Papers

We prove that every number field of degree $n$ and absolute discriminant $D$ has a defining integral polynomial of degree $n$ whose coefficients are all $O_n(\sqrt D)$ in size, thus resolving a 1998 conjecture of Ruppert. Our proof compares ideal lattices with their trace duals, via which we handle all number fields with the exception of CM fields whose maximal totally real subfields have small discriminant. We then treat the remaining fields by averaging completed Epstein zeta functions over all possible ideal classes and archimedean scalings, for which we make critical use of Hecke's formula and the subconvexity theorem of Michel and~Venkatesh. Our proof does not use the Quasi-Riemann Hypothesis.

We associate a canonical operator to every linear recurrence sequence admitting a distinct-root Binet representation \(\mathrm{U}_n=\sum_{j=1}^{k}A_j\lambda_j^n.\) The construction is based on the spectral Binet operator \(\mathcal{B}_\mathrm{U}=\sum_{j=1}^{k}A_jT_{\lambda_j}, \) where \(T_{\lambda_j}f(x)=f(\lambda_jx)\) is the dilation operator associated with the characteristic root \(\lambda_j\). The corresponding recurrence derivative is defined by \(\mathrm{D}_\mathrm{U}=x^{-1}\mathcal{B}_\mathrm{U}.\) Its fundamental property is the monomial identity \(\mathrm{D}_\mathrm{U}(x^n)=\mathrm{U}_nx^{n-1},\) which establishes a direct correspondence between recurrence sequences and degree-lowering operators. We develop the operator theory of \(\mathrm{D}_\mathrm{U}\), including iteration, a spectral product rule, the associated Leibniz defect, and a quotient rule. The construction reveals a natural bifurcation determined by the value \(\mathrm{U}_0\), producing the Ward and non-Ward regimes as intrinsic operatorial phenomena. The product structure is shown to be governed by \(k-2\) residual spectral channels, leading to notions of spectral cancellation and spectral balance. As an application, we obtain a complete qualitative classification of second-order recurrence calculi. Using Horadam sequences as a universal second-order model, we prove that every such calculus is determined by the invariant triple \((\mathrm{W}_0,\Delta, Z_\mathrm{W}),\) where \(\mathrm{W}_0\) identifies the Ward/non-Ward type, \(\Delta\) determines the spectral geometry, and \(Z_\mathrm{W}\) determines regularity or singularity.

We show that the percolation model induced by the double random current representation of the Ising model on $\mathbb Z^d$, with $d\geq 3$, satisfies local uniqueness of the macroscopic cluster with high probability for every $\beta>\beta_c$, uniformly in boundary conditions (i.e. sources). Such a result is sometimes called supercritical sharpness in percolation theory and yields a detailed description of the model via standard renormalisation techniques. As applications, we show the exponential decay of truncated correlations for the Ising model and exponential mixing for the FK-Ising measure, results previously obtained by Duminil-Copin, Goswami and Raoufi. However, our proof is considerably simpler, as it replaces the use of a highly technical multi-valued map principle with a simple sprinkling argument.

We show that any Serrin domain in $\mathbb{R}^2$ with two boundary components is a member of the canonical unduloidal family

Given a sequence $\mathbf{a}=(a_i)_{i\geq 1}$ of positive real numbers, we say that a colouring $c:E(K_n)\to \mathbb{N}$ is $\mathbf{a}$-bounded if the proportion of edges coloured $i$ is at most $a_i$, for all $i$. We write \[ \mathbf{a}\, \to\, K_k \] if, for all sufficiently large $n$, all $\mathbf{a}$-bounded colourings $c:E(K_n)\to \mathbb{N}$ contain a monochromatic $K_k$. We say that the sequence $\mathbf{a}$ is Decay-Ramsey if $\mathbf{a}\to K_k$ for all $k\ge 2$. In this short paper we introduce the question of determining which sequences are Decay-Ramsey and present some initial results.

We prove the full generating series arithmetic Siegel–Weil formula on the product of two modular curves, with arbitrarily singular coefficients. This is the case $\operatorname{GSpin}(2,2)$ in Kudla's program.

