Papers

We describe a polytopal realization of categorical $n$-associahedra. The normal fan of this polytopal realization is a modification of the authors' velocity fan and was found by OpenAI's Astra model.

We establish an upper bound for the ground state energy density of a dilute Bose gas, interacting through a general short-range, repulsive potential. To reach this goal, we extend the approach recently introduced in [1] for hard spheres.

In this paper, we study the deformed Hermitian-Yang-Mills (dHYM) equation on the non-compact Calabi-Yau manifold $Z = {\rm{Tot}}({\bf{K}}_{X})$, where $X$ is a rational homogeneous variety. Since $X$ is a Fano variety, $Z$ is a resolution of the singularity at the vertex of the affine cone ${\rm{Aff}}(X)$, provided the cone is built using the anticanonical polarization ${\bf{L}} = {\bf{K}}_{X}^{-1}$. Using the cohomogeneity-one symmetry of the Ricci-flat Kähler metric obtained via the Calabi ansatz on $Z$, we reduce the fully nonlinear PDE underlying the dHYM equation to a scalar, asymptotically autonomous ordinary differential equation (ODE). From this, we determine the exact condition on the topological phase that guarantees global existence of solutions. As an application, we show that every holomorphic line bundle over $Z$ admits a smooth, globally defined Hermitian connection solving the dHYM equation, provided the total phase lies in an explicit open interval determined by the Lie-theoretic data. Also, we prove a rigidity result classifying the exact geometric conditions under which the dHYM solution collapses into a classical Hermitian-Yang-Mills (HYM) connection. The results established generalize previous constructions and provide a substantial new class of examples. Furthermore, the approach presented allows one to study the behavior of the dHYM solutions through ODE methods. Using this approach, we construct the first explicit non-trivial example of a Hermitian-Einstein connection on a line bundle over a non-toric Calabi-Yau manifold which is not dHYM.

We study determinantal planar Coulomb gases with a smooth radial potential, a smooth radial perturbation, jump-type singularities (including the bulk, semi-hard, hard, and soft-edge regimes), and a hard wall region given by a centered disk, a disk complement, or an annulus. We obtain explicit asymptotic expansions of the partition functions up to and including the term of order $1$. Our results unify and generalize many of the earlier works.

This paper establishes a mesoscopic central limit theorem for linear statistics of $\beta$-ensembles or log-gas, as the dimension $N\to\infty$, in the temperature regime $1/N\ll\beta(N)\le 1$. For simplicity, we assume that the potential is one-cut regular and analytic. In this regime, the size of the fluctuations depends on $\beta(N)$ and the mesoscopic scale. We show that there is a transition at a critical $\eta\asymp 1/ N\beta(N)$ between a Random Matrix regime, where the limiting variance is given by the $\mathsf{H}^{1/2}$-norm and a Poisson regime where the limiting variance is given by the $L^2$-norm. We also describe the critical regime. The proof of the CLT relies on optimal local laws at intermediate temperatures and Stein's method for $\beta$-ensembles. In particular, in this regime, it is necessary to construct new correction terms to the classical equilibrium measure to obtain a suitable re-centring of linear statistics and describe their fluctuations. We also obtain a free energy expansion.

We characterise exacting, rank-Berkeley, Berkeley, and club Berkeley cardinals by compactness properties of logics and, equivalently, by compactness properties of certain topological spaces. This raises the strength known to be obtainable by statements about compactness into the realm of choiceless large cardinal axioms, the strongest known statements in terms of consistency strength. Moreover, it shows that compactness assumptions can directly violate the axiom of choice and the axiom $V=\text{HOD}$.

Let $(M^n,g)$ be a complete, connected, noncompact Riemannian manifold, and let $f\ge0$ be a proper $C^2$ weight. We assume polynomial bounds on $f$, $|\nabla f|$, and $\Delta f$ (Assumptions H1-H3), with growth exponent $\alpha>0$. We study the drift Laplacian $L_f=\Delta-\langle\nabla f,\nabla\cdot\rangle$ on $L^2(M,e^{-f}dV)$. We assume that its eigenfunctions have finite polynomial growth orders $\gamma_k$, measured on the level sets of $f^{1/\alpha}$. We also assume that $\gamma_k\to\infty$ in spectral order (Assumption H4). We compare this growth filtration with the filtration by eigenvalue $\lambda_k$. For $\alpha>1$, we study the relation $\lambda_k\asymp\gamma_k^{(2\alpha-2)/\alpha}$, called compatibility. An explicit rotationally symmetric example shows that this relation can fail for the full spectrum, even when $\alpha=2$. We prove discreteness and weighted Agmon estimates for $\alpha>1$. A moment estimate bounds the concentration radius in terms of the growth order and a finite-scale prefactor. A lower localization condition gives one direction of the spectral comparison. For exact warped products with a one-dimensional base, we impose the radial growth condition and a regularity assumption on the radial drift. These give $\alpha=2$ and $\lambda_k\asymp\gamma_k$ in the radial sector. For general $(M,g)$, we assume a transitive isometric symmetry of the level sets instead of a warped-product structure. The invariant-sector growth condition and radial-drift regularity then give $\alpha=2$ and $\lambda_k\asymp\gamma_k\asymp k$. Finally, exact polