Papers

We show that there exists a set $S \subset \mathbb{Z}^2$ containing no four points on a circle or a line such that $|S \cap [n]^2| = \Omega(n)$ as $n \rightarrow \infty$. Since any no-four-on-a-circle set in $[n]^2$ has size $O(n)$, this resolves (up to a constant) a question raised by the current authors and Keevash concerning the density of extensible no-four-on-a-circle constructions. Our construction is based on weighted random sampling from the integer lattice followed by careful deletion.

We prove that the Bézout inequality for mixed volumes characterizes simplices among full dimensional convex bodies in every dimension, resolving a conjecture of Soprunov and Zvavitch. We give two separate proofs of the conjecture. Along the way, we also prove characterizations of simplices in terms of longest chords or relative inradii.

In 2025, the affine dual curvature measures $\widetilde{C}_m^a(K,\cdot)$ of convex body $K$ were introduced by Cai, Leng, Wu, and Xi, and the even Minkowski problem for the affine dual curvature measures was solved. In this paper, the necessary and sufficient condition of affine dual Minkowski problem for general measures with $m>1$ are proposed.

We study metrics whose distances on distinct two-point subsets are algebraically independent over the rationals. We prove that every compatible metric on a strongly zero-dimensional metrizable space of cardinality at most continuum can be uniformly approximated by compatible metrics with this property. If the space is completely metrizable, the approximating metrics can also be chosen complete. For every $\sigma$-compact metrizable space, the metrics with algebraically independent distances form a $G_\delta$ set in the uniform topology. We also study rigid metrics, whose only bijective self-isometry is the identity. On every locally compact Polish space, rigid proper metrics form a $G_\delta$ set among proper compatible metrics. Using a theorem of Niemiec, we obtain uniform density of rigid metrics on compact metrizable spaces with at least three points. Passing to compact completions then yields rigid approximations of every totally bounded compatible metric on any space with at least three points.

Wasserstein barycenters provide a notion of weighted mean for probability measures on a metric space. We prove that the barycenter curvature-dimension condition ${\rm BCD}(K,N)$ is equivalent to ${\rm RCD}(K,N)$ for $K\in \mathbb R$ and $1<N<\infty$. This gives a new characterization of Riemannian curvature-dimension spaces by entropy inequalities at barycenters of finite families of measures. As a byproduct, we introduce an almost ${\rm BCD}$ condition that allows an additive error in the entropy inequality. It is stable under measured Gromov-Hausdorff convergence and implies essential non-branching when the error is sufficiently small. The resulting class contains non-Riemannian Finsler spaces. Within the ${\rm BCD}$ framework, this answers an open problem posed by Ambrosio in his 2018 ICM survey. We further bound the failure of the parallelogram identity for cotangent norms in terms of the entropy error.

A unit ball is surrounded with a ring of $n$ equal infinite circular cylinders. Each cylinder touches the ball and its two neighbors. Can one more cylinder of the same radius be added if the configuration may be rearranged? Can the $n$ cylinders instead be enlarged, and by how much? We prove that insertion is impossible for $3\le n\le7$ and possible for $n\ge9$. The case $n=8$ remains open. The case $n=6$ is Kuperberg's problem from 1990, which was resolved recently. We prove that the ring of four cylinders cannot be expanded. This settles a conjecture of Ogievetsky and Shlosman from 2018. For $n\ge5$ the largest possible expansion radius remains unknown.

