Papers

We resolve the long-standing problem of Rudin concerning uniformly bounded orthonormal bases of homogeneous polynomials on complex unit spheres. Specifically, we show that for every $K \geq 2$ and $N \geq 0$ there exists a basis of degree $N$ homogeneous polynomials in $\mathbb{C}^K$ that is orthonormal with respect to the surface measure of the unit sphere of $\mathbb{C}^K$ and uniformly bounded by a constant depending only on $K$ on the unit ball of $\mathbb{C}^K$. Bourgain resolved the case $K = 2$ in 1985 and the case $K = 3$ in 2016. No cases $K \geq 4$ were previously known.

We study the geometry of semi-infinite geodesics in the Airy line ensemble. We establish sharp estimates for disjointness probabilities involving geodesics with nearby directions and endpoints in the top line of the Airy line ensemble. Moreover, we obtain a power law upper bound on the coalescence depth of two semi-infinite geodesics with same asymptotic direction and establish exponential concentration estimates for geodesic jump times more generally. We also establish pathwise Hölder $1/2^-$-continuity of Busemann profiles associated to semi-infinite geodesics with common asymptotic directions (in terms of the asymptotic direction). We characterize exceptional behaviour of semi-infinite geodesics through their Busemann profiles and establish geometric properties of the associated sets of non-coalescence and non-uniqueness. Finally, we determine the Hausdorff dimensions of collections of semi-infinite geodesics: collections with a fixed top-line endpoint and arbitrary positive asymptotic direction almost surely have Hausdorff dimension $1$, while collections with a fixed positive direction and arbitrary endpoint almost surely have Hausdorff dimension $3/2$.

We present a proof of the two-token graph reconstruction conjecture. Every finite simple graph $G,H$ of order at least three, $F_2(G) \cong F_2(H)$ if and only if $G \cong H$. We also classify the automorphism groups of two-token graphs whose connected roots have order at least five. Combining these results, we show that, for connected graphs $G,H$ of order at least five, every isomorphism $\phi:F_2(G)\to F_2(H)$ admits a factorization $\phi=\eta\circ\sigma^{(2)}$, where $\sigma^{(2)}$ is induced by a graph isomorphism $\sigma:G\to H$, and $\eta\in \operatorname{Aut}(F_2(H))$ is a product of twin-cut switches or Cartesian exchanges.

We study one-dimensional cascades of heat equations controlled through their last component. For a fixed time $T>0$, we consider the Hilbert space $\mathcal{C}_T$ of all initial data that can be steered exactly to zero at time $T$, endowed with the norm of the minimal $L^2$ control. This space can be strongly anisotropic: in explicit examples, it is incomparable with every isotropic spectral Sobolev or sub-Gaussian Gevrey scale associated with an adapted basis. For two equations, under the block Fattorini–Hautus condition, we give an almost sharp spectral description of $\mathcal{C}_T$. More precisely, in an adapted Riesz basis of generalized eigenvectors we define an explicit increasing scale $(\mathcal{V}_\tau)_{\tau\in\R}$ of Hilbert spaces such that $\mathcal{V}_{T-\varepsilon}\hookrightarrow\mathcal{C}_T\hookrightarrow\mathcal{V}_T$ for every $\varepsilon>0$. The weights of $\mathcal{V}_T$ combine heat dissipation with the loss of observation inside each double generalized eigenspace. When the coupling and control regions are disjoint, this recovers the known minimal null-control time $T^\star$ and identifies it as a transition time for controllable data: for $TT^\star$, the natural state space is properly contained in $\mathcal{C}_T$. We extend the analysis to arbitrary finite upper triangular heat cascades. The observation losses of the different directions in each spectral block may have distinct exponential orders, leading to a hierarchy of critical times rather than a single threshold. This hierarchy describes successive gains of null controllability on adapted modal subspaces and accounts for the interactions and cancellations between different coupling paths.

