Problems

number-theory

Riemann Hypothesis

Every nontrivial zero of the Riemann zeta function has real part 1/2.

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theoretical-computer-science

P versus NP

Determine whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time.

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combinatorics

Erdős discrepancy problem

For every sequence of signs, some homogeneous arithmetic progression has unbounded discrepancy.

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algebraic complexity / matrix multiplication

Algebraic continuation around rational rank-48 schemes for 4×4 matrix multiplication

Characterize exact algebraic families, boundary degenerations, and lower-bound obstructions around known rational rank-48 decompositions of 4×4 matrix multiplication.

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number-theory / Analytic number theory

The Proportion of Zeta Zeros on the Critical Line

The Riemann hypothesis asserts that every nontrivial zero of the zeta function lies on the critical line. Short of proving it, the standard measure of progress is the proportion of zeros known unconditionally to lie there: Selberg established a positive proportion, Levinson reached a third in 1974, Conrey two fifths in 1989, and the record stood at $\tfrac{5}{12}$ for zeros that are simple and on the line, and $0.6603$ for distinct zeros. Under the Riemann hypothesis, Montgomery deduced $\tfrac23$ simple from the pair-correlation second moment in 1973. His prime-side evaluation was already unconditional; RH entered only to read the zero side as a positive sum over real ordinates. Goldston and Suriajaya isolated that termwise positivity as the remaining obstacle and asked what would follow if it could be removed. This removes it, proving unconditionally that at least $\tfrac23$ of zeros are simple and on the line and at least $\tfrac56$ are distinct - $67.25\ldots\%$ and $0.83625$ with the Montgomery-Taylor window.

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algebra / Algebraic Geometry

Jacobian Conjecture

Every polynomial map $\mathbb{C}^n \to \mathbb{C}^n$ with constant nonzero Jacobian determinant is invertible, with a polynomial inverse.

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geometry-topology / Complex geometry; differential topology

The $(3,4,\infty)$ Modular Family of 2-Tori as a Complex Structure on $S^6$

Hopf's problem, posed in 1948: does the six-sphere $S^6$ admit an integrable complex structure? $S^6$ is one of only two spheres carrying an almost complex structure at all (the other is $S^2$), from the octonions' multiplication, but almost complex structures need not be integrable, and whether that one - or any other - integrates has stood open for 78 years through a history of disputed attempts, including a widely discussed 2016 argument by Atiyah that did not hold up. This paper claims yes: it builds an explicit compact complex threefold $X$, fibred over $\mathbb{P}^1$ by complex 2-tori degenerating at three points, and argues $X$ is simply connected with the integral homology of $S^6$, hence diffeomorphic to it.

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algebraic complexity / matrix multiplication

Low-memory scheduling of the public rank-49 4×4 matrix multiplication circuit

Determine low-peak-live-storage topological schedules for the fixed public 49-product, 156-addition arithmetic DAG under the stated scalar overwrite-on-final-use model.

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algebra / Geometric group theory

Gromov and Weiss's Question on Sofic Groups

Is every group sofic - does every group admit approximate finite permutation representations? A central open question of geometric group theory since Gromov introduced soficity: soficity implies Gottschalk's surjunctivity conjecture, Kaplansky's stable finiteness and more, and no non-sofic group was known. An explicit construction now establishes that non-sofic groups exist.

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geometry-topology / Kähler geometry

The Yau–Tian–Donaldson Conjecture for Constant Scalar Curvature Kähler Metrics

The Yau–Tian–Donaldson conjecture predicts that a polarized manifold carries a canonical Kähler metric in its polarization class exactly when it is K-polystable. Settled for Kähler–Einstein metrics on Fano manifolds, a proof of the uniform version of such a conjecture for general constant scalar curvature Kähler metrics has been showed in a recent preprint (arxiv:2605.30063v2). While the original, non-uniform Yau-Tian-Donaldson conjecture has been proved to be false: there is a polarized smooth projective fivefold that is K-polystable but admits no extremal Kähler metric in $c_1(A)$, so K-polystability does not imply existence. This makes the uniform version of the Yau-Tian-Donaldson Conjecture optimal as experts expected.

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geometry-topology / Differential geometry

The $C^\infty$ Carathéodory Conjecture on Umbilic Points

Carathéodory's conjecture, Problem 8.1 of Ghomi's list and traceable to 1922, asks whether every closed convex surface in $\mathbb{R}^3$ has at least two umbilic points. Hamburger settled the real-analytic case in 1940-41 and it stands. The $C^\infty$ case is false: an explicit support function gives a smoothly embedded two-sphere bounding a convex body with exactly one umbilic point. The same family disproves the smooth Loewner conjecture, whose member at $k=1$ has an isolated trace-free Hessian zero of winding number three.

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theoretical-computer-science / Algebraic complexity

The Matrix Multiplication Exponent

The matrix multiplication exponent $\omega$ is the infimum of all $t$ for which two $n \times n$ matrices can be multiplied in $O(n^t)$ arithmetic operations. Strassen showed in 1969 that $\omega < 3$, and sixty years of work has driven the upper bound down without anyone knowing the true value. Whether $\omega = 2$ is one of the central open questions of algebraic complexity. The current bounds come from the laser method as refined by combination loss analysis. This paper attacks the optimization problem at the core of that refinement, reformulating it so it can be solved in a larger setting, designing a new optimization algorithm for it, and then refining that algorithm with AlphaEvolve. The result is $\omega < 2.371177$, improving the previous best of $2.371339$.

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combinatorics / Graph Theory

Cycle Double Cover Conjecture

Conjectures that every bridgeless graph has a collection of cycles covering each edge exactly twice.

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combinatorics / Additive combinatorics

The Sum-Product Conjecture over the Reals

Erdos and Szemeredi conjectured that every finite set of reals satisfies $\max(|A+A|,|AA|) \ge |A|^{2-o(1)}$. False: there are arbitrarily large $A \subset \mathbb{R}$, of algebraic integers in a number field of degree $\asymp \log|A|$, with $\max(|A+A|,|AA|) \le |A|^{2-c}$ for an absolute $c > 0$. Variants give counterexamples in function fields of fixed positive characteristic.