We prove that every Gromov elliptic domain $\Omega$ in a complex manifold has the following property, which we call meromorphic pseudoconvexity: each boundary point has a neighborhood $V$ such that $\Omega\cap V$ becomes Stein after removing a set which is, locally in $\Omega\cap V$, the union of the polar sets of finitely many meromorphic functions on $\Omega\cap V$. Since meromorphic functions on Hartogs figures extend to their hulls, it follows that, for a domain $\Omega=\operatorname{int}\overline\Omega$ in a Stein manifold, Gromov ellipticity is equivalent to being Oka and Stein, and also to being Oka and admitting a complete Kähler metric. This complements a theorem of Matsumura and the author that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic. As applications, we show that the Oka manifolds $\mathbb C^n\setminus\overline{\mathbb B^n}$ and $\mathbb C^n\setminus\mathbb R^n$, $n\geq2$, are not elliptic, and that no proper domain with Lipschitz boundary in $\mathbb P^n$, $n\geq2$, is elliptic.

Onsager solved the two-dimensional Ising model in zero field, but no solution is known in a magnetic field. We prove that in any dimension $d\ge2$, at every fixed dimensionless field outside a countable set, the free energy obeys no algebraic differential equation in the coupling, even locally. The obstruction arises from droplet geometry: the maximal droplet volume grows as the $d/(d-1)$ power of its boundary size, while an integer band in the pressure coefficients protects the extremal contributions against cancellation. Our result excludes Onsager-type solutions and closed forms in elliptic, hypergeometric, or Painlevé functions of the coupling at these fields.

We classify all complex generalized spin-c Killing spinors with complex Killing function $i\mu-\alpha$ on complete Riemannian surfaces, presenting an explicit list of all possible cases. We show that, contrary to previous statements in the literature, there exist complex generalized spin-c Killing spinors on Riemannian surfaces for which $\alpha\mu\not\equiv0$. We illustrate the classification by providing several explicit local, compact, and complete examples, including a complete example for which $\alpha\mu>0$ everywhere. We then move to three dimensions, proving that a spin-c Riemannian three-manifold admits a complex generalized spin-c Killing spinor with complex Killing function $i\mu-\alpha$ if and only if it is a $\mu$-exact trans-Sasakian manifold of type $(\alpha,\mu)$. In addition, we construct a geodesically incomplete Riemannian three-manifold equipped with a complex generalized spin-c Killing spinor for which neither $\mu$ nor $\alpha$ vanishes identically, and then prove that this is not possible if the characteristic vector field of the associated trans-Sasakian structure is complete. These results settle a conjecture by Große and Nakad (Results in Mathematics, 2015) stating that on two- and three-dimensional Riemannian manifolds the complex Killing function of any complex generalized spin-c Killing spinor must be either globally real or globally imaginary.

We prove that completely parabolic translation surfaces are Veech, answering a question of Smillie-Weiss.

We introduce invariants we call $k$-divisibility and $d$-dominance, which are combinatorial constraints on the minimal generators of monomial ideals. We then prove that $k$ (respectively, $d$) is an upper bound (respectively, lower bound) for the projective dimension of a $k$-divisible (respectively, $d$-dominant) monomial ideal. This leads us to an explicit minimal resolution of $k$-divisible, $d$-dominant ideals when $d=k-1$ via a Lyubeznik resolution. This framework allows us to prove a conjecture made by Alesandroni. Using these invariants, we also provide a characterization of monomial ideals whose minimal free resolution is attainable from the Taylor resolution by performing exactly one consecutive cancellation. We also prove that for all $q,k,d$ in the range $1\leq d<k \leq q-1$, there exist monomial ideals that are $k$-divisible and $d$-dominant with $q$ minimal generators.

For $H$ a hypergraph, the Hall ratio $\rho(H)=\text{max}_{X\subseteq V(H)}\frac{\lvert{X}\rvert}{\alpha(H[X])}$ is a natural lower bound for its chromatic number, and among non-empty $k$-uniform hypergraphs the smallest possible Hall ratio is $k/(k-1)$. For graphs ($k = 2$), having minimum Hall ratio $2$ forces chromatic number $2$. Nešetřil, Rödl, and Sales asked whether a similar phenomenon holds at higher uniformities. We answer this question negatively in a strong form, showing that for every $k \geqslant 3$ there are $k$-uniform hypergraphs with Hall ratio exactly $k/(k-1)$ and arbitrarily large fractional chromatic number.

