Recent items from Terence Tao's blog and Mathstodon, Quanta, the lab blogs and arXiv. They are not claims and are not graded. 0 of 77 name a statement the register holds.
A brief post to note two reports that just came out: The Advisory Group on Mathematics and Artificial Intelligence (AGMAI) has posted its general recommendations to AI companies with regards to generation of solutions to mathematical problems. A working group report “AI and TCS: The Next Six Months” arising from a two-day event at the […]
arXiv:2609.35836v1 Announce Type: new Abstract: For an $r$-uniform hypergraph $H$, let $\nu(H)$ be the maximum number of edges no two of which share $r-1$ vertices, and $\tau(H)$ the minimum number of $(r-1)$-sets such that every edge contains one of them. Aharoni and Zerbib conjectured that $\tau(H)\le\lceil\frac{r+1}{2}\rceil\,\nu(H)$, which for $r=3$ generalizes Tuza's conjecture on triangles.
arXiv:2609.35842v1 Announce Type: new Abstract: The representation number of a graph is the least positive integer $k$ for which its vertices can be arranged in a word, each occurring $k$ times, so that two distinct letters alternate precisely when the corresponding vertices are adjacent. We prove that every bipartite graph on $N\ge9$ vertices has representation number at most $\lceil N/4\rceil$.
arXiv:2609.35920v1 Announce Type: new Abstract: Fishburn's latent-subset conjecture, proposed in 1987 and revisited in 1988, asserts that for every dual intersecting family $\mathcal F\subseteq 2^{[n]}$, there exists $i\in[n]$ such that $|\mathcal F^L(i)|\geq|\mathcal F(i)|$. Here $\mathcal F^L$ is the family of the subsets of members of $\mathcal F$ that do not belong to $\mathcal F$. In this pap
arXiv:2609.36022v1 Announce Type: new Abstract: Let $d \geq 5$, $0 < \gamma < d - 2$, and $\Omega_N$ be the binomial random subset of $Q_N = [-N,N]^d \cap \mathbb Z^d$ with retention probability $p_N = N^{-\gamma}$. We prove that, with failure probability of optimal exponential order, every subset $B \subseteq \Omega_N$ of fixed positive relative density realizes, at each scale $p_N^{-2/(d - 2)} \
arXiv:2609.36186v1 Announce Type: new Abstract: For a graph $H$, $\langle H \rangle$ denotes the class of all subdivisions of $H$ and $tw(H)$ denotes the treewidth of $H$. In this paper, we prove the following. For $k\geq 1, R\geq 1$, let $G,H$ be two graphs such that strong isometric path complexities (Chakraborty et al. [\textsc{Disc. Math., 2026}]) of both $G$ and $\langle H \rangle$ are at mos
arXiv:2609.36220v1 Announce Type: new Abstract: The classical Hadwiger-Nelson problem asks for the chromatic number of the unit distance graph of the Euclidean plane. Over the years, this problem has been considered for a variety of other normed spaces, with higher-dimensional Euclidean space $\mathbb{R}^d$ being perhaps the most natural and well-studied. Alon, Buci\'c, and Sauermann proved that,
arXiv:2609.36242v1 Announce Type: new Abstract: We prove a dimensional strengthening of the colorful Helly theorem. Let $\mathcal{C}_1,\dots,\mathcal{C}_{d+1}$ be finite nonempty families of convex sets in $\mathbb{R}^d$. If $C_1\cap\cdots\cap C_{d+1}\neq\varnothing$ for every choice of $C_i\in\mathcal{C}_i$, then $\sum_{i=1}^{d+1}\dim(\bigcap\mathcal{C}_i)\geq0$, where $\dim\varnothing=-1$. We al
arXiv:2609.36389v1 Announce Type: new Abstract: Binomial coefficients for multidimensional arrays count occurrences of particular configurations. For the sake of presentation, the emphasis is put on column-binomial coefficients for two-dimensional finite arrays. In this article, we first show that these coefficients can be computed through some Magnus transform. To get structural and combinatorial
arXiv:2609.36415v1 Announce Type: new Abstract: We give a self-contained elementary proof of the known result that every set $A\subset\R$ of cardinality seven satisfies $|A+A|\le |A-A|$. The argument adapts Hegarty's method, using representation counts and the largest positive differences. An exact counting identity, two applications of the Cauchy--Schwarz inequality, and an analysis of a symmetri
arXiv:2609.36551v1 Announce Type: new Abstract: Jackson and Thomassen conjectured that every $2k$-strong digraph contains a spanning $k$-strong oriented subdigraph. We prove sharp degree conditions for the existence of such a subdigraph. For every fixed positive integer $k$ and all sufficiently large $n$, every $n$-vertex digraph $D$ with $\delta^0(D)\ge\lfloor(n+k-1)/2\rfloor$ admits a $k$-strong
arXiv:2609.36725v1 Announce Type: new Abstract: We prove two arithmetic nonexistence criteria for tight spherical $5$-designs in dimension $(2m+1)^2-2$ with $m$ even. Together, they recover the applicable criteria of Bannai-Munemasa-Venkov and Nebe-Venkov while covering parameters excluded by neither earlier result. Examples include $m=16,100$ under the first criterion and $m=96,168$ under the sec
arXiv:2609.36895v1 Announce Type: new Abstract: Let $G$ be a graph of order $n$ with adjacency matrix $A_G$, let $\mathbf e$ denote the all-one vector, and let $W_G=[\mathbf e,A_G\mathbf e,\ldots,A_G^{n-1}\mathbf e]$ be its walk matrix. We consider the case $\operatorname{rank}W_G=n-2$, the first corank for which distinct graphs can have the same walk matrix. We give a complete structural descript
arXiv:2609.37159v1 Announce Type: new Abstract: We prove the equivalence of Hindman's finite sums theorem with a natural extension of Ramsey's theorem.
arXiv:2609.37208v1 Announce Type: new Abstract: Fang and Lin [J. Algebraic Combin. 63 (2026), Art.~58] asked whether, whenever $F$ is edge-color-critical with $\chi(F)=r+1$, every non-$r$-partite, $F$-free graph of maximum adjacency spectral radius must also maximize the number of edges. We give a negative answer. Let $F=K_1\vee\mu(K_3)$, where $\mu(K_3)$ is the Mycielskian of a triangle. This gra
arXiv:2609.37262v1 Announce Type: new Abstract: Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. We prove that $d(E_2\times C_3^r)=2r+10$ for $0\le r\le3$. The upper bounds follow from signed zero-block identities and two finite statements in the four-dimensional symplectic space over $\mathbb F_3$. The first supplies edge weights for all completable balanced triangles on any
arXiv:2609.37290v1 Announce Type: new Abstract: A balanced $k$-partite $k$-graph is a $k$-uniform hypergraph whose vertex set is partitioned into $k$ classes of the same size and whose edges meet every class in exactly one vertex. Lo and Markstr\"om (2014) determined the minimum vertex-degree threshold for perfect matchings when $k=3$, and Lu, Wang and Yuan recently determined it when $k=4$. We pr
arXiv:2609.37390v1 Announce Type: new Abstract: We present a determinantal evaluation formula for the very-well-poised ${}_8\Phi_7$ basic hypergeometric Askey-Wilson function in terms of elementary functions. The formula in question is valid for discrete values of the four permutation-symmetric Askey-Wilson parameters; specifically, these consist of all quadruples of parameter values that are give
arXiv:2609.37410v1 Announce Type: new Abstract: Let $s(N)$ denote the smallest side length of a square containing $N$ unit squares with arbitrary orientations and pairwise disjoint interiors. Nagamochi's Packing Unit Squares in a Rectangle (2005) states a rectangle packing bound from which he deduces two infinite families of exact values: $s(k^2-1) = k$ and $s(k^2-2) = k$ for every integer $k \geq