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analysis / Spectral geometry

Schiffer's Conjecture and the Pompeiu Problem

If a smooth bounded domain in $\mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian that is constant on the boundary, must the domain be a ball? Pompeiu posed an equivalent integral-equation form in 1929; Schiffer's 1957 reformulation via Neumann eigenfunctions is the version on Yau's 1982 list (Problem 80), and Williams proved the two formulations logically equivalent for simply connected domains in 1976. Cao-Labora and de Dios Pont construct infinitely many planar domains with large $N$-fold symmetry that are not balls and admit such an eigenfunction, disproving Schiffer's conjecture; applying Williams' classical reduction to the same domains (their Corollary 1.2) disproves Pompeiu's problem as well.

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number-theory / Elliptic curves

A Rank-$31$ Record for an Elliptic Curve over $\mathbb{Q}$

How large can the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$ be? Whether ranks are unbounded is open, and progress is measured by explicit records, tabulated by Dujella: rank $\ge 28$ from 2006, raised to $\ge 29$ by Elkies and Klagsbrun in 2024, and to $\ge 30$ three days before this one by the same team (see the related entry). Now $\ge 31$, witnessed by an explicit curve $y^2 + xy + y = x^3 + x^2 + a_4 x + a_6$ with $a_4$ of 67 digits and $a_6$ of 99, carrying thirty-one independent rational points.

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theoretical-computer-science / Network information theory

Whether Marton's Inner Bound Achieves the Broadcast Channel Capacity Region

Marton's inner bound, proposed in 1979, is the best known achievable region for a general discrete memoryless broadcast channel, and whether it always achieves the capacity region had been open ever since. It does not: there is a finite two-receiver discrete memoryless broadcast channel whose two-letter Marton value strictly exceeds twice its one-letter value, so the complete one-letter Marton region is strictly contained in the capacity region.

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geometry-topology / Discrete geometry

Upper Bounds for High-Dimensional Sphere Packing

How dense can a sphere packing in $\mathbb{R}^n$ be as $n \to \infty$? The Kabatiansky-Levenshtein upper bound stood for almost fifty years; the new proof improves the asymptotic upper bound all the way down to the Cohn-Elkies linear-programming threshold.

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number-theory / Elliptic curves

Record Rank for an Elliptic Curve over $\mathbb{Q}$

How large can the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$ be? Whether ranks are unbounded is open, and progress is measured by explicit records, tabulated by Dujella: rank $\ge 28$ from 2006, raised to $\ge 29$ by Elkies and Klagsbrun in 2024. Now $\ge 30$, witnessed by an explicit curve $y^2 + xy = x^3 + a_4 x + a_6$ with $a_4$ of 63 digits and $a_6$ of 94, carrying thirty independent rational points.

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geometry-topology / Asymptotic convex geometry

KLS Conjecture for Quadratic Forms

Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is $\operatorname{Var}\langle MX, X\rangle \le C\, \mathbb{E}|\nabla\langle MX, X\rangle|^2$ for every symmetric $M$?

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analysis / Operator algebras

Connes' Rigidity Conjecture

Are ICC property (T) groups remembered by their von Neumann algebras - if $L(\Gamma) \cong L(\Lambda)$ for such groups, must $\Gamma \cong \Lambda$? A counterexample refutes Connes' conjecture that these groups are uniquely determined by their group von Neumann algebras.

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analysis / Complex analysis

Nevanlinna’s half-plane omitted-values problem

We construct a real meromorphic function $F$ on $\mathbb{C}$ such that $F^{-1}(\{0,1,\infty\})\subset\mathbb{R}$, while $F$ is not of bounded type in either half-plane. More strongly, for every $a\in\widehat{\mathbb{C}}\setminus\{0,1,\infty\}$, the $a$-point divisor in either half-plane fails the Blaschke condition. Thus the construction provides an independent negative answer to a question going back to Nevanlinna’s 1925 work that had remained open for over a century. Postcomposition gives the analogous counterexample for any prescribed triple of distinct values in the Riemann sphere. The core construction and proof were generated during an autonomous run of GPT-5.6 Sol Ultra.

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geometry-topology / Combinatorial Geometry

Erdős's Planar Unit Distance Conjecture

Conjectured upper bound on how many pairs among $n$ points in the plane can be exactly one unit apart.

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combinatorics / Graph theory

Petersen Coloring Conjecture

Jaeger conjectured that every bridgeless cubic graph $G$ admits a Petersen coloring: a map $\varphi\colon E(G)\to E(P)$ into the edges of the Petersen graph $P$ such that, for every vertex $v$ of $G$, the three edges at $v$ are sent to three edges meeting at a common vertex of $P$. Equivalently, by Jaeger's theorem, every bridgeless cubic graph has a normal 5-edge-coloring. The conjecture implies both the Berge-Fulkerson conjecture and the 5-cycle-double-cover conjecture. False: there is an explicit simple connected bridgeless cubic graph on $112$ vertices, of girth five and edge- and vertex-connectivity three, with no Petersen coloring.

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combinatorics / Matroid theory

White's Conjecture on Matroids

White conjectured that the symmetric exchange binomials generate the toric ideal of a matroid. This is now known to be false; a rank $9$ binary matroid constitutes a counterexample.

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analysis / Functional analysis

Banach's isometric conjecture

Banach asked in 1932 whether a real Banach space $X$ whose $n$-dimensional subspaces, for some fixed $1 < n < \operatorname{dim}X$, are all isometric must be a Hilbert space. Gromov proved the conjecture for even n, and subsequent work settled several odd-dimensional cases. We prove the conjecture for every odd n, including all previously unresolved cases. Together with Gromov’s even-dimensional result, this completes Banach’s isometric conjecture in the real case. The proof combines bundle topology with Brouwer degree theory.

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analysis / Complex analysis

Sendov's Conjecture

Let $p$ be a complex polynomial of degree $n \ge 2$ whose zeros all lie in the closed unit disk. Then for every zero $a$ of $p$, there exists a critical point $\zeta$ of $p$ such that $|\zeta-a| \le 1$. This is the standard Sendov statement and exactly matches the theorem Mazur formalized.

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theoretical-computer-science / Coding theory

Upper Bounds for Binary and Spherical Codes

What is the maximum size of a binary code of given minimum distance? The linear-programming bounds of McEliece, Rodemich, Rumsey and Welch (1977) resisted improvement for half a century. The new upper bounds are exponentially stronger at every prescribed distance, with analogous results for high-dimensional spherical codes.

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analysis / Harmonic analysis

Stein’s dimension-free weak-(1,1) Riesz transform problem

The Riesz transforms $R_1,\ldots,R_n$ on $\mathbb{R}^n$ are the Fourier multipliers $-i\xi_j/|\xi|$, the natural higher-dimensional Hilbert transforms. Stein proved in 1983 that their $L^p$ bounds can be taken independent of the dimension for every $1 < p < \infty$. At the 1986 ICM he asked whether the same holds at the endpoint $p=1$: is there an absolute constant $C$, independent of $n$, with $$|\{x : |Rf(x)| > \lambda\}| \le \frac{C}{\lambda}\,\|f\|_{L^1(\mathbb{R}^n)}$$ for every $\lambda > 0$? The Calderon-Zygmund route gives a constant that grows with the dimension, and the best known was Janakiraman's $c\log n$. This paper answers yes, with $C = 2$, for the vector transform $R = (R_1,\ldots,R_n)$ - so the same constant serves every single component $R_j$ uniformly in $n$.

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combinatorics

Asymptotically attaining the Moore bound

For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. It is shown that $$\lim_{d \to \infty}\frac{n_k(d)}{d^k} = 1$$ for every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter. Also proved a similar lower bound on the edge-variant of the problem, and a tight asymptotic for the bipartite variant of the edge problem.

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theoretical-computer-science / Algebraic complexity

Lower Bounds for the Permanent in Arithmetic Circuits

How large must arithmetic circuits and formulas computing the $n \times n$ permanent be? New lower bounds include an arithmetic-formula bound of order $n^4/\log n$, far beyond the quadratic barrier that stood for decades.

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geometry-topology / Differential geometry

Universal volume growth bounds from positive intermediate curvature

In 1986 Gromov asked whether every complete $n$-dimensional Riemannian manifold with $\mathrm{Ric} \ge 0$ and $\mathrm{Scal} \ge 1$ satisfies $$\mathrm{Vol}\,B_R(p) \le C(n)\,R^{n-2}$$ for every $p$ and every $R > 0$. The three-dimensional case had been settled, and higher dimensions were known only under extra hypotheses such as nonnegative sectional curvature, noncollapsing or an injectivity-radius bound. This paper answers the question affirmatively, as the case $m = n-2$ of a uniform family: for every $0 \le m \le n-2$, if $\mathrm{Ric} \ge 0$ and the $(m{+}1)$-intermediate curvature of Brendle-Hirsch-Johne is at least 1, then $\mathrm{Vol}\,B_R(p) \le C(n,m)\,R^m$. At $m = 1$ this gives linear volume growth under positive biRicci curvature in every dimension.

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probability-statistics / Analysis of Boolean functions

Talagrand’s convolution conjecture

On the Boolean hypercube $G = \{-1,1\}^n$ with uniform measure $\lambda$, let $T_\mu f(x) = \int_G f(x \odot y)\,d\mu(y)$ be convolution by a finite positive measure $\mu$, and set $$\psi_\mu(u) = \sup\{u\,\lambda(\{T_\mu f \ge u\}) : f \ge 0,\ \|f\|_1 = 1\},$$ which measures how much better than Markov's inequality convolution makes the tail. In 1989 Talagrand conjectured that for the biased-coin product measure $\mu_a = (\tfrac{1+a}{2}\delta_1 + \tfrac{1-a}{2}\delta_{-1})^{\otimes n}$ with $0 < a < 1$, $$\psi_{\mu_a}(u) \le \frac{C_a}{\sqrt{\log u}} \qquad (u > 1),$$ with $C_a$ depending on $a$ alone and not on the dimension $n$. He offered a \$1000 prize for a proof. The Gaussian analogue was settled by Eldan and Lee; the hypercube case, the original, stayed open. This paper claims the conjectured bound.

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probability-statistics / High-dimensional probability

Talagrand's Convexity Problem

Talagrand's convexity problem asks whether a universal number of Minkowski sum operations turns any set of large Gaussian measure into one containing a convex body of comparable measure. It is equivalent to a question about subgaussian vectors: is every centered $1$-subgaussian random vector in $\mathbb{R}^n$ the sum of a universal number of standard Gaussian vectors? Both are answered affirmatively, via the sharper statement that any random vector dominated in convex order by a standard Gaussian is the sum of three standard Gaussian vectors.

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combinatorics / Matroid theory

Rota's Unimodality Conjecture for Matroid Flats

Is the sequence $W_0, W_1, \dots, W_n$ counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.

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geometry-topology / 4-manifold topology

The Kinoshita Conjecture and Kirby Problem 4.37

Kinoshita conjectured that every embedded projective plane in $S^4$ is reducible. False: an irreducible embedded projective plane exists in $S^4$. The construction also answers both parts of Problem 4.37 of the Kirby problem list.

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theoretical-computer-science / Lattices & cryptography

Polynomial-Factor Hardness for the Closest Vector Problem

Is the closest vector problem NP-hard to approximate within polynomial factors $n^c$? Yes for some $c > 0$: hardness of approximation reaches polynomial factors, with consequences for decoding and related lattice problems - a foundational question underpinning post-quantum cryptography where hardness had stalled at almost-polynomial factors since the late 1990s.

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probability-statistics / Probability

Krauth-Mezard Storage Capacity of the Ising Perceptron

Krauth and Mezard predicted in 1989 that the storage capacity of the Ising perceptron at zero margin is an explicit constant $\alpha_\star \approx 0.8330786$. Ding and Sun proved the matching lower bound and Huang the upper bound, but each was conditional on a global sign condition nobody had verified. Both conditions now hold rigorously, so $M_N/N \to \alpha_\star$ in probability with $\alpha_\star \in [0.833078599, 0.833078600]$.

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quantum-information-computing / Entanglement theory

Two-Copy Distillability of Werner States

Is a Werner state that is not one-copy distillable ever two-copy distillable? The first open rung of the NPT bound-entanglement ladder, open since 2000.