We prove the spacetime Penrose inequality for asymptotically flat initial data sets $(M^n,g,k)$ in all dimensions $n\geq 3$, under the dominant energy condition, for data admitting the 'maximally parameterized form' defined in equation (4.7). In this case, the second fundamental form $k$ is locally parameterized by $n(n+1)/2$ functions, giving the same formal local functional degrees of freedom as an arbitrary symmetric $2$-tensor. The proof is based on a new 1-parameter family of generalized Schoen-Yau identities and their gauge-invariant specialization, inspired by the generalized Jang framework of Bray-Khuri [8, 9]. For maximally parameterized data, these identities produce a Riemannian comparison metric \[ \bar g=g+\beta\otimes\beta \] with nonnegative scalar curvature $R_{\bar{g}} \geq 0$. Under a natural nondegeneracy condition (5.7) at the apparent horizon, the comparison metric admits a smooth minimal completion up to $\partial M^n$. Together with suitable asymptotic control, this reduces the Penrose inequality to its Riemannian counterpart. The all-dimensional conclusion uses the Riemannian Penrose theorem of Bi-Zhu [6], together with a minimizing enclosure argument that allows singular boundaries. More generally, in the spirit of [8, 9], we note that the Penrose conjecture follows whenever a certain system of PDEs (3.18) involving $k$ can be solved, in which case our new identity (3.15) immediately implies $R_{\bar{g}} \geq 0$. A natural problem is thus to regularize our degenerate system to prove a general existence theory, thereby proving the Penrose conjecture. Essentially, we believe this is what the OpenAI paper [27] posted last week on October 6, 2026 claims to have done.

We compute minimal $\mathfrak{sl}_N$ complexes for oriented rational tangles, denoted $[\![\tau_{u/v}^{O}]\!]_N$, in terms of the topological Fukaya category of the $3$-punctured disk. This demonstrates that the complexes stabilize, allowing us to define a HOMFLY-PT homology for rational tangles.

Building on the large sieve, we prove a new sieve estimate for sets which are missing various residue classes modulo prime squares. For certain interesting residue class configurations, our bound is a significant improvement over the corresponding Johnsen-Selberg sieve inequality. Our application of this sieve estimate is motivated by Ostmann's problem. We study additive decompositions of sums of squares and show that any non-trivial sumset decomposition must consist of two sets of roughly equal size: If $A_1, A_2 \subseteq \mathbb{Z}$ with $A_1 + A_2 = \{x^2 + y^2 : x, y \in \mathbb{Z}\}$ and $|A_1|, |A_2| \geq 2$, then $\sqrt{x}/(\log x)^{9/2} \ll |A_i \cap [1, x]| \ll \sqrt{x}(\log x)^4$ for $i \in \{1, 2\}$.

The $k$-color Ramsey number of a poset $P$ is the smallest dimension $d$ such that any $k$-coloring of the $d$-dimensional Boolean lattice contains a monochromatic induced copy of $P$. For $ j\ge 2$ the $j$-diamond is the poset that consists of a maximum element, a minimum element and an antichain of size $j$. We prove that the $k$-color Ramsey number of the $j$-diamond is at most $jk$. Colorings of the Boolean lattice that assign sets of the same size the same color provide matching lower bounds for $j \in \{2,3,4\}$.

Rockafellar showed that every cyclically monotone operator is contained in the subdifferential of a convex function, and characterized the maximal ones. We establish the quasi-convex analogue of this theorem, with no assumption on the operator: in any separable Banach space, a setting we show to be sharp, every cyclically quasi-monotone operator is contained in the normal cone operator of a quasi-convex function. In finite dimension, we characterize maximal cyclically quasi-monotone operators and show that they are maximal quasi-monotone. Unlike in the convex case, the potential is not unique, and we build it from a preference relation obtained through successive preorders. As applications, we extend Afriat's theorem on revealed preferences in economics to infinite data sets, and obtain Kantorovich-type potentials for a supremal optimal transport problem.

We study the long-time validity of the inviscid limit problem by measuring when the two-dimensional ideal fluid ceases to accurately approximate a slightly viscous fluid with the same initial datum. We construct a stationary solution of the Euler equation from which the Navier–Stokes solution separates at a time scale proportional to the logarithm of the inverse viscosity, achieving the fastest separation from Euler steady states.