Generative modelling involves two complementary tasks: learning a probabilistic approximation of an unknown data distribution from observations, and designing a tractable mechanism for sampling from this approximation. We present a stochastic-analysis perspective on score-based generative models, connecting ideas from classical sampling theory, score estimation, and diffusion time reversal. We first explain how score functions link Langevin sampling with score matching, and how Gaussian smoothing leads to denoising score matching and multiscale sampling methods. We then emphasize the distinction between dynamics preserving a fixed distribution and dynamics transporting a prescribed family of marginals. This motivates a path-space treatment of diffusion time reversal. Finally, we connect the ideal reverse diffusion with practical generative algorithms.

We study the size of sets $\mathcal{B}\subseteq\mathbb{Z}_{\ge0}$ for which $\mathbb{P}+\mathcal{B}$ contains every sufficiently large integer. We obtain $B(X)\gg\log X\log\log X$, where $B(X)=\#(\mathcal{B}\cap[0,X])$ and the implied constant is absolute. This gives a quantitative form of Conjecture 2 of Ruzsa and excludes additive complements of size $O(\log X)$, as asked in one part of Erdős Problem 32. More generally, we construct a block of $\lfloor c\log X\log\log X/(B(X)+1)\rfloor$ consecutive integers outside $\mathbb{P}+\mathcal{B}$ in $[X/2,X]$, whenever this number is positive.

Quantum annealing (QA) offers a physics-inspired approach for such subproblems by mapping them onto an Ising Hamiltonian, which is algebraically equivalent to a Quadratic Unconstrained Binary Optimization (QUBO) model executable on quantum annealers or hybrid solvers. However, their practical employment in wireless systems critically depends on the specific problem formulation, on the particular variable encoding, constraint handling, solver selection, hardware embedding, and fair benchmarking. This paper develops wireless-oriented QUBO modeling guidelines for QA-assisted resource allocation. We first present a beginner-friendly toy example in which a terrestrial-aerial resource-allocation problem with a fractional signal-to-interference-plus-noise ratio (SINR) maximization objective is reformulated into a two-bit QUBO via Boolean interpolation, converted into an Ising representation, and decoded back into a wireless scheduling decision. The same scheduling is applicable to UAV, aircraft, satellite, and terrestrial links in SAGIN systems. We then review the QA foundations and solver ecosystems, including binary quadratic, discrete quadratic, constrained quadratic, and hybrid models from the perspective of NG wireless optimization. Furthermore, we discuss both binary and discrete encoding, penalty-based constraint conversion, embedding overhead, solution decoding, and feasibility verification. A QUBO Compatibility Index is introduced as a lightweight diagnostic metric for assessing the associated QUBO formulation readiness, rather than a predictor of runtime or solution quality. Finally, a case study on max-min uplink power control in cell-free massive MIMO illustrates how a fairness-oriented wireless problem can be discretized, constrained, converted into the concept of a QUBO, and evaluated against classical baselines. A host of key challenges and future directions are also discussed.

For either real or complex scalars, we improve the known upper bound for the Banach–Mazur distance between $L_\infty[0,1]$ and $\ell_\infty$, proving $d_{\operatorname{BM}}(L_\infty[0,1],\ell_\infty)\leqslant (2+\sqrt5)^2=9+4\sqrt5=17.944\ldots$. The additional input is a stabilised kernel estimate: the copy of $\ell_\infty$ in $L_\infty$ is chosen so that its kernel is an $\ell_\infty$-sum of singular two-point hyperplanes, and Cohen's bound-two construction is then applied coordinatewise. The quantitative part of the proof is an asymmetric Pełczyński-type factorisation, with the elementary optimisation included below.

In this paper we examine the special class of bipartition complete matroids, which includes free products and principal sums of matroids (Crapo and Schmitt, 2005; Bonin and Kung, 2015). A matroid $\mathcal{M}_{SQ}$ on $S \uplus Q$ is said to be $\{S,Q\}$–complete iff whenever $b^1_S \uplus b^1_Q, \, b^1_S \uplus b^2_Q,$ and $b^2_S \uplus b^1_Q$ are bases of $\mathcal{M}_{SQ},$ we also have $b^2_S \uplus b^2_Q$ as a base of $\mathcal{M}_{SQ}.$ We use the linking operation $\mathcal{M}_{SP} \leftrightarrow_{\mathcal{M}} \mathcal{M}_{PQ} :\equiv (\mathcal{M}_{SP} \vee \mathcal{M}_{PQ}) \times (S \uplus Q)$ (Theja and Narayanan, 2014) for this study. We construct the $\{S, Q\}$–augmentation of an arbitrary matroid and show that this augmentation is again a matroid, that the dual of the augmentation is the augmentation of the dual matroid and that the $\{S, Q\}$–augmentation of an $\{S, Q\}$–complete matroid leaves it unchanged. We also show that bipartition completeness is stable under minors in the appropriate bipartitioned sense. Finally, we show that free products and principal sums of matroids $\mathcal{M}_{S},\mathcal{M}_{Q}$ are $\{S, Q\}$–complete and give explicit minimal decompositions for them.

An $(r,z,g)$-mixed cage is a mixed graph that is $r$-regular in edges, $z$-regular in arcs, of girth $g$ and of minimum order $n=n(r,z,g)$. We give a general lower bound for $n(r,z,g)$. In the case $g=3$, we determine the exact value $n(r,z,3)=r+2z+1$. For $g=4$ we give two improved lower bounds and we determine the exact value $n(r,z,4)=2r+4z$ for $r\geq 2z-2$. We also present computational results, obtaining mixed cages and computational lower bounds by means of exhaustive backtracking searches with symmetry reductions. Moreover, we give an upper bound for $n(1,z,g)$ that matches all our computed values.

In the abelian sandpile model on $\mathbb{Z}^d$, a site with at least $2d$ grains topples by sending one grain to each neighbor. In a box $[-N,N]^d$ whose boundary absorbs grains, adding grains at random sites and toppling defines a Markov chain. Athreya and Járai (2004) constructed the stationary state on $\mathbb{Z}^d$ as the limit of its stationary distributions as $N\to\infty$. They asked whether, when a grain is added at the origin of a sandpile in the stationary state, the resulting avalanche of topplings is finite. Járai and Redig (2008) proved that the answer is yes for $d\geq 3$, using the transience of random walk. We prove, using a different method, that avalanches are also finite for $d=2$.

We use the theory of canonical lifts to prove the integral geometric Tate conjecture for ordinary Calabi-Yau threefolds over finite fields, and also for divisor classes on certain ordinary Calabi-Yau varieties.

We determine a universal sector of the spectrum of light operators in four-dimensional $\mathcal{N}=1$ class $\mathcal{S}$ superconformal field theories arising from M5-branes wrapped on a compact hyperbolic Riemann surface $\Sigma_2$. We work at large $N$, analysing Kaluza–Klein fluctuations about the dual Bah–Beem–Bobev–Wecht anti-de Sitter backgrounds for general values of the twist parameter $z$. To obtain these spectra, we specialise recently derived mass operators to modes that are constant on $\Sigma_2$ by imposing a suitable $z$-dependent generalised $G$-structure. We then organise the resulting eigenstates into long and protected $\mathcal{N}=1$ supermultiplets with definite flavour quantum numbers. As $z$ varies, rich patterns emerge in the spectrum, including both isolated multiplets and infinite towers of multiplets, which we characterise analytically.

We develop and analyze a projected extragradient method, preconditioned by a Riesz map, for a first-order price-formation mean field game in one space dimension. The model couples Hamilton–Jacobi–Bellman and transport equations through a global market-clearing constraint. Finite propagation justifies truncating the first-order transport equation to a bounded interval. A monotone weak formulation incorporates the mixed initial–terminal data. Its Riesz map uses $L^2$ norms for density and price directions and a space-time $H^1$ norm for value-function directions, preconditioning the residual without adding higher-order regularization terms to the MFG equations. A density-only argument rules out cocoercivity on the localized admissible set in every equivalent Hilbert metric, motivating extragradient updates. For $H(x,p)=p^2/2$, we use a monotone upwind discretization with adjoint-consistent transport and a state-constraint boundary closure. We establish a mesh-independent Lipschitz bound on bounded convex admissible sets with uniform density, slope, and price bounds. With exact metric projection onto these sets, the iteration converges to a solution of the discrete variational inequality on each fixed grid, with an admissible step-size threshold independent of both mesh sizes. Convergence under mesh refinement is not proved. On a quadratic benchmark, all reported runs reach the stopping tolerance with one fixed step size across a factor-of-16 reduction in both mesh sizes. Price errors and value-function errors measured away from low-density regions decay approximately at first order under refinement, while density errors decrease more slowly.