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mathematical-physics

Full-RSB in the Sherrington–Kirkpatrick spin glass

This work proves full replica symmetry breaking for the zero-field Sherrington–Kirkpatrick model at zero temperature $\beta=\infty$: the Parisi minimizer is absolutely continuous, has a smooth density, and has support $[0,1)$, thereby confirming the prediction by Parisi.

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logic-foundations / Model theory

$SOP_2 = SOP_3$

The classes of SOP_2 and SOP_3 first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.

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theoretical-computer-science / Streaming algorithms

Optimality of Greedy for Single-Pass Semi-Streaming Matching

Can any single-pass semi-streaming algorithm beat the naive greedy $1/2$-approximation for maximum matching? No. No single-pass semi-streaming algorithm, deterministic or randomized, achieves a better-than-half approximation, so greedy is optimal. The same construction settles the optimal competitive ratio of online matching with preemption at $1/2$.

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number-theory / Number theory

Erdős Problem #387

Erdős and Graham asked whether $\binom{n}{k}$ with $1 \le k \le n/2$ must always have a divisor $\le n$ that is close to $n$, meaning bigger than a fixed constant times $n$. Settled in both directions: true when $k$ is large enough as a function of $n$, but false in general, since there are $\binom{n}{k}$ with $k$ small compared to $n$ having no such divisor.

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probability-statistics / Probability

Feige's Conjecture

Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb{E}X_i \le 1$, and let $S$ be their sum. Is $\mathbb{P}(S < \mathbb{E}S + 1) \ge 1/e$? Feige proved the constant $1/13$ and conjectured the sharp $1/e$. Three independent July 2026 proofs settle it, both building on the Vlassis-Thomas calibration theorem; the sharper one determines the optimal small-deviation bound for every deviation $\delta \ge 1$.

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analysis / Spectral Geometry, Laplace Eigenvalues

Pólya's Conjecture for Neumann Balls in Dimensions Three and Higher

Pólya conjectured in 1954 that the Weyl-law expression bounds the eigenvalue counting function of the Laplacian. The paper proves the Neumann case for Euclidean balls in dimensions three and higher, extending the authors' earlier planar and Dirichlet results. Key difficulty: estimating zeros of derivatives of ultraspherical Bessel functions rather than of Bessel functions themselves.

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analysis / Calculus of variations and geometric measure theory

Gamow liquid-drop minimizer conjecture

For a measurable set $\Omega\subset\mathbb R^3$, let $$\mathcal E(\Omega)=P(\Omega)+\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|},$$ where $P$ is De Giorgi perimeter, and set $$V_*=5\frac{2-2^{2/3}}{2^{2/3}-1}\approx3.51.$$ The conjecture asks for the complete fixed-volume minimization picture. Chodosh and Gianocca prove that, for every $0<V\le V_*$, balls of volume $V$ uniquely minimize $\mathcal E$ among all measurable $\Omega$ with $|\Omega|=V$, up to translation and null sets; for $V>V_*$, no minimizer exists. Consequently, $$\inf_{0<|\Omega|<\infty}\frac{\mathcal E(\Omega)}{|\Omega|}=3\left(\frac{9\pi}{5}\right)^{1/3}=\frac92\left(\frac{8\pi}{15}\right)^{1/3},$$ with equality exactly for translates, modulo null sets, of the ball of volume $5/2$, equivalently radius $(15/(8\pi))^{1/3}$.

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algebra / Operator algebras

Connes' Rigidity Conjecture for ICC Property (T) Groups

Connes' rigidity conjecture asks whether an ICC group with Kazhdan's property (T) is determined by its group von Neumann algebra. Disproved for this class: two explicit countable discrete groups $\Gamma_1$ and $\Gamma_2$, both ICC and property (T), are non-isomorphic as groups while $L(\Gamma_1) \cong L(\Gamma_2)$.

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combinatorics / Zero-error information theory

Record Lower Bounds for the Shannon Capacity of Odd Cycles

Determine the Shannon capacities of odd cycles beyond $C_5$, or improve the best explicit bounds. Lovasz's theta function settled $C_5$ in 1979 and every longer odd cycle has stayed open since. The current records, all obtained with model assistance and formally verified, are $\Theta(C_7) \ge 3.258805369885$, $\Theta(C_{11}) \ge 5.294502522149$, $\Theta(C_{13}) \ge 6.302455083464$, $\Theta(C_{15}) \ge 7.301600534487$, $\Theta(C_{19}) \ge 9.357192705918$, $\Theta(C_{21}) \ge 10.342455853338$ and $\Theta(C_{23}) \ge 11.328224257774$.

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analysis / Matrix analysis

Crouzeix's Conjecture

Crouzeix conjectured in 2004 that for every square complex matrix $A$ and every polynomial $p$, $\lVert p(A)\rVert \leq 2 \max_{z \in W(A)} |p(z)|$, where $W(A)$ is the numerical range of $A$ - that is, the numerical range is a 2-spectral set. Crouzeix proved a constant of 11.08 in 2007 and Crouzeix and Palencia lowered it to $1+\sqrt{2}$ in 2017; the conjectured constant 2 is attained by $2\times 2$ matrices. Jin proves the sharp bound by a function-theoretic route whose key theorem reduces the problem, via a sampling strategy, to a positivity condition; Lorist and Schwenninger independently prove it days later by combining double-layer potential machinery with a perturbation lemma for 2-dilations.

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analysis / Harmonic Analysis

HRT Conjecture

Heil, Ramanathan and Topiwala conjectured in 1996 that any finite set of time-frequency shifts of a nonzero square-integrable function is linearly independent. This refutes it: there is a Schwartz function admitting 12 linearly dependent time-frequency shifts.

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probability-statistics / Random combinatorial optimization

Central limit theorem for the random assignment problem

Let $C_n$ be the minimum cost of a perfect matching in an $n\times n$ matrix of independent uniform random variables. Aldous proved in 1992 that $\mathbb{E}[C_n]$ converges, later identifying the limit as $\zeta(2)$ via the Poisson-weighted infinite tree; Parisi's exact finite-$n$ formula for exponential costs was then proved by Linusson-Wästlund and independently by Nair, Prabhakar and Sharma. The fluctuations resisted. Talagrand applied product-space concentration, Wästlund computed the exponential model's variance as $4\zeta(2)-4\zeta(3)+O(n^{-2})$, and Chatterjee proved an order-$n^{-1/2}$ lower bound under tail hypotheses that exclude the bounded uniform law - but no central limit theorem for $C_n$ was known. This paper claims one: $\sqrt{n}\,(C_n-\zeta(2)) \Rightarrow \mathcal{N}(0,\,4\zeta(2)-4\zeta(3))$.