We prove that every equal-weight spherical $4$-design on $\mathbb{S}^2$ has at least twelve points. Since the regular icosahedron is a spherical $5$-design, this determines the exact minimum$$N_4(\mathbb{S}^2)=12;$$equivalently, no such design has $9$, $10$ or $11$ points.The proof is part of a finite-defect theory. If a spherical $2m$-design has corank $c=N-\dim P_m$, its Naimark complement consists of unit vectors $u_x\in\mathbb{S}^{c-1}$ forming a spherical $2$-design and satisfying the exact coupling$$u_x\cdot u_y=-\frac{K_m^{(d)}(x\cdot y)}{c}\qquad(x\ne y).$$This gives a pairwise kernel bound, an antipodal lower bound, Cayley-Bacharach information, and uniform lower bounds for multiplicative-relation spaces. In corank one the design splits into two equal spherical $m$-designs. We derive a residue formula for its signed Schoenberg coefficients; the coefficient of degree $m+3$ is negative exactly when $3\le d\le m+1$, excluding corank one throughout that range. At strengths four and six the only examples in any dimension are the regular hexagon and octagon, respectively.In corank two the complement is a circle Gale frame. Multiplication by its phase forces at least $d-1$ linear-quadratic aliases and yields exact norm and socle identities in every dimension. For eleven nodes on $\mathbb{S}^2$, two aliases produce a real harmonic cubic and a Hermitian quartic matrix. A matrix-valued Cayley-Bacharach argument eliminates the generic branch; the exceptional branch reduces to a Pauli normal form and contradicts the second moments. In dimensions $d\ge4$ the corank-two problem remains open; we identify a forced quadratic socle as the obstruction to extending the present argument.

We study the maximal multiplicative persistence of $k$-cycles in random Čech complexes. Let $f:\mathbb{R}^d\to\mathbb{R}$ be a probability density function, let ${P}_n$ be a Poisson process with intensity $nf$, and let $\Pi_{k,n}$ denote the largest death-to-birth ratio among all non-essential $k$-cycles ($1\le k \le d-1$). For the uniform distribution in the unit hypercube, it was proved in [9] that $\Pi_{k,n} = \Theta\left( \left(\frac{\log n}{\log\log n}\right)^{1/k}\right)$. In this paper we sharpen and extend this result to a law of large numbers, for a broad class of distributions. Most significantly, we show that the limiting constant depends only on $d$ and $k$, and not on the probability density $f$. Thus the extreme value of multiplicative persistence exhibits a universality phenomenon. We show that the limiting constant is determined by the asymptotic covering density of the $k$-dimensional sphere. Our proof identifies the geometric mechanism underlying maximal cycles, a persistent isoperimetric inequality, which gives sharp bounds on the number of points needed to generate a highly persistent cycle. By combining covering-density estimates with isoperimetric inequalities, we show that this minimum is asymptotically attained by efficient coverings of a $k$-sphere. A key ingredient is a geometric measure theory argument that uses compactness to relate discrete covering counts to the volume of a limiting cycle.

The Pachter-Sturmfels phylogenetic rank of a graph G is the minimal number of metric trees needed to embed G isometrically. Here, all edges of G are of length one and the product of metric trees is endowed with the supremum norm. We develop both a greedy and an exact algorithm for computing phylogenetic ranks. Using our algorithms, we construct a database of phylogenetic ranks which includes all graphs on 6 and 7 vertices. In particular, we exhibit examples disproving that the phylogenetic rank is hereditary, bounded by $\lceil \frac{n}{2} \rceil$, and a generalised 4-point conjecture by Pachter and Sturmfels. In addition, we show that the phylogenetic rank is subadditive under 1-sums and certain 2-vertex-sums, and that it is trivially upper bounded by n-1, where n denotes the number of vertices. We also provide a complete classification of graphs with phylogenetic rank 1 and construct several infinite families with phylogenetic rank $\lceil \frac{n}{2} \rceil$.

We use metric functionals to build a weak topology associated with $d$-weak convergence on metric spaces. We also show the existence of a sequence in the space $C[0,1]$ that is unbounded in the sup-norm yet converges $d$-weakly. This points out an error in an earlier work.

We study the Hausdorff dimension of projections of level sets of Lyapunov exponents for affine iterated function systems. Assuming that the tuple of linear parts is typical, we obtain a variational formula in terms of pressure, topological entropy, and the Lyapunov dimensions of invariant measures corresponding to Lyapunov exponents. In $\mathbb{R}^3$, we show that this value equals the Hausdorff dimension of the canonical projection of the level set under the strong open set condition. We further extend the result to arbitrary dimensions, showing that the same formula holds for Lebesgue-almost every choice of translation vectors under a suitable contraction assumption.