Many scientific and engineering applications require the global optimization of expensive black-box functions. In high-dimensional settings, however, achieving computational scalability is a major challenge. To address this challenge, we propose a distributed spatial branch-and-bound framework that exploits partially separable structure for the global optimization of box-constrained black-box problems. The proposed framework adopts a two-layer structure in which branching is performed only on a lower-dimensional set of coupling variables in an outer layer. Within each outer node, the remaining non-overlapping (private) variables, which appear independently across blocks, have their domains partitioned and explored in parallel within an inner layer, resulting in a naturally parallelizable framework that is well suited for implementation on multi-core CPU architectures. At each node, valid upper bounds are obtained via sampling, with optional refinement using a local solver, while lower bounds are computed using data-driven convex underestimators constructed over partitions of the private variables. A residual-based correction and finite-sample validity analysis provide conditions under which the partitionwise underestimators are valid lower bounds, while the proposed branching and refinement mechanism establishes convergence to an $\epsilon$-global optimum. Numerical studies on benchmark and ODE-constrained parameter-estimation problems demonstrate improvements in both sampling efficiency and computational performance relative to monolithic approaches.

Let theta be an ordinary character or a Brauer character of a finite group N. We show that theta extends to every overgroup $G=G_\theta\trianglerighteq N$ if and only if Z(N) <= Ker(theta) and the deflated character of Inn(N) extends to Aut(N)_theta. This equivalence is compatible with fields of values. A similar investigation for extensions from subnormal subgroups leads to a new characteristic subgroup of N.

For a fixed matrix-polynomial trajectory in $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$, we give a necessary and sufficient intrinsic Diophantine condition for its time averages to equidistribute at a power rate on its homogeneous orbit closure. The condition uses a lattice in the space of polynomial functions on the normalized orbit-closure group, each admitting a representative of total degree at most two in the matrix entries. These polynomial tests have a common degree bound of two, independent of the curve degree and optimal in general. The proof begins with a classification of the possible orbit-closure groups and derives lower bounds for polynomial coefficient vectors from the intrinsic condition. It combines quantitative nilmanifold equidistribution, unipotent tangent approximation, and effective flow estimates, with arithmetic separation and quantitative nondivergence controlling cusp excursions and periodic-orbit obstructions.

Edge deletion problems ask for the fewest edges whose deletion gives a graph with a specified property. We study two such problems on block graphs. First making the graph claw-free, and second making it a line graph. Since every line graph is claw-free, the second problem is at least as difficult as the first. We show that on block graphs the two optimum values are always equal, and both are attained by the line graph of a forest. The main step is a completion argument, in which starting from any claw-free solution, we put back edges to get some nice structure inside each block still maintaining the claw-free property. This reduces both edge-deletion problems to the same problem of selecting vertices within the blocks of the input graph. We solve it by dynamic programming in $O(n+m)$ time, which also produces an optimal deletion set and a forest whose line graph is what remains after the completion argument. We also extend the method to $K_{1,r+1}$-free edge deletion for every fixed $r$.

We study disjoint solution pairs $(u,D)$ to the minimal surface equation in $\mathbb{R}^n$, with $u=0$ on $\partial D.$ A conjecture of W. Meeks states that at most two such pairs can exist in $\mathbb{R}^n.$ We confirm the conjecture in two complementary cases. For $n\in\{2,3,4,5,6\},$ we prove it under a superlinearity assumption. In dimension $n=2,$ we establish the result under a suitable sublinearity assumption. We also derive estimates on the asymptotic angle of domains supporting nontrivial solutions for $n\geq 3.$

We extend Hintz's separation of variables for radiative gravitational and electromagnetic perturbations from Kerr-Newman to Kerr-Newman-de Sitter black holes. On subextremal backgrounds with nonzero rotation and charge, an explicit algebraic matrix transformation yields coupled two-component radial and angular ordinary differential systems linked by a common separation constant. The transformation is independent of the frequency and azimuthal number and applies to both signs of spin weight. For real frequencies, we establish a complete angular mode decomposition. We also translate regularity at the future black hole outer and cosmological horizons into ingoing and outgoing boundary conditions, respectively, for the separated radial systems. These results show that exact separation of the coupled radiative equations extends beyond asymptotically flat Kerr-Newman spacetime.