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algorithms-optimization / Convex optimization

Point Convergence of Nesterov's Accelerated Gradient Method

Nesterov's accelerated gradient method (1983) is a cornerstone of optimization, yet whether its iterates themselves converge to a minimizer, rather than just the function values, stayed open for over forty years. Jang and Ryu resolve it in the affirmative. Ryu first announced the continuous-time result on X; Bot, Fadili and Nguyen's concurrent human proof of the critical-regime case (answering a decade-old conjecture of Attouch and co-authors) explicitly credits that AI-assisted announcement as what it discretizes.

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combinatorics / Graph theory

Seymour's Second Neighborhood Conjecture

Seymour conjectured that every oriented graph has a vertex $x$ with $|N^{++}(x)| \ge |N^{+}(x)|$. It holds for oriented graphs of minimum out-degree exactly $7$, the first improvement to the out-degree threshold since Kaneko and Locke settled degree $6$ in 2001.

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mathematical-physics / Classical electrostatics

Maxwell's Conjecture on Point-Charge Equilibria

Do $n$ point charges whose electrostatic potential has only non-degenerate critical points always have at most $(n-1)^2$ of them? A configuration of five charges - three at the vertices of an equilateral triangle plus two small central charges pulled apart into a shallow bipyramid - has at least $24 > 16$ non-degenerate critical points, so the conjecture is false.

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algebra / Affine Algebraic Geometry, Polynomial Maps

A Five-Variable Counterexample to the Hessian Conjecture

The paper exhibits an explicit integer polynomial in five variables, of total degree 14 with constant Hessian determinant 128, whose gradient is not injective. Its formal Legendre transform is therefore not a polynomial, so the Hessian conjecture $\mathrm{HC}_5$ is false. The counterexample comes from a one-variable Schur descent applied to the six-variable doubling of Alpöge's 2026 Jacobian counterexample.

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quantum-information-computing / Entanglement theory

Finite-Copy Distillability of NPT States in the DiVincenzo Family

Whether negative-partial-transpose states undistillable from one copy become distillable from finitely many copies is a basic open problem in entanglement theory. In the canonical two-parameter DiVincenzo family used as its symmetry-reduced testbed, a distinguished one-copy-undistillable state is shown to be two-copy distillable in every local dimension d >= 3, disproving the conjecture that the family's whole one-copy-undistillable region stays undistillable for arbitrarily many copies.

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probability-statistics / Random matrix theory

The Ellipsoid Fitting Conjecture

Given $n$ independent standard Gaussian vectors in $\mathbb{R}^d$, an ellipsoid fit is a positive semidefinite matrix $S$ with $x_i' S x_i = d$ for every $i$. Saunderson, Parrilo and Willsky conjectured that this semidefinite feasibility problem has a sharp threshold at $n \sim \frac{d^2}{4}$. Proved: below the threshold a fit exists with probability tending to one, above it none does.

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combinatorics / Additive combinatorics

Ben Green's Open Problem 90

For $A \subset \mathbb{F}_p$ of density $1/2$, call $A$ almost affine invariant under $\varphi(x) = ax+b$ if $|A \triangle \varphi(A)| = o(p)$. Problem 90 asks for the threshold $K$ below which $A$ can be almost affine invariant simultaneously under all such $\varphi$ with $|a|, |b| \le K$ and $a \ne 0$. The threshold is $K = o(\log p)$.

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mathematical-physics / Spin glass theory

Talagrand's critical Sherrington-Kirkpatrick overlap conjecture

At the critical inverse temperature $\beta=1$ in the Sherrington-Kirkpatrick spin glass model, Talagrand conjectured that the expected squared overlap of two independent Gibbs replicas has an exact $N^{-2/3}$ scaling: there exists a constant $a>0$ such that $$ \lim_{N\to\infty} N^{2/3}\mathbb{E}\langle R_{1,2}^2\rangle=a. $$ Du and Huang prove that this limit exists and is positive and finite. More strongly, they determine the full limiting quenched distribution of the rescaled overlap $N^{1/3}R_{1,2}$ in terms of the reflected $\mathrm{Airy}_1$ point process.

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combinatorics / Polyhedral combinatorics

The Mihail-Vazirani Conjecture

Mihail and Vazirani conjectured that the graph of every $0/1$-polytope has edge expansion at least one. Disproved by a family of $0/1$-polytopes whose edge expansion decreases exponentially in the dimension.

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probability-statistics / Markov chain mixing time

Kannan–Tetali–Vempala conjecture (bipartite/binary-matrix case)

The swap chain flips checkerboard 2×2 blocks to sample 0/1 matrices with fixed row and column sums. Kannan, Tetali and Vempala conjectured in 1997 that it mixes in polynomial time for all feasible margins; the lazy chain is shown to have spectral gap at least $\binom{m}{2}^{-1}\binom{n}{2}^{-1}$ on $m \times n$ matrices, which is worst-case tight and settles the bipartite case.

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geometry-topology / Birational geometry

Shokurov's Global Index Conjecture for Foliations

Shokurov's global index conjecture, in the setting of foliations. Proved for foliations in dimension at most three, which also answers a question of Liu, Meng and Xie in dimension three.

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combinatorics / Graph theory

Albertson–Berman Induced Forest Conjecture

Albertson and Berman conjectured that for every simple planar graph $G$ on $n$ vertices, the largest vertex set inducing a forest has size at least $n/2$. The standing lower bound since the same year has been Borodin's $2n/5$, from his acyclic five-colour theorem. False: there is an explicit $31$-vertex simple $3$-connected maximal planar graph $T$ whose largest induced forest has exactly $15$ vertices, and an infinite family $M_k$ on $31k$ vertices with induced-forest number exactly $15k$, giving the ratio $15/31 < 1/2$ even for triangulations of minimum degree five.