We prove that there is an absolute constant $c>0$ such that for every finite $(\alpha,M)$-Ahlfors–David regular set $A\subseteq[N]$ with $0\leq\alpha<1$ and $M\geq2$, we have the sumset estimate \[|A+A|\geq |A|^{1+c(1-\alpha)/\log M}.\] We also prove a corresponding statement for Ahlfors–David regular sets in $[0,1]$. The proof combines a multiscale entropy decomposition with inverse results from additive combinatorics, showing that Ahlfors–David regularity forces a definite entropy gain at each scale. The dependence $1/\log M$ in the exponent is of optimal order.

Socolar and Taylor asked for a single, simply connected three-dimensional prototile that forces nonperiodicity by shape alone, admitting no weakly nonperiodic tiling; the Schmitt-Conway-Danzer biprism and the three-dimensional Socolar-Taylor tile admit screw motions or a periodic stacking direction. We exhibit a rational polyhedral $3$-ball $Q$, which we call Chair44 (R44): a seven-cube chair whose $24$ exposed unit panels carry tiny square-pyramid features, and prove, as a proof submission, that $Q$ admits tilings of $\mathbb{R}^3$ by congruent copies, reflections allowed, and that every such tiling has no translational period and a symmetry group of order at most $24$; every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The solid was designed to a reading of the aperiodic-monotile phenomenon reached with the Six Birds emergence calculus (Section 3.3), and the construction turns on a single finite test, checked by machine: the tile's own contact rule survives coarsening, so that the decoded parent tiling obeys the tile's rule and no other. The proof combines a written geometric argument, that the features force every tiling onto a registered lattice, with exhaustive finite enumerations; the companion census is replayed by two independent implementations, every finite gate is kernel-checked in Lean 4 (modulo a named compiler hook per native_decide theorem), and the written geometric lemmas and the logical assembly are Lean theorems as well, so that the theorem is kernel-checked modulo the named compiler hooks; the written proofs remain as exposition.

We consider the integrable dynamics of a planar Kepler billiard in the plane bounded by a branch of a hyperbola focused at the Kepler center. As a sequel to our previous work \cite{JZ2}, we use the existence of the foci-caustic circle to identify an elliptic curve on which the dynamics is linearized and we analyze Cayley's condition on $n$-periodic orbits in this setting. By taking various limits also we discuss the dynamics of planar Kepler billiards with boundary being a focused parabolic arc, as well as a straight line. In this last case, our approach offers a different way to deduce the elliptic curve as in \cite{Felder}.

We introduce a codimension one construction of coarse Ricci curvature in Lorentzian geometry. Using the $1$-Lorentz-Wasserstein distance $\ell_1$, we compare probability measures supported on small spacelike hypersurfaces through nearby events and recover, in a precise asymptotic regime, the ambient Ricci curvature in future-directed unit timelike directions, with a universal dimensional prefactor reflecting this codimension one construction. The construction echoes the Raychaudhuri equation, which relates the evolution of spatial volume expansion along timelike geodesics to Ricci curvature. Our quantitative estimates rely on transport maps with controlled displacement, constructed through a Moser-type flow. At leading order, the slice construction is insensitive to smooth weights. By smearing the spacelike slices into thin timelike tubes and calibrating the temporal and spatial scales, we recover the timelike Bakry-Émery tensor $\mathrm{Ric}+\mathrm{Hess}\,V$ associated with the weighted reference measure $\smash{\mathfrak{m}=\mathrm{e}^{-V}\,\mathrm{vol}_g}$.

We introduce centro-sectional measures with parameters q,m for log-concave functions on Rn, defined in terms of the q-th moments of their Radon transforms with respect to the Haar measure on m-dimensional subspaces, where m=1,...,n-1, and establish the corresponding variational formulas. Our measures generalize the notion of dual curvature measure if q=1, and are related to the Sine transform if q=2. In line with the coarea formula for log-concave functions as BV functions, the variational formulas give rise to the Euclidean centro-sectional measures and the spherical centro-sectional measures. In the symmetric setting, we solve the associated even functional centro-sectional Minkowski problem, which asks which pairs of measures can arise as the centro-sectional measures of an even log-concave function.

We study the maximizers of the generalized volume product \[ \gamma_\sigma^n(A)\,\gamma_\sigma^n(A^\circ) \] among all measurable subsets $A\subset\mathbb{R}^n$, where $A^\circ$ denotes the polar set of $A$, and where $\gamma_\sigma^n$ denotes the centered Gaussian probability measure on $\mathbb{R}^n$ with covariance $\sigma^2 I_n$, $\sigma>0$. It turns out that the maximizers depend on $\sigma$. We prove that they exist and are convex bodies. In dimension $n=1$, we find the exact form of the maximizers. In dimension $n\ge 2$, we show that they are smooth bodies of revolution whose support function satisfies a certain differential equation. Moreover, for $\sigma^2 \le \frac{1}{{n}}$ we show that the Euclidean unit ball is the unique maximizer, while this is no longer the case for $\sigma^2\ge {\frac{2}{n+1}}$. In dimension $n=2$, we close the gap by showing that the Euclidean unit ball is the unique maximizer for $\sigma^2 \le \frac{2}{3}$.

Chatterjee and Sloman proved that a bounded measurable similarity function with sufficiently small average Gromov hyperbolicity admits a tree representation with small mean approximation error. Their argument uses a weighted version of Szemerédi's regularity lemma and does not yield explicit quantitative bounds. Here, we establish a tighter relation between average hyperbolicity and mean tree approximation error. For a similarity function $s:S\times S\to[0,b]$, we prove that $$\operatorname{Tree}(s) \leq (63/e)^{1/3} \sqrt[3]{b^2 \operatorname{Hyp}(s)} \leq 2.8512 \sqrt[3]{b^2 \operatorname{Hyp}(s)}.$$ The proof uses a simple pivoting construction inspired by \KwikCluster. We also discuss the optimal dependence on average hyperbolicity, including a square-root lower bound, and connections with ultrametric fitting.

We give an elementary and combinatorial proof of the spherical Hadwiger theorem, which states that every continuous, rotation-invariant valuation on the space of closed convex cones in $\R^n$ is a linear combination of the conic intrinsic volumes. Our proof uses a signed orthoscheme decomposition. Its vanishing step requires only integrability of an orthoscheme restriction of the residual valuation, a weaker regularity condition than continuity.

Answering a question of Chepoi and Hagen, we give two constructions of bounded degree CAT(0) square complexes that cannot be isometrically embedded into any finite product of trees. The first is based on Burling graphs and has degree at most five, which is optimal. These square complexes topologically embed into $\R^3$ since we further prove that every CAT(0) square complex whose vertices have degree at most five can be topologically embedded into the product of a line and a star. It turns out that in this setting, Burling graphs are the only obstruction: we prove that CAT(0) square complexes that topologically embed into $\R^3$ and whose crossing graph forbids some induced Burling graph can be isometrically embedded into a finite product of trees. Our second CAT(0) square complex has degree at most six and its crossing graph forbids an induced Burling graph. Of independent graph-theoretic interest, along the way we construct intersections of chordal and interval graphs with arbitrarily large girth and chromatic number. We also discuss connections to nice labellings of event structures.

The Thom-Levine theorem is a classical method for proving the triviality of deformations by integrating suitable vector fields. In this note, we show that the converse of this criterion does not hold in the bi-Lipschitz category. More precisely, we give an example of a smooth one-parameter deformation of a real function germ in two variables which is bi-Lipschitz trivial but not strongly bi-Lipschitz trivial. The central germ has an isolated critical point, though it is not finitely determined.