We announce an explicit construction of a unitary, strongly rational vertex operator algebra $U$ of central charge $8$ inside $V_{E_6\oplus A_2}$, with a ribbon equivalence $\mathrm{Rep}(U)\simeq Z(\mathcal H_3)$ realizing the Haagerup double. The vertex algebra is generated by six affine currents and three specified weight-two fields; its weight-one Lie algebra is $\mathfrak{sl}_2\oplus\mathfrak{sl}_2$, at levels $1$ and $39$. We outline the argument, which combines exact lattice calculations, an all-weight strong-generation induction and $C_2$-cofiniteness, semisimple descent for a full corner of Zhu's algebra, modular Galois symmetry, and holomorphic $E_8$ extensions.

The expressivity of a neural network $\Phi$ describes the class of functions that a network can express. The term topological expressivity refers to a family of measures particularly suited to understand binary classification and is often formulated in terms of the Betti numbers of a decision boundary $\beta (\Phi^{-1}(0))$. Despite its popularity, this measure is not as useful for regression and other problems that do not involve a decision boundary. Instead, we propose to study topological expressivity in terms of the Betti numbers of the output of a neural network $\beta(\Phi(X))$. We formally prove that a ReLU network with architecture $(d,\sum\limits_{k=1}^{d } \beta_k(3k+2),\sum\limits_{k=1}^{d}\beta_k(k+1))$ can express spaces homotopy equivalent to wedge sums of spheres with arbitrary Betti numbers. This gives upper bounds for any arbitrary connected ($\beta_0=1$) sequence of Betti numbers, but by constructing spaces homotopy equivalent to products of $N$ spheres of dimension $d_1,...,d_N$ we achieve much better bounds $(\sum\limits_{n=1} d_n, 3\sum\limits_{n=1}\limits^N d_n + 2N, \sum\limits_{n=1}\limits^N d_n + N) $ for Betti numbers given by $\prod_n (1+x^{d_n})$. Finally we conjecture a much stronger tight bound which gives the architectural bound $(d,d+1,d+1)$ for a space homeomorphic to the sphere.

The main result is an obstruction to the presence of contact regions for c-symplectic confoliations in any odd ambient dimension. As a first application, we prove a 5-dimensional analogue (under stronger assumptions) of a Reeb stability theorem for confoliations in dimension 3 due to Eliashberg and Thurston. Refining work of the first author with Cardona, we further give two new notions of contact approximations of c-symplectic confoliations in any odd ambient dimension generalizing the 3-dimensional notion of Eliashberg-Thurston. As a second application of our main theorem, we then give an example, in any ambient odd dimension, of a c-symplectic (and in fact strong symplectic) foliation that cannot be contact approximated, in either of these two senses. This relies on the property, of independent interest, that contact structures approximating (in these senses) a weakly c-symplectic confoliation are also weakly fillable. The proofs of all these statements rely on pseudo-holomorphic curve theory. Lastly, we describe several natural classes of examples of these types of contact approximations of c-symplectic confoliations.

A parallel sampler can choose at random which coordinates to generate together. For a target $p$ and the output law $q_U$ of a schedule $U$ drawn afresh for each output, we compare the schedule-averaged divergence $\mathbb{E}_U D_{\mathrm{KL}}(p\|q_U)$ with the output-law divergence $D_{\mathrm{KL}}(p\|\mathbb{E}_U q_U)$, in nats. Rounds commit independent draws from exact conditional marginals frozen at the round's start, each output's schedule is drawn before sampling from a finite mixture of schedules, and every scheduled round is charged. For the uniform distribution on binary strings of even length $n$ with $n/2$ ones, the least expected number of rounds at a fixed tolerance $\varepsilon>0$ is $\Theta_\varepsilon((\log n)^2)$ under the first criterion but $(e^{-\varepsilon}+o(1))\log_2\ln n$ under the second, and the double-logarithmic order is attainable with the same round count on every run. For this target, a count-state recursion gives the exact least output divergence at every round cap over all causal randomized policies. The separation extends to weighted fixed-count targets with bounded weight ratios and quota fractions bounded away from zero and one, without permutation symmetry, and to the full-support Beta$(1,1)$–Bernoulli target, where the sharp expected cost under the output-law criterion is $(e^{-\varepsilon}/2+o(1))\log_2\log_2n$, attained with rare $n$-round runs. Prior work implies the order of the schedule-averaged cost on the balanced target and its $O_\varepsilon((\log n)^2)$ upper bounds on both extensions.