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geometry-topology / Convex geometry

The Generalized Busemann-Petty Problem in Dimensions 2 and 3

If origin-symmetric convex bodies $K, L \subset \mathbb{R}^n$ satisfy $\mathrm{vol}_m(K \cap E) \leq \mathrm{vol}_m(L \cap E)$ for every $m$-dimensional subspace $E$ with $1 < m < n$, does $\mathrm{vol}_n(K) \leq \mathrm{vol}_n(L)$ follow? Answered affirmatively for subspace dimensions $m = 2$ and $m = 3$.

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algebra / Algebraic geometry

The Period-Index Conjecture

For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field $k$ of characteristic $0$ and any $d \geq 3$ there is a $d$-dimensional variety over $k$ carrying a Brauer class that violates it, for Hodge-theoretic reasons. For $d = 3$ the construction needs no uncountability, so the conjecture fails already over $\overline{\mathbf{Q}}$.

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algebra / Algebraic Geometry

Grothendieck's Finite Flat Group Scheme Order Question

Grothendieck asked whether every finite locally free group scheme of order $n$ is killed by $n$ (its $n$-th convolution power map equals the unit). The counterexample is an order-4 group scheme not killed by 4 (killed only by 8); since Deligne settled the commutative case, it is necessarily non-commutative over a non-reduced base.

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geometry-topology / Discrete geometry

Kusner's Conjecture on Equilateral Sets in $\ell_p^n$

Kusner conjectured in 1983 that the maximum number of points in $\mathbb{R}^n$ that are pairwise at $\ell_p$-distance one is exactly $n+1$ for every $2 < p < \infty$, as in the Euclidean case. False: an explicit configuration of $n+2$ equilateral points exists for some exponent, placing the infimum of exponents at which the conjecture fails in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.

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analysis / Complex analysis

Phelps–Rodriguez Conjecture

Let $p$ be a complex polynomial of degree $n\ge2$ whose zeros all lie in the closed unit disk. For every zero $a$ of $p$, there is a critical point $\zeta$ satisfying $|\zeta-a|<1$, except when $|a|=1$ and $p$ is a nonzero scalar multiple of $z^n-a^n$.

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combinatorics / Additive combinatorics

The Largest Sum-Free Subset of the Lattice Cube

How dense can a sum-free subset of the lattice cube $\{1,\dots,n\}^d$ be? Aydinian and Cameron asked for the limiting density, which is also Problem 6 in Ben Green's list of 100 open problems. The natural conjecture is that the optimum is a slice $\{x : 1 \le L(x) < 2\}$ for a linear map $L$, previously known only for $d \le 4$. Proved for all $d$. The paper also shows the same phenomenon fails if the cube is replaced by an arbitrary convex set avoiding the origin.

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combinatorics / Glauber dynamics

Rapid mixing for spin systems on graphs of girth at least five

It is proved that, for every $\delta\in(0,1)$, the Glauber dynamics for the uniform distribution on proper $q$-colorings is rapidly mixing when $q\geq(1+\delta)\Delta$ and the underlying graph has girth at least $5$ and maximum degree $\Delta=\Omega_{\delta}(1)$. This result also extends to general multi-spin systems satisfying a local spectral contraction condition, including the anti-ferromagnetic Potts model with $q\geq(1+\delta)(1-\beta)\Delta$. These results are achieved by a new spectral local-to-global principle on graphs with girth at least five for general multi-spin systems, and a novel Fourier analysis for Glauber dynamics on a star. The main ideas behind all the proofs were developed through several rounds of interaction with GPT-5.6 Sol Ultra.

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combinatorics / Graph theory

Seymour's Second Neighborhood Conjecture

Seymour conjectured that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Known cases include tournaments (Fisher 1996) and minimum outdegree at most six (Kaneko-Locke 2001), and for dense incomplete graphs a series of results restricting the structure of the missing edges. This work proves the conjecture for every oriented graph of order $n = 2\delta + 2$, where $\delta$ is the minimum outdegree, with no prescribed structure on the missing edges; with Fisher's tournament theorem this gives every oriented graph satisfying $n \le 2\delta + 2$.

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combinatorics / Matroid theory

Log-Concavity of Flats of Matroids

Mason conjectured the following: let $M$ be a matroid of rank $r$, and let $W_i$ denote the number of flats of $M$ of rank $i$. Is it true that for all $1 \leq i \leq r - 1$, we have $W_i^2 \geq W_{i + 1}W_{i - 1}$? This is false; a counterexample is given by a graphic matroid whose graph is a generalized theta graph with $79$ edges.

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probability-statistics / Discrepancy theory

The Matrix Spencer Conjecture for Finite Groups

The group version of the Matrix Spencer conjecture holds: for every finite group $G$ there are signs $\varepsilon \in \{\pm 1\}^G$ with $\left\|\sum_{g \in G} \varepsilon_g \rho(g)\right\| \le C\sqrt{|G|}$, where $\rho$ is the left regular representation and $C$ is universal.

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analysis / Dynamo theory

Smooth Random Fast Dynamo on the Three-Torus

Arnold's fast-dynamo problem asks for a smooth divergence-free velocity field on $\mathbb{T}^3$, chosen independently of the magnetic diffusivity, that drives exponential growth of the magnetic field at every sufficiently small diffusivity. This constructs a genuinely $C^\infty$ field with that behaviour: random and time-dependent, refreshing iid on finite time blocks, for which the almost sure exponential growth rate is at least $1/2$ at each fixed small enough resistivity, with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment bound. The field is neither autonomous nor deterministic, so Arnold's smooth autonomous problem on $\mathbb{T}^3$ remains open.

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combinatorics / Combinatorial design theory

Hadamard Matrix of Order 668

There exists a Hadamard matrix of order $668$: a matrix $$ H\in\{-1,1\}^{668\times668} $$ such that $$ HH^{\mathsf T}=668I_{668}. $$ Equivalently, the $668$ rows of $H$ are pairwise orthogonal.

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geometry-topology / Convex geometry

Bellman's Lost-in-a-Forest Problem for the Golden Gnomon

What is the shortest curve guaranteed to reach the boundary of the golden gnomon - the isosceles triangle with equal sides $1$ and apex angle $108^\circ$ - from an unknown starting position and heading? The optimum is a symmetric seven-piece path of segments, circular shoulders and tangents, of exactly determined transcendental length $C = 1.282676\ldots$ - the first proved exact optimum for an isosceles triangle with base angle below $45^\circ$.

