotlight at ICML 2026 (top ~2.2% of submissions). In a nutshell, we move Prover-Verifier Games from pixel space to concept space: a prover selects a sparse set of human-readable concepts, a verifier must classify using only those concepts, and an adversarial prover tests whether the verifier can be fooled. The result is a scalable route to verifiable, nonlinear concept-based classification on datasets such as CLEVR-Hans, CIFAR-100, ImageNet-1k, and COCOLogic.
TL;DR: Seven years ago I argued that AI could enable a zeroth world: economies operating at a multiple of first-world productivity, the way the first world operates at a multiple of the third. The update: the technology arrived faster than I expected, the productivity gains are real but unevenly distributed, and the zeroth world is being built right now, in the US and in China. Europe is perfectly on track to watch from the sidelines, which is particularly bitter because, demographically, Europe needs it more than anyone else. The newest twist: frontier cyber-defense capability is now allocated by invitation and Europe is mostly not on the list.
TL;DR: This is a short summary of our position paper The Hidden Cost of Tokenization: Why (most) Non-English Speakers Pay More for Less by Jennifer Haase and Sebastian Pokutta. The basic point is simple: tokenization is not a neutral preprocessing step. It determines how much users pay, how much context they get, how much compute is burned, and potentially how well a model can reason in a language. The same semantic content can require 1.3x, 5x, or even more than 10x as many tokens depending on the language-tokenizer pairing.
TL;DR: AI-driven scientific discovery with systems such as AlphaTensor, AlphaEvolve, etc is en vogue and we are also heavily invested in that space. However, these LLM- and RL-driven approaches, while impressive often in themselves, are not always the right tool to get the job done. In this post I will talk about three recent projects from our group that take different angles on LLM- and RL-driven mathematical discovery systems. Two of them revisit classical problems made famous through these AI systems and show that classical structured search matches or beats them on these specific problem classes; with a tiny fraction of the compute. The third example goes the other way around: AI not as discovery engine, but an AI-accelerated subroutine sits inside a classical structured search and lets us settle a thirty-year-old conjecture on real algebraic plane curves of degree seven. This points to an interesting shift in the recent AI vs. classical discussion: it is not so much about whether or not to use AI, but where in the stack AI belongs.
TL;DR: This is a short summary of our paper Lower Bounds for Frank-Wolfe on Strongly Convex Sets by Jannis Halbey, Daniel Deza, Max Zimmer, Christophe Roux, Bartolomeo Stellato, and Sebastian Pokutta. We prove a matching $\Omega(1/\sqrt{\varepsilon})$ lower bound for Frank-Wolfe on strongly convex sets, showing that Garber and Hazan’s 2015 upper bound is tight. The construction of the lower bound deviates from the standard route quite a bit: instead of searching for worst-case initializations, we build them backward from the optimum.
TL;DR: This is a summary of our recent paper The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learning by Max Zimmer, Nico Pelleriti, Christophe Roux, and Sebastian Pokutta augmented with some personal perspectives and thoughts. The main point is not that AI can now “do research” in some vague science-fiction sense. The more useful point is that, with the right workflow, general CLI coding agents can already act like research associates: they can write proofs, formulate conjectures, implement ideas, run experiments, document failures, verify intermediate claims, and keep going for hours, while the researcher remains responsible for idea generation, creativity, direction, judgment, and final verification.
TL;DR: Newcomb’s paradox — should you take one box or two? — splits rational decision-makers almost evenly. There are four natural mathematical frameworks (causal inference, algorithmic self-reference, statistical counterfactuals, and online learning) that give different answers, and the disagreement reveals deep structural tensions in what it means to choose rationally.
TL;DR: How faithfully does an LLM’s text output reflect the semantic geometry encoded in its hidden states? Forced-choice behavioral probing recovers substantially more internal similarity structure than open-ended generation, and behavioral features improve prediction of unseen hidden-state similarities above lexical and cross-model baselines.
TL;DR: Rising classroom heterogeneity and workload make AI in schools inevitable; with FACET we explore how evidence-based, teacher-centered AI can support meaningful differentiation and AI literacy without replacing human judgment.
TL;DR: SCIP Optimization Suite 10.0 brings a numerically exact solving mode for rational MILPs, noticeable performance gains for MILP/MINLP, stronger presolving and symmetry handling, better heuristics and conflict analysis, IIS detection, and major updates to GCG, PaPILO, PySCIPOpt, and MIP-DD.
TL;DR: The 2025 Nobel Prize in Economics honors Joel Mokyr, Philippe Aghion, and Peter Howitt for explaining how innovation drives sustained growth. Mokyr identifies the historical preconditions that allow innovation to accumulate; Aghion and Howitt formalize how creative destruction underpins modern growth. Together, their work clarifies why growth cannot be taken for granted—and what kinds of poli