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Open problems, frontier research, and arXiv summaries. Updated daily.

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Quanta Magazine
Is AI Reasoning Right for the Wrong Reasons?
# Is AI Reasoning Right for the Wrong Reasons? Artificial intelligence systems have become remarkably good at solving complex problems, including adv...
3h ago
Aperiodical
Double Maths First Thing: Issue 63
I'd be happy to help summarize this article, but it looks like the content was cut off after "I've…" — I'm only seeing the introduction and not the ac...
Jul 30
Quanta Magazine
Physicists Solve a Big Quantum Mystery. Now, Old Results Don’t Add Up.
Here is a 3-paragraph summary of the article for mathopen.com: For 25 years, physicists have wrestled with a confusing puzzle in particle physics, an...
Jul 30
Aperiodical
\(-e^i\) to Watch: MathsWithKala
Here is a 3-paragraph summary for mathopen.com: MathsWithKala is an Instagram channel created by Kala, dedicated to sharing mathematics with audience...
Jul 28
Quanta Magazine
A New Way That a Cow’s Inner World Shapes Earth’s Atmosphere
# How a Tiny Organelle Inside a Cow's Gut Influences Climate Change Deep inside the digestive system of a cow lives a microscopic world that plays a ...
Jul 28
MAA
2025 MAA Impact Report
# 2025 MAA Impact Report Summary The Mathematical Association of America (MAA) has released its 2025 Impact Report, showcasing a remarkable year of g...
Jul 24

Does the Carlson-Simpson lemma for 1-variable words have strictly lower proof-theoretic strength than the full Carlson-Simpson theorem, and can it be characterized by a precise arithmetical conservation class below Pi-1-1?

The Carlson-Simpson theorem is a powerful Ramsey-type result about variable words over finite alphabets, and it is known to require substantial set-existence assumptions. Recent work establishes that a restricted 1-variable version satisfies Pi-0-4 conservation over RCA-0, placing it in a specific r...

Mathematical Logic / Reverse Mathematics
Open ProblemsView all →
Probability / Random Matrix Theory
Does the pair correlation function of the Sine-beta process converge to that of a Poisson process as beta approaches zero, and if so, at what precise rate?
The Sine-beta process is a canonical point process on the real line that interpolates between highly structured eigenvalue repulsion at large beta and complete ...
Jul 30
Mathematical Logic
Does there exist a complete and decidable axiomatization of the modal logic governing the forcing relation over all models of ZFC simultaneously, rather than over any fixed ground model?
The internal modal logic of forcing assigns modal operators to set-theoretic statements by interpreting possibility as truth in some forcing extension and neces...
Jul 29
Random Matrix Theory / Probability
Does the Marchenko-Pastur law hold for tensor powers of exchangeable vectors that are not unconditional, and if so under what moment or dependence conditions?
The Marchenko-Pastur law describes the limiting spectral distribution of sample covariance matrices formed from high-dimensional data. The paper under considera...
Jul 27
Probability Theory
For a general smooth stationary Gaussian process on the real line, can one derive the exact asymptotic joint distribution of the location of the minimum and the overshoot above a high threshold, conditioned on that minimum being unusually low?
Resolving this problem would provide a complete extreme value theory for smooth Gaussian processes analogous to the classical Fisher-Tippett-Gnedenko theorem fo...
Jul 26
Mathematical Logic
Does there exist a complete classification of all Post complete extensions of the basic conditional logic system CK, analogous to the classical Post completeness theorem for propositional logic?
A complete Post-style classification would provide a definitive map of the expressive boundaries of conditional logic, telling us exactly which collections of c...
Jul 26
arXiv TodayAll →
Logic
conservation of a Carlson-Simpson lemma for 1-variable words
The paper studies a combinatorial theorem about coloring words over a finite alphabet. Imagine you have an alphabet like...
arXiv:2607.28116
Logic
On the Definability of Strong Negation in Bilateral Logics
The paper investigates a specific question in philosophical logic: can a particular kind of negation, called "strong neg...
arXiv:2607.27252
Logic
Hadwiger's conjecture for hypergraphs
Hadwiger's conjecture is a famous unsolved problem in graph theory from 1943. It proposes a deep connection between two ...
arXiv:2607.27243
Probability
Small value probabilities of additive and derivative martingales in supercritical branching Brownian motions and super Brownian motions
Branching Brownian motion is a probabilistic model where particles move randomly through space while simultaneously spli...
arXiv:2607.27612
Probability
Universal topological statistics on triangulated singular spaces
Persistent homology is a tool from topology that tracks how geometric features like holes, loops, and voids appear and d...
arXiv:2607.27535
Probability
The stochastic Keller--Segel system in critical spaces
The Keller-Segel system is a set of equations originally developed to describe how biological cells, such as bacteria, m...
arXiv:2607.27472
Research BlogsAll →
Tanya Khovanova
Coin Parity
Peter Cameron
Entropy, partitions, groups and AI
Gil Kalai
Various News Items
Peter Cameron
AI for the working mathematician
Peter Cameron
I read the news today, oh boy