We give an exposition of a proof, discovered by Astra, of an Aleksandrov estimate for kinetic equations with rough coefficients. We explain this method in the simpler elliptic and parabolic settings as well.

Euler's constant $\gamma$, also known as the Euler-Mascheroni constant, remains one of the most elusive constants in mathematics. Despite extensive study, its irrationality is still unproven, and Apéry-like approximations of it are remarkably scarce. We present a novel identity that expresses $\gamma$ in terms of a Meijer G-function and its first few derivatives. This identity is then generalized and used to systematically generate infinitely many Apéry-like approximations of $\gamma$ using a Conservative Matrix Field construction based on the Meijer G-function.

We study the convergence of Markov chains in the geometry induced by Hilbertian embeddings associated with positive definite kernels, also called Maximum Mean Discrepancy (MMD). Under a contraction assumption in the kernel-induced geometry, and a standard spectral gap assumption, we establish non-asymptotic, high-probability bounds for the convergence of the empirical occupation measure to the invariant measure. The obtained bounds separate the roles of the two conditions: the spectral gap determines the effective sample size in the leading term, while the kernel contraction enters only through a linear correction. As an application, we study one-step prediction of dynamical systems from a single trajectory, deriving fast learning rates for a nonlinear least-squares estimator. The rates highlight the interplay between mixing properties in the kernel-induced geometry and the classical source and capacity conditions used in the analysis of least-squares.

We consider the demand-private coded caching problem with heterogeneous file sizes and cache capacities, where each user must recover its requested file without learning any information about the demands of the other users. We first extend the virtual-user reduction technique to accommodate heterogeneous cache sizes, thereby enabling the construction of demand-private coded caching schemes from non-private schemes. Our main contribution is an exact characterization of the memory-rate trade-off for the fundamental setting of two users and two files with arbitrary file sizes and cache capacities. Specifically, we characterize the optimal trade-off through eight linear inequalities and establish achievability by constructing explicit coded caching schemes for the non-trivial corner points, followed by memory sharing. These constructions include novel asymmetric cache placement strategies tailored to heterogeneous file sizes and cache capacities. For the converse, we exploit the distributional equivalences induced by demand privacy to derive information-theoretic lower bounds that explicitly account for the underlying heterogeneity. Our results generalize the exact memory-rate trade-off characterization for the homogeneous demand-private coded caching setting with two users and two files, while quantifying the additional transmission cost imposed by demand privacy compared with the non-private heterogeneous setting.

In every sub-Riemannian Heisenberg group $\mathbb H^n$, with $n\geq 2$, we prove the existence of a global perimeter minimizer whose boundary is a smooth, non-characteristic, entire intrinsic graph but it is not a vertical hyperplane.

Consider a queueing system operated by Bob, which is intended to serve only jobs generated by a legitimate user, Willie. Willie's jobs arrive according to a Poisson process, and the service times are exponentially distributed. Bob seeks to covertly serve additional jobs arriving according to an independent Poisson process, without Willie detecting their presence. As the legitimate user, Willie is provided with certain statistics of each busy period by Bob. In particular, we consider two statistics: (i) the number of jobs served during each busy period and (ii) the duration of each busy period. We establish that, under either statistic, the maximum expected number of additional jobs that Bob can covertly serve obeys a square-root law. Specifically, over n busy periods, Bob can covertly serve additional jobs in the order of square root of n, whereas any scaling which is above square root of n enables Willie to asymptotically detect the presence of these additional jobs with vanishing error probability. Furthermore, we characterize the scaling constant associated with the square-root law, which we refer to as the covert queueing constant, for each of the two statistics. We compare the scaling constants corresponding to the two statistics and argue that the scaling constant under statistic (i) is no greater than that under statistic (ii).