arXiv:2609.37439v1 Announce Type: new Abstract: We prove that the Ehrhart $h^*$-polynomial of a rank-two matroid with exactly three parallel classes is real-rooted whenever its smallest parallel class has size at most three. This bound is sharp: the rank-two matroids with parallel-class sizes $(4,561,600)$ and $(4,a,a+29)$, for all sufficiently large integers $a$, have $h^*$-polynomials that are n
arXiv:2609.37553v1 Announce Type: new Abstract: In this paper, we first construct two graphs $\mathcal{F}(l,m,t,s)$ and $\mathcal{G}_4(l,m,t,s)$. Then we introduce the graph $\mathcal{F}_n(l,m,t,s)$ and the operator $E_{\mathcal{G}_4(l,m,t,s)}$, where $\mathcal{F}_n(l,m,t,s)$ is defined by identifying the vertex $c$ of $n$ copies of $\mathcal{F}(l,m,t,s)$, and $E_{\mathcal{G}_4(l,m,t,s)}$ is defin
arXiv:2609.37744v1 Announce Type: new Abstract: In 1977, Erd\H{o}s posed the problem of determining the maximum number $f_r(n)$ of edges in an $n$-vertex $r$-uniform hypergraph in which all disjoint pairs of edges have distinct unions. F\"uredi later conjectured that, for every fixed $r\ge 4$ and all sufficiently large $n$, $f_r(n)=\binom{n-1}{r-1}+\lfloor \frac{n-1}{r}\rfloor$. In this paper, we
arXiv:2609.37748v1 Announce Type: new Abstract: We study weak and strong forbidden subposet problems in the linear lattice $L_n(q)$. We show that the $q$-analogues of the Bukh--Griggs--Lu conjecture fail for every prime power $q$. If $q\ge 3$ and $n$ and $d$ have opposite parity, then $La_q(n,L_d(q))=La_q^*(n,L_d(q))=\Sigma_q(n,d)$. We also show that $La_q(n,D_s)=La_q^*(n,D_s)=\Sigma_q(n,2)$ for $
arXiv:2609.37774v1 Announce Type: new Abstract: Let $G$ be a connected graph, and let $\lambda_1(G) > \lambda_2(G)$ denote its two largest adjacency eigenvalues. The spectral gap of $G$ is defined as the difference $\lambda_1(G) - \lambda_2(G)$. For integers $r\geq 2$ and $s\geq 0$, the double kite $DK(r,s)$ is formed by taking two vertex-disjoint copies of the complete graph $K_r$ and joining one
arXiv:2609.37778v1 Announce Type: new Abstract: With a view toward applications in Riemannian geometry, we explore coloop splitting properties of regular matroids. Nienhaus showed by classification in rank four that a regular matroid has a cocircuit whose deletion yields two coloops unless the matroid takes a particular form. In the latter case, one can split off any element of the ground set as a
arXiv:2609.37876v1 Announce Type: new Abstract: The B\'ar\'any-Larman conjecture states that for any $d+1$ sets of $r$ points each in $\mathbb{R}^d$, considered as color classes, we can partition their union into $r$ rainbow $(d+1)$-tuples whose convex hulls intersect. We prove that the topological version of this conjecture holds when $r$ is a prime number. We also show that the optimal colorful
arXiv:2609.37877v1 Announce Type: new Abstract: A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols so that each symbol appears exactly once in every row and column. A partial Latin square of order $n$ is an $n \times n$ array whose cells are either empty or filled in such a way that each symbol appears at most once in every row and column, and at most $n$ distinct symbols
arXiv:2609.37912v1 Announce Type: new Abstract: We show that low discrepancy is equivalent to spectral gap for directed $k$-uniform hypergraphs. For undirected hypergraphs this recovers a theorem of Lenz and Mubayi, with a considerably shorter proof. At the heart of our argument is a Frieze--Kannan type spectral regularity lemma for hypergraphs based on the variational notion of hypergraph eigenva
arXiv:2609.37961v1 Announce Type: new Abstract: Given an $n\times n$ symmetric matrix $M$ with largest eigenvalue $\lambda_1\geq 0$, it is easy to show that the solution of the optimisation problem $\max_{v\in [-1,1]^n}v^TMv$ is at most $\lambda_1 n$. We prove the following converse: if every $n'\times n'$ principal submatrix of $M$ has maximal eigenvalue at least $\lambda$, then $\max_{v\in [-1,1