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combinatorics / Matroid theory

The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture

The Kajitani–Ueno–Miyano conjecture asserts that every finite uniformly dense matroid has a cyclic basis ordering. The conjecture is proved for all matroids of rank three. The new result establishes the previously unresolved divisible case, where the ground-set size is a multiple of three, without assumptions of simplicity, representability or paving. Together with the previously published coprime-case theorem of van den Heuvel and Thomassé, this covers every finite uniformly dense rank-three matroid. The unrestricted conjecture remains open in higher rank.

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combinatorics / Design theory

Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically

Kotzig conjectured that for every even $n \ge 4$ the complete graph $K_n$ decomposes into $n-1$ perfect matchings such that every pair of them forms a Hamilton cycle. An asymptotic version holds: $K_n$ decomposes into $n-1$ perfect matchings of which $(1-o(1))n$ have the property that any pair forms a Hamilton cycle.

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probability-statistics / Discrepancy theory

The Matrix Spencer Conjecture for C*-Algebra Contractions

A structured special case of the Matrix Spencer conjecture, reached through the representation theory of finite-dimensional C*-algebras: the conjectured discrepancy bound holds for every family of contractions contained in a suitable algebra.

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combinatorics / Discrete geometry

Borsuk Conjecture lowest-ever counterexample (N=63)

Borsuk's conjecture asked whether every bounded set in $\mathbb{R}^n$ can be partitioned into $n+1$ subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in $\mathbb{R}^{63}$ whose smaller-diameter subsets have at most 5 points, so at least $\lceil 321/5\rceil = 65 > 64$ parts are required. The previous record dimension was 64 (Jenrich-Brouwer, 2014), and the first failing dimension remains open for $4 \le n \le 62$. The construction modifies Bondarenko's $G_2(4)$ two-distance set: a 320-point rank-63 subconfiguration plus one added scaled projected point, which makes the set three-distance - precisely why it was not reachable inside the two-distance framework in which all previous work took place.

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number-theory / Diophantine approximation

The Lonely Runner Conjecture for Nine and Ten Runners

The Lonely Runner Conjecture of Wills and Cusick states that among $k+1$ runners at distinct constant speeds on a unit circle, each runner is at some time at distance at least $1/(k+1)$ from all others. Following Rosenfeld's computer-assisted proof for 8 runners, the paper refines his approach with a sieve and proves the cases of 9 and 10 runners.

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combinatorics / Extremal graph theory

Tuza's Conjecture for Maximum Degree at Most Seven

Tuza conjectured that every finite simple graph satisfies $\tau(G) \leq 2\nu(G)$, where $\nu$ counts pairwise edge-disjoint triangles and $\tau$ is the fewest edges whose deletion leaves the graph triangle-free. Puleo had proved it for maximum average degree below 7. Proved here for every graph of maximum degree at most seven, crossing the equality boundary of Puleo's sparsity theorem.

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probability-statistics / Statistics

Universal Multiplicative FDR Bound for Benjamini-Hochberg

The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal multiplicative constant. It does not: there are finite Gaussian models whose FDR divided by $q$ diverges as $q \downarrow 0$, with an explicit two-sided lower bound $q\sqrt{\log(1/q)}/(2\sqrt{\pi}) + 0.6493 q + o(q)$.

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analysis / Partial Differential Equations

Rivière’s regularity question for critical $n$-Laplace systems with antisymmetric potentials

Let $n>2$. We construct a map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$ that is discontinuous at the origin and smooth on the punctured ball $B^n \setminus \{0\}$, together with an antisymmetric potential $\Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n)$ such that $-\mathrm{Div}(|\nabla U|^{n-2}\nabla U)=\Omega\cdot |\nabla U|^{n-2}\nabla U$ in $D'(B^n)$. This gives a negative answer to a regularity question posed by Rivière. Our potential admits the Lorentz-space regularity $\Omega \in \bigcap_{q>2}L^{(n,q)} \setminus L^{(n,2)}$. In addition for given $1<p<\infty$ we can enforce $\nabla U \in L^{(n,p)}$ but $\nabla U \notin L^{(n,1)}$. The construction does not give a counterexample to regularity for weakly $n$-harmonic maps or for higher-dimensional $H$-systems. The example was generated by ChatGPT 5.6 Sol on August 5, 2026. The work itself was written by the author and thoroughly reviewed to ensure its correctness.

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theoretical-computer-science / Average-case complexity

The Polynomial-Time Low-Degree Conjecture

The low-degree conjecture predicts that when the low-degree advantage between a planted distribution and a uniform null distribution stays bounded, no polynomial-time algorithm can distinguish them. It is false. There is a planted distribution that agrees with the null through the relevant degree, is invariant under vertex relabeling, and is nevertheless distinguished in polynomial time by a rank argument.

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combinatorics / Algebraic graph theory

Babai's Minimal Cayley Graph Problem

A Cayley graph is minimal when no proper subset of its connection set generates the group. Babai asked whether minimal Cayley graphs have bounded chromatic number. Resolved negatively: finite minimal Cayley graphs exist with arbitrarily large chromatic number.

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mathematical-physics / Dynamo theory; spectral PDE

Autonomous Lipschitz Fast Dynamo on the Three-Torus

Does there exist a single real-valued, divergence-free, time-independent Lipschitz velocity field $u\in W^{1,\infty}(\mathbb T^3;\mathbb R^3)$, chosen independently of magnetic diffusivity, that is a fast dynamo for the kinematic induction equation on the flat three-torus? The author constructs such a field and constants $\varepsilon_0,\gamma_0>0$ such that, for every $0<\varepsilon\le\varepsilon_0$, the induction operator has an eigenvalue $\lambda_\varepsilon$ with $\operatorname{Re}\lambda_\varepsilon\ge\gamma_0$. Thus every sufficiently small diffusivity admits a nonzero real divergence-free magnetic field with exact exponential $L^2$ growth. The velocity is Lipschitz but not $C^1$, so this settles only the Lipschitz regularity variant; Arnold's smooth autonomous fast-dynamo problem on $\mathbb T^3$ remains open.