We survey recent developments on the null energy condition (NEC) from the perspective of optimal transport, with emphasis on non-smooth Lorentzian geometry. After recalling the role of classical energy conditions in general relativity, we discuss the relation between Ricci curvature and entropy convexity in optimal transport. The main focus is the authors' optimal-transport characterization of the NEC. In the smooth setting, the NEC is equivalent to displacement concavity of the Shannon entropy power along suitable null-geodesic transport plans supported on null hypersurfaces and measured with respect to rigged measures. We review the geometric ingredients of this characterization and applications, including a weighted Hawking area theorem. We then turn to the synthetic theory introduced in a second paper of the authors. Synthetic null hypersurfaces are defined using achronal boundaries, gauge functions, and reference measures in topological causal spaces. In this framework, the displacement-concavity characterization becomes a synthetic curvature condition, $\mathsf{NC}^{e}(N)$, meaningful without differentiable structure. We discuss compatibility with the smooth theory, invariance under natural changes of gauge and reference measure, and stability under convergence. Finally, we present a synthetic version of Penrose's singularity theorem for continuous Lorentzian metrics. These results indicate that geometric and causal consequences of the NEC persist beyond the smooth setting and highlight optimal transport as a framework connecting curvature, causality, and singularity formation in Lorentzian geometry.

Grünbaum asked whether every planar convex body admits, for every $0\leq t\leq 1/4$, two orthogonal lines cutting it into pieces with cyclically ordered areas $t,t,1/2-t,1/2-t$. Bárány posed the analogous question for well-behaved planar measures and conjectured that the answer there is negative. We confirm Bárány's conjecture in a particularly robust form: for every fixed $0<t<1/4$ we construct smooth, strictly positive, centrally symmetric, strongly log-concave measures arbitrarily close to the standard Gaussian for which the prescribed partition does not exist. In contrast, we prove that the partition exists for every $t$ whenever the measure is invariant under an orientation-reversing affine involution. We also exhibit a $96$-point counterexample for which no pair of perpendicular lines produces cyclic counts $8,8,40,40$.

Using a novel effective non-asymptotic concentration inequality, we establish a distribution-uniform generalization of Sturm's strong law of large numbers for inductive means of $L^1$-sequences of i.i.d. random variables on (separable) Hadamard spaces. Building on that, we establish a strong law of large numbers for integrable random monotone vector fields on (separable) Hilbert-Hadamard spaces, that is Hadamard spaces where all tangent cones isometrically embed into Hilbert spaces, extending a previous result of Salim set in Hilbert spaces. We use this latter strong law to establish a probabilistic Lie-Trotter-Kato formula for the gradient flow generated by an integral function over (separable) Hilbert-Hadamard spaces where all tangent cones are actually full Hilbert spaces. This application leverages the previous Lie-Trotter-Kato formula established for gradient flows of sums of convex functions by Stojković together with a result relating resolvent and gradient flow convergence established by Bačák, as well as a new result on the interchangeability of the subdifferential and the integral, which we establish for $L^2$-Lipschitz integrands. The last ingredient provides, to our knowledge, the first nonlinear version of (a particular case of) a previous result due variously to Ioffe and Tikhomirov, Levin, Hiriart-Urruty, Thibault as well as Rockafellar and Wets. Throughout the paper, we highlight various remaining open problems.

We give an explicit example of a $W^{*}$-bundle $M$ over a compact, metrizable space $K$, which has each fiber $M_{p}$ a $\textrm{II}_{1}$-factor with separable predual, and which satisfies the following negation of the generator problem: given any finite family $a_{1},\cdots,a_{n}\in M$ of continuous sections, there is a $p\in K$ (depending upon that family) so that $a_{1,p},\cdots,a_{n,p}$ do not generate $M_{p}$ as a von Neumann algebra. More generally, if $N$ is a sub-bundle of $M$ with the property that each fiber is hyperfinite, or has a Cartan, or is generated by two commuting diffuse subalgebras, or has diffuse central sequence algebra, or is generated by a single sequential commutation orbit, then given any finite family $a_{1},\cdots,a_{n}\in M$ of continuous sections, there is a $p\in K$ (depending upon that family) so that $a_{1,p},\cdots,a_{n,p}$ together with $N_{p}$ do not generate $M_{p}$. We discuss implications for the generator problem for von Neumann algebras: e.g. there is no “continuous" way to take countably many generators for a tracial von Neumann algebra with separable predual and produce a single generator, at least if such a procedure works for all von Neumann algebras simultaneously.