arXiv:2609.35893v1 Announce Type: new Abstract: Bringmann, Han, Heim, and Kane recently investigated periodic sign patterns in the Fourier coefficients of weakly holomorphic modular forms arising from eta-quotients and posed numerous conjectures in the appendix to their work. Using classical theta-function identities, dissections, and coefficientwise positivity, we prove 72 of these conjectural si
arXiv:2609.36045v1 Announce Type: new Abstract: We study a $p$-adic rigidity phenomenon suggested by rank two attractors. Near an ordinary special fiber of a Calabi--Yau family, the excellent Frobenius of Beukers and Vlasenko acts on the parameter disk. Achinger--Zdanowicz construct a canonical $W_2$-lift of the Frobenius twist, while Brantner--Taelman construct a canonical formal lift. We conject
arXiv:2609.36109v1 Announce Type: new Abstract: In a recent paper arXiv:2503.20727, Ichino and Prasanna proved a supercongruence for binomial coefficients and related it to representations of $\mathrm{GL}_2$ over $p$-adic rings. We prove a $q$-analog of this supercongruence using the $q$-Lucas theorem, from which a strengthening of the main result of loc. cit. can easily be deduced. Our proof also
arXiv:2609.36300v1 Announce Type: new Abstract: We expand on the concept of \emph{admissible pairs} from \cite{Levin:GCD} and explore its relation with the main Diophantine arithmetic inequalities, with discriminant term and for points of bounded degree, that have been predicted by Vojta \cite{Vojta:1998}. In this context, among other new results, we prove a \emph{harder implication} which is in t
arXiv:2609.36411v1 Announce Type: new Abstract: We establish an exact zero-one law for twisted Diophantine approximation with an arbitrary fixed real matrix and a positive non-increasing approximation function satisfying dyadic regularity. The criterion is the convergence or divergence of a dyadic series whose summands are Khintchine-Groshev block volumes divided by homogeneous lattice-point count
arXiv:2609.36811v1 Announce Type: new Abstract: The study of monogenic polynomials is a classical problem in algebraic number theory. Existing criteria for deciding whether a polynomial is monogenic typically rely on discriminant computations together with methods such as Dedekind's criterion, Newton polygons, or valuation-theoretic techniques. In this paper, we develop a general local criterion f
arXiv:2609.37252v1 Announce Type: new Abstract: Let $p$ be an odd prime number. Based on recent work of Zhang, we prove that for a finite set $S$ of primes of $\Q$ containing $\infty$ but not $p$, any continuous odd representation $G_{\Q,S} \to \GL_2(A)$ over a complete Noetherian local ring $A$ with finite residue field of characteristic $p$ has finite image, where $G_{\Q,S}$ denotes the Galois g
arXiv:2609.37332v1 Announce Type: new Abstract: Let $f\colon\mathbb{N}\to\mathbb{U}$ be a non-pretentious multiplicative function in the sense of Granville and Soundararajan. We study the convergence of the autocorrelation averages of $f$. In particular, we investigate the relationship between the local pretentious behaviour of $f$ on intervals of the form $[x^{\varepsilon},x]$ and its deviation f
arXiv:2609.37434v1 Announce Type: new Abstract: Erd\H{o}s, Odlyzko and S\'ark\"ozy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$; their conjecture remains open even under the GRH. We prove that its analogue holds, unconditionally and in the wider range $q^{\Theta}$ for any fixed $\Theta > 3/4$, in Granville's random model of the primes.