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algorithms-optimization / Exact exponential algorithms

Counting Linear Extensions Below the $2^n$ Barrier

Koivisto asked at Dagstuhl in 2013 whether the linear extensions of an arbitrary $n$-element poset can be counted exactly in time $O^*(c^n)$ for some $c < 2$. Yes: a deterministic exact algorithm runs in $O^*(1.89^n)$, breaking the $2^n$ barrier for the general problem.

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mathematical-physics / Spin glasses; probability

The Gardner Transition in the Ising pure $p$-spin glass

For the Ising pure $p$-spin glass with $p\ge3$, Gardner predicted in 1985 that the Parisi measure passes through two transitions as the inverse temperature $\beta$ grows: replica symmetric (RS), then one-step replica symmetry breaking (1-RSB), then full replica symmetry breaking (FRSB). The author's earlier paper established the RS phase for $0<\beta\le\beta_1^p$ and the 1-RSB phase on a nonempty interval immediately above $\beta_1^p$, leaving the rest of the phase diagram open. This sequel claims the remainder: a unique second critical inverse temperature $\beta_2^p>\beta_1^p$, with the measure 1-RSB throughout $\beta_1^p<\beta\le\beta_2^p$, and for $\beta>\beta_2^p$ supported on $\{0\}\cup[\underline q,\overline q]$ with a smooth density on the interior, hence FRSB.

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combinatorics / Discrete geometry

The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique

The big-line-big-clique conjecture of Kára, Pór and Wood asserts that for all $k, \ell$ there is an $n$ such that every finite point set of at least $n$ points contains $\ell$ collinear points or $k$ points that pairwise see each other. True for $\ell = 4$, $k = 6$, the first case left open: every finite point set of size at least $10^{11055931}$ has four collinear points or six pairwise visible points.

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algorithms-optimization / Approximation algorithms

The Optimal Approximation Ratio for Permanents of PSD Matrices

What is the best deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix? Resolved up to lower-order terms in the exponent: an explicit concave maximisation $\widehat P(A)$ satisfies $e^{-\gamma n}\widehat P(A) \le \mathrm{per}(A) \le \widehat P(A)$, giving a deterministic $e^{(\gamma+\varepsilon)n}$-approximation for every $\varepsilon > 0$ and matching the known $e^{(\gamma-\varepsilon)n}$ hardness, where $\gamma$ is the Euler-Mascheroni constant.

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analysis / Harmonic analysis

Sparse domination implies convex body domination

Nazarov, Petermichl, Treil and Volberg conjectured that scalar sparse domination should imply convex body domination for the corresponding coordinate-wise vector-valued extension. More precisely, if a bilinear form $\Lambda$ admits an $(r,s)$-sparse bound, then its extension to $\mathbb C^n$-valued functions should admit an $(r,s)$-convex body sparse bound. Laukkarinen and Lorist prove this implication for $1\leq r,s<\infty$ with $1/r+1/s>1$, in particular for the classical $(1,1)$-sparse setting.

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quantum-information-computing / Quantum Information, Bosonic Channels

Bosonic Quantum Communication Beyond the Thermal Threshold

Holevo and Werner's 1999 lower bound on the quantum capacity of the bosonic thermal attenuator comes from thermal inputs. Is it optimal? The paper proves it is exactly the supremum over single-mode Gaussian states, then exhibits a non-Gaussian state that beats it, giving positive quantum capacity in a region where every single-mode Gaussian input yields none.

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number-theory / Number theory

The Banks-Martin Conjecture on Primitive Sets

Banks and Martin conjectured in 2013 that for a primitive set $A$ and any set $Q$ of primes, the Erdos sum of the members of $A$ composed only of primes in $Q$ is at most the corresponding sum over $Q$ itself. The unrestricted form turned out to be false once $Q$ is allowed to contain $2$; Lichtman proposed a revised form restricted to odd primes. That revised conjecture, long viewed as a unifying master theorem for the area, is proved here.

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combinatorics / Symbolic Dynamics, Tilings

A Counterexample to Nivat's Conjecture for a Non-Convex Window

The paper constructs an exact cluster $F\subseteq\mathbb{Z}^2$ of cardinality 8 with full affine span and an $F$-tiling whose orbit closure contains no 1-periodic $F$-tiling, giving a non-degenerate counterexample to Nivat's conjecture for non-convex windows. This answers negatively a question of Kari and Moutot from 2023.

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theoretical-computer-science / Algorithms; derandomization

Bipartite Exact Matching in P

The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly $t$ red edges. Introduced by Papadimitriou and Yannakakis in 1982, it has been in randomized polynomial time since Mulmuley-Vazirani-Vazirani (1987) while membership in P stayed open for four decades. The paper claims a deterministic polynomial-time algorithm, replacing probabilistic amplification with deterministic evaluations.

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combinatorics / Extremal combinatorics

The Frankl-Peng-Rodl-Talbot Question on Turan Density Intervals

Frankl, Peng, Rodl and Talbot asked in 2007 whether the set of Turan densities of families of $r$-graphs contains intervals. It does: for every $r \ge 3$ the set contains non-degenerate intervals, including one of the form $[1-\delta_r, 1]$.

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combinatorics / Graph theory

The Erdos-Hajnal High-Girth Subgraph Conjecture

Erdos and Hajnal asked whether $h_r(G) = \max\{\chi(H) : H \subseteq G,\ \mathrm{girth}(H) \ge r\}$ tends to infinity as $\chi(G)$ does, for every fixed $r \ge 4$. It does in every fixed polynomial edge-density regime.

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differential-equations / Evolution equations

Lions' Maximal Regularity Problem at the Half-Holder Endpoint

Lions asked whether the variational solution of a non-autonomous divergence-form problem has maximal L2-regularity under Holder continuity in time of the coefficients. Disproved at the half-Holder endpoint: a bounded, uniformly elliptic, real scalar coefficient, half-Holder in time and arbitrarily close to the heat equation, whose Lions solution has a time derivative that is not square integrable.

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combinatorics / Extremal graph theory

Erdős Problem #146: Degeneracy Conjecture

If $H$ is bipartite and $r$-degenerate, is $\mathrm{ex}(n;H) \ll n^{2-1/r}$ (a \$500 Erdős-Simonovits prize conjecture)? A counterexample refutes the degeneracy conjecture.

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