We construct a metric 2-sphere of finite Hausdorff 2-measure that is doubling, linearly locally connected and Loewner but cannot be mapped to the standard sphere by a quasisymmetric mapping. Our result gives a negative answer to a question of Ntalampekos.

We prove that if a geodesic metric space contains no $c$-fat theta curve for some $c>0$, then it is $(1,K)$-quasi-isometric to a cactus graph, where $K$ depends only on $c$. Using a coarse characterization of cacti in terms of $c$-fat theta curves this implies that every geodesic metric space quasi-isometric to a cactus is $(1,K)$-quasi-isometric to a cactus graph.

In 2006, Alexander Soifer conjectured that one cannot fully cover a square of side length $> n$ with $n^2 + O(1)$ unit squares. For small $n \in \{2, 3\}$, in 2009, Janusz Januszewski proved that it is impossible to fully cover a square of side length $> n$ with exactly $n^2 + 1$ unit squares. Recently in 2023, Baek and Lee proved that it is impossible to fully cover an equilateral triangle of side length $> n$ with exactly $n^2 + 1$ unit equilateral triangles whose sides are parallel to it. There were also some progress on a related problem: what is the largest square with side length $S(k)$ that can be fully covered by $k$ unit squares. However, there have been no improvement on the original conjecture. In this work, new tools have been developed that led to the proof of this conjecture for $n = 4$, with potential applications to related problems.

We study packings of annuli of a common width, allowing each ring to nest inside the hole of a larger one. The objectives of maximizing contact area and cardinality diverge: area is superadditive in the radius, cardinality is not. Under superincreasing radii, every descending greedy maximizes every positive, strictly increasing, superadditive objective. More strongly, any choice among feasible containers yields the lexicographically maximal feasible set, for containers of arbitrary shape in every dimension. This placement irrelevance holds unconditionally for at most three rings and fails at four in disks and squares; twin instances exclude every universal rule based only on the observable state. Write $\rho=\max_i(\sum_{j>i}r_j)/r_i$. The additive model has threshold exactly $1$. For disks we prove the exact global threshold $\tau=\varphi$, with no failure at $\rho\le\varphi$, for every finite inventory, even with independent hole radii. The key geometric theorem states that, under golden tail bounds, an entire disk list fits a circular container if and only if its three largest disks fit; this supplies the uniform exchange of parents that the threshold proof needs. The Tribonacci constant $T\approx1.83929$ remains the exact floor of a rigid subfamily. A dimension-reduction lemma transfers spherical sharpness results to all dimensions $d\ge2$, and a separate argument proves the golden threshold for at most five rings in those dimensions. For square pans, a Cartesian confinement criterion gives twins and a family proving $\tau_{\square}\le Y\approx1.6845$; its optimality is open. For independent holes, the exact universal area guarantee under $\rho\le\kappa<1$ is $\min(1,\kappa^{-2}-1)$, with threshold $1/\sqrt2$. The repository has 122 Lean theorems. Euclidean geometry, forest assembly and continuity remain written proofs; numerical checks do not substitute for them.

We prove that every closed Lipschitz curve in Euclidean space extends to a Lipschitz disc of least area, with Lipschitz constant controlled by that of the boundary curve. Our proof gives a new approach to the classical Douglas–Radó solution of the Plateau problem but accommodates arbitrary self-intersections and avoids Sobolev maps. The argument extends to metric spaces with curvature bounded above in the sense of Alexandrov.