arXiv:2609.37450v1 Announce Type: new Abstract: In this paper, we study a comparison between $\infty$-adic and finite multiple zeta values in positive characteristic. We construct a surjective comparison map from the algebra of $\infty$-adic multiple zeta values to a quotient of the algebra of finite multiple zeta values. This map factors through the quotient by $\zeta_A(q-1)$, thereby establishin
arXiv:2609.37459v1 Announce Type: new Abstract: Salikhov (2008), Zeilberger-Zudilin (2020) and Bai (2026) all bound the irrationality measure of pi with the same shape of complex contour integral, differing only in three exponents. We give one closed-form expression for the resulting bound as a function of those exponents. The saddle-point data come from an explicit cubic; the arithmetic factor --
arXiv:2609.37466v1 Announce Type: new Abstract: Let p be a prime, q = p, A = F_p[T], and let P be a monic irreducible of degree d with cyclotomic function field K_P and mod-P Teichmuller character omega_P. For a character chi = omega_P^i, Goss defined g(X, chi), the "congruent to one modulo p" part of the Artin L-function L(X, chi), and conjectured that deg_X g(X, omega_P^i) = 5 and 3 = 2, contrad
arXiv:2609.37471v1 Announce Type: new Abstract: Call a set $A$ of positive integers admissible if, whenever $a_1\cdots a_r=b_1\cdots b_s$ with $a_1,\dots,a_r$ distinct elements of $A$ and $b_1,\dots,b_s$ distinct elements of $A$, necessarily $r=s$. Erd\H{o}s asked whether admissible sets can have density $1-\varepsilon$ for every $\varepsilon>0$, and whether $\{1,\dots,N\}$ always contains an admi
arXiv:2609.37507v1 Announce Type: new Abstract: We study the distribution of integers with a large smooth part. We obtain a uniform estimate for the number of integers up to \(x\) whose \(y\)-smooth part exceeds a given threshold \(z\), in the range \(\log z\le y\le z\le x/2\). Our estimate has the expected exponential order governed by the Dickman function and holds uniformly throughout this rang
arXiv:2609.37508v1 Announce Type: new Abstract: Pan and Zhang proved that a positive proportion of positive integers can be represented in the form $a^2+b^2+2^{c^2}$, where $(a,b,c)$ are nonnegative integers. In this note, we give a short proof of their result. The proof uses Romanoff's method and an elementary divisor argument to control the relevant second moment.
arXiv:2609.37595v1 Announce Type: new Abstract: We prove an analog of the Ax-Schanuel theorem for the Drinfeld $j$-function in odd characteristic. Roughly speaking, if the graph of $\boldsymbol{j}\colon\Omega^n\rightarrow\mathbb{A}^n_{\mathbb{C}_\infty}$ and its derivatives has an atypical intersection $\mathcal{V}$ with an algebraic variety, then $\mathcal{V}$ projects to a weakly-special subvari
arXiv:2609.37615v1 Announce Type: new Abstract: Let the theta and Epstein zeta functions be $\Theta(\alpha,L)=\sum_{v\in L}e^{-\pi\alpha|v|^2}$ for $\alpha>0$ and $E(L,s)=\sum_{v\in L\setminus\{0\}}{|v|^{-2s}}$ for $s>2$, respectively. We consider full-rank lattices $L\subset\mathbb R^4$. Let the covolume of $L$ be one, and let $\mathcal D_4=2^{-1/4}D_4$ be the root lattice $D_4$ rescaled to covol
arXiv:2609.37736v1 Announce Type: new Abstract: The 290-Theorem of Bhargava-Hanke completely classifies all universal quadratic forms over the rational integers in terms of a finite criterion set: a positive definite quadratic form is universal if and only if it represents every integer in this finite set. In this article, we study diagonal forms of higher degree over a totally real number field a
arXiv:2609.37754v1 Announce Type: new Abstract: For integers $m\geq 2$ and $m\mid N$, let $G(N;m)=\gcd\{\binom{N}{k}:0<k<N,\ m\mid k\}$. We prove a complete $p$-adic valuation formula for $G(N;m)$ at primes $p\equiv -1\pmod m$, under the hypotheses $m\geq 3$, $m\mid N$, and $m<N$. Writing $N=\sum_i d_i p^i$, put $A=\sum_{i\text{ even}}d_i$ and $B=\sum_{i\text{ odd}}d_i$. Then $v_p(G(N;m))$ is $2$
arXiv:2609.37886v1 Announce Type: new Abstract: We prove a conjecture of Smyth on the Mahler measure of non-reciprocal trinomials of height $1$, determining exactly when their Mahler measures are greater than or less than $M(x+y+1)=\rho=1.381356\ldots$, conjectured to be the smallest limit point of $M(P)$ for non-reciprocal polynomials. Conjecturally, our result gives the full spectrum of $M(P)<\r
arXiv:2609.36022v1 Announce Type: cross Abstract: Let $d \geq 5$, $0 < \gamma < d - 2$, and $\Omega_N$ be the binomial random subset of $Q_N = [-N,N]^d \cap \mathbb Z^d$ with retention probability $p_N = N^{-\gamma}$. We prove that, with failure probability of optimal exponential order, every subset $B \subseteq \Omega_N$ of fixed positive relative density realizes, at each scale $p_N^{-2/(d - 2)}
arXiv:2609.37368v1 Announce Type: cross Abstract: A phase that reaches a fault-tolerant processor in additive pieces can be remembered until the last piece arrives, or executed on arrival: the first option costs classical memory carried across rounds, the second costs magic states committed before the phase is known. We determine the exchange rate $\alpha$, committed $T$ gates per bit of memory fo
arXiv:2609.38001v1 Announce Type: cross Abstract: We prove a geometric version of Poonen's "Mordell--Lang plus Bogomolov" theorem for semi-abelian varieties in characteristic zero. This is a generalization of the Mordell--Lang conjecture and the geometric Bogomolov conjecture.