We prove that every mod $k$ integral cycle $T$ of sufficiently high codimension in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by a cycle which is a $\mathbb{Z}_k$-smooth submanifold $\Sigma$ of nearly the same area, up to a singular set of codimension 5. If in addition $T$ belongs to a mod $k$ homology class admitting an embedded $\mathbb{Z}_k$-smooth manifold representative, then $\Sigma$ can be taken fully regular, up to the codimension one stratum of classical singularities. This extends the smooth approximation theory to cycles with coefficients in $\mathbb{Z}/k\mathbb{Z}$ in stable regimes. Moreover, we provide examples showing the impossibility of locally approximating minimizing cycles by smooth or $\mathbb{Z}_k$-smooth submanifolds, even in homology classes with smooth or $\mathbb{Z}_k$-smooth representatives.

We establish a smoothness criterion for Fano varieties with quotient singularities. As an application, we show that if $X\to B$ is a Lagrangian fibration from a compact hyper-Kähler manifold and $B$ has quotient singularities, then $B$ is a projective space.

We prove that, for every real Lipschitz function $A$ and every integer $n\ge1$, $\|T_{A,n}\|_{L^2(\mathbb{R})\to L^2(\mathbb{R})}\le C\,\|A'\|_\infty^n$, where $T_{A,n}$ is the $n$-th Calderón commutator and $C$ is an absolute constant. The essential tool is the Coifman–McIntosh–Meyer theorem, applied to a holomorphic family of complex curves, together with the Cauchy integral formula on a contour surrounding the interval of secant slopes. We also include a simpler version of the argument that gives $O(\log n)$ and explains why the contour has the shape it has.

We prove a maximal diameter rigidity theorem for closed 3-dimensional Riemannian manifolds under a spectral lower bound for the Ricci curvature. We show that if $0<\gamma<2$, $K>0$ and $\lambda_1(-\gamma\Delta+\operatorname{Ric})\geq 2K$, then the optimal bound $\operatorname{diam}(M,g)\leq\frac{\pi}{\sqrt{K}}$ is attained if and only if (M,g) is isometric to the round 3-sphere of constant sectional curvature $K$.

In this article, we study the fractal refinement of the Carleson's problem: (1) We introduce an equivalence relation on linear dispersive PDEs on $\mathbb{R}^{n}$ based on the high-frequency oscillation of their Fourier multipliers, proving that equations in the same class share the same Hausdorff dimension of divergence sets. As applications, we estimate these dimensions for abstract dispersive families. Consequently, we recover known $1$-dimensional results for linearized Boussinesq and Beam propagations, while establishing new results for both in higher dimensions and for Klein-Gordon propagations in all dimensions. (2) Next, we study the effect of radial symmetry by obtaining dimension estimates for fractional Schrödinger equations with concave phase and radial initial data. Applying a transference principle, dimension estimates are extended to families of abstract dispersive equations. As special cases, we recover the known result for the Beam propagation and also obtain new results for the linearized Boussinesq and the Klein-Gordon propagations in the radial setting. (3) We study analogues of (2) for the $3$-dimensional real hyperbolic space by obtaining dimension estimates for a wide class of dispersive propagations: the Schrödinger, fractional Schrödinger (with convex and concave phases), Beam, linearized Boussinesq, Wave and Klein-Gordon equations. This is the first result of its kind, outside the setting of flat Euclidean geometry. Our arguments are based on an inter-spatial transference principle of geometric-analytic flavour. In the process, we also obtain a quantitative result on the singular sets of Sobolev functions, in the spirit of Žubrinić, which may be of independent interest.

For $n\geq3$, let $\overline B_n=B_n/[P_n,P_n]$ be the crystallographic quotient of the Artin braid group, with translation subgroup $\overline P_n=P_n/[P_n,P_n]\cong\mathbb Z^N$, where $N=\binom n2$. We prove that every compact connected flat $3$-manifold $M$, considered up to affine equivalence, is the base of a geometric torus bundle whose total space is a compact flat manifold with fundamental group a Bieberbach subgroup of some $\overline B_n$. More precisely, for every $n\geq n_M$, where $n_M\in\{3,4,5\}$ is an explicit optimal threshold depending on $M$, we construct a Bieberbach subgroup $\Gamma_n(M)\leq\overline B_n$ of dimension $N$ and a complete normal subgroup $K_n(M)\cong\mathbb Z^{N-3}$ consisting of translations such that $$ \Gamma_n(M)/K_n(M)\cong\pi_1(M). $$ We also determine which of the associated torus-bundle group extensions split: for the realizations constructed here, splitting occurs for exactly four of the ten affine-equivalence classes of compact flat $3$-manifolds.