The $(0,x,1,2)$-hiccup sequence starts at $x$ and increases by 1 when the current index has already appeared as a value, and by 2 otherwise. We prove directly that for $x\geq3$ the gaps between its missing values are $1,2,3,\ldots$, each repeated $x-2$ times, and we derive closed formulas, an explicit error bound, and a generating function in terms of Ramanujan's theta functions. The counting functions of the complements satisfy nested recurrences. The sequences of missing values are the partial complements of polygonal numbers. In particular, the $(x-2)$-fold composition of the enumeration of the positive integers that are not $x$-gonal is the hiccup sequence minus 1. We extend the nested recurrence and a tree interpretation to every miss increment $z\geq2$ and every initial value $x\geq2$. Further interpretations involve polyomino vertices, monochromatic pairs, and, by a theorem of Bickle, Ramsey core numbers.

We present several new and surprising results on the theory and practice of Arbitrary Precision Approximate (APA) matrix multiplication, which refers to algorithms that output merely approximations to matrix multiplication even in exact arithmetic. While APA algorithms have been around for nearly half a century, and despite the fact that they have potentially lower (asymptotic) cost than many fast exact matrix multiplication schemes, they have not been studied as closely, to the extent that relatively little is known about their numerical properties. In this paper, we shine a spotlight on these algorithms by presenting three important results. First, we provide a new search algorithm for APA routines that is easy to use, automatically generates encoding matrices with functional entries, and can match or beat the best known APA algorithms for several sizes. In particular, we improve the upper bound on the border rank of $\langle 2,4,5 \rangle$ matrix multiplication. Second, we combine this insight with alternative bases to minimize the number of additions in an APA algorithm, thereby improving its leading coefficient. Finally, we prove the same kind of stability that characterizes fast exact matrix multiplication for an algorithm of Bini [Calcolo 1980] that computes an exact product of two matrices using a modified APA scheme (with or without alternative bases).

We prove that every length-preserving automorphism of a nonspherical Artin or Coxeter group has infinite Reidemeister number. As a consequence, we establish property $R_\infty$ for large-type free-of-infinity Artin groups, infinite strongly rigid Coxeter groups, and infinite virtually abelian Coxeter groups, including all affine Coxeter groups. We also prove that the automorphism group of every infinite irreducible non-affine strongly rigid Coxeter group is acylindrically hyperbolic.

The $q$-deformed Weil-Petersson volumes of Do and Norbury are the top-degree parts of Okuyama's discrete volumes, which are defined by topological recursion on the spectral curve of the double-scaled SYK (DSSYK) model. We show that the same q-deformed volumes can be obtained from a double-scaled matrix model whose disk density is found by zooming in on the soft edges of the DSSYK density. Applying the same procedure to the hard edge of $\mathcal{N}=1$ DSSYK, we obtain candidate $q$-deformed $\mathcal{N}=1$ super Weil-Petersson volumes. They are polynomials of fixed degree, and they reduce to the $\mathcal{N}=1$ super Weil-Petersson volumes in a scaled $q\to1$ limit. These candidate volumes differ from the $\mathcal{N}=1$ volumes of Do and Norbury in (arXiv:2510.12431) up to what we call "irrelevant" contributions which exactly vanish in the scaled $q\to 1$ limit. Together, these results demonstrate how q-deformed volumes can be naturally associated to the spectral edges of various DSSYK models and provides an intuitive way to construct them in supersymmetric contexts.

We consider solutions to the two-dimensional Navier-Stokes and Euler equations with the velocity in $L^{\infty}(\mathbb{R}^2)$ and the vorticity in the uniformly local Yudovich space $Y^{\theta}_{ul}(\mathbb{R}^2)$. We show that if $\theta$ grows sufficiently slowly, then for every $T>0$, solutions of the Navier-Stokes equation converge to solutions of the Euler equation in $L^{\infty}([0,T];L^2_{ul}(\mathbb{R}^2))$ as $\nu \to 0^+$, provided they are generated from the same initial data.

We formalize the Hamilton–Perelman Ricci-flow proof of Thurston's geometrization conjecture. The public Lean declaration is reproduced in this paper. Its geometric interpretation follows, and verification is recorded in 4.