[This is a guest post by a coalition of Caltech mathematicians: the original organizers of Mathathon (Alvan Arulandu, Andrea Li, Avni Garg, Brian Zhao, Caiman Moreno-Earle, Sathvik Redrouthu), two coauthors of the Open Letter about the Mathathon (Dylan King, Tasmin Chu), and other members of the Caltech community (Mayla Ward, Merrick Hua, Robert Joseph George, […]
After a long hiatus, the problem, which was likely inspired by juggling, has finally been resolved by a group of young mathematicians. The post Mathematicians Harness Randomness To Crack a 55-Year-Old Conjecture first appeared on Quanta Magazine
[This is a guest post by Jess Werk. This blog post was initially written in a different file format and converted using AI. — T.] Hello, I am Jess Werk, professor and chair of Astronomy at the University of Washington. Astronomy may interest mathematicians right now because our field has already been reshaped by supercomputers […]
[This is a guest post by Ivan Corwin. This blog post was initially written in a different file format and converted using AI. — T.] Mathematics and those mathematicians who guide its development have been central to societal advancement since antiquity. AI, like many tools that were invented by mathematicians (e.g., the abacus, slide rule, […]
Just a quick post to note that the International Council for Industrial and Applied Mathematics (ICIAM) has released a statement on mathematics and artificial intelligence (as well as a longer version), which makes many points echoing several already made recently here and elsewhere. The longer statement also makes reference to a recent statement by the […]
[This is a guest post by Bryna Kra and Rachel Ward. -T.] The Summit on PhD Math Education in the Age of AI was held September 17–18, 2026, jointly hosted by the Harvard Department of Mathematics and the Center of Mathematical Sciences and Applications (CMSA). It brought together twenty-four senior mathematicians from across the discipline, […]
[This is a guest post by Amit Sahai. This blog post was initially written in a different file format and converted using AI. — T.] When I was an undergraduate student, I remember talking with several students who felt that the pace at which the top students could understand new math concepts was far too […]
[This is a guest post by Tapio Schneider. This blog post was initially written in a different file format and converted using AI. — T.] [This will be cross-posted on the CliMA blog.] The apparent proof of finite-time blow-up of the forced Navier-Stokes equation, announced by OpenAI on September 8, has brought into focus a […]
[This is a guest post by Martin Hairer, cross-posted from Proofs and Prompts. — T.] There has been a lot of speculation regarding the “Advisory Group on Mathematics and Artifical Intelligence” agmai.org since it was announced on Monday. In this short blog post, I would like to explain in a bit more detail how the […]
OpenAI is working with an independent Advisory Group on Mathematics and Artificial Intelligence to guide the review and communication of emerging AI results.
The proof of a decades-old conjecture has given researchers a new way to understand complex networks. The post Mathematicians Build Long-Awaited Graph Sandwich first appeared on Quanta Magazine
A group of 25 Fields Medalists, including myself, have made a joint declaration on Math and AI: https://mathandai.org/ . We welcome additional signatories (similar to the Leiden declaration), as can be seen on the page. See also this article in the Economist announcing the declaration: https://www.economist.com/science-and-technology/2026/09/11/top-mathematicians-are-outraged-by-openais-methods
Many in my profession are drawn to it in part because of this ability to direct these childlike impulses to productive use, which leads surprisingly often (by an extremely indirect pipeline) to actual advances in science, technology or engineering as per Eugene Wigner's "unreasonable effectiveness of mathematics". But, perhaps in order to impress the other adults observing (and, through public fun
Children eventually become adults, and "put away childish things". Significant portions of our time become devoted to optimizing "important" and immediate goals: weighing the costs and benefits of various decisions, and acting on them accordingly. Unstructured play and exploration become restricted to one's hobbies or personal activities only. This is so normalized in modern adult life that most o
"Toutes les grandes personnes ont d'abord été des enfants. (Mais peu d'entre elles s'en souviennent.)" [All grown-ups were once children... but only few of them remember it.] Antoine de Saint-Exupéry, "Le Petit Prince." An impromptu game of "who can name the largest number?" is an example I often point to as a way in which young children teach each other about non-trivial mathematical ideas and fa
Working out whether a question is actually worth highlighting is a lengthy, deliberate, and subjective process, often informed by historical experience on what good mathematics was generated (or not generated) while working on earlier problems of this type. In particular, being aware of the "difficulty landscape" in a field - what questions are very easy to answer with known methods, which ones ca
I wrote recently about how the collection of good, fruitful open problems is now being mined in a non-renewable fashion, leading to the potential scenario of these problems becoming scarce. This may seem unintuitive at first, since the set of possible problems one could ask is infinite. Perhaps the following analogy can help: a country or region can suffer a critical shortage of drinking water whi
By sheer coincidence, another completely independent result on the Euler blowup question has just been released by Ganeshram, Duruisseaux, and Anandkumar https://anima-ai.org/2026/09/07/stable-singularity-of-the-euler-equations-on-r3-without-forcing/, relating to the more "mainstream" approach to finite time blowup for Euler or Navier-Stokes, in which one uses numerical simulation or machine learn
For the specific ansatz of perturbations chosen here, the amplitude-frequency dynamics turn out to be a remarkably simple ODE, at least for the model case of the Boussinesq equation, though there are still many technical details (involving spatial cutoffs etc.) that enlarge the size of that paper to 76 pages in length, even after the authors efforts to simplify the arguments. There does not seem t
RE: https://mastodon.social/@tristanbuckmaster/117233413705701198 A remarkable achievement: Alpöge and Buckmaster have managed to push one of the major promising approaches towards constructing blowup solutions to fluid equations --- as developed by Cordoba and Martınez-Zoroa --- to establish finite time blowup for many key fluid equations, including 3D incompressible Euler, with a smooth forcing
However, as observed in Section 14.1, there were noticeable opportunity costs with moving too rapidly. We noted a few cases where a key implication stumped us for a while, and forced us to come up with interesting new techniques to construct infinite counterexamples to that implication; but subsequently it was found via an automated theorem prover that the implication in question actually had a re
While working on the revision of the Equational Theories Project (ETP) report at https://arxiv.org/abs/2512.07087 I came across an observation that we quietly made at the back of that report (see Section 14.1) but has become more relevant today. The ETP was a successful project, running over several months, to resolve over 22 million implications in universal algebra (and formalize them in Lean) t
A new proposed competition for AI companies: rather than being the first to announce solutions to unsolved math problems, be the first to announce a new mathematical insight.
In such an alternate scenario, there would likely be more numerical progress on the bounded gaps between prime problem by 2026 than in the current timeline. However, as per Goodhart's law, optimizing for such a metric does not necessarily move one closer to other valuable goals, such as advancing the understanding of mathematics as a whole. As I mentioned in a previous post, these negative effects
2005-2010 The bound occasionally drops over the years, as bored individuals with extra compute credits point their latest AI models at the problem to squeeze a bit more out of the proof techniques. However, these improvements gather almost no attention, either within the professional mathematics community or on social media. ~2010: One of the few mathematicians remaining in the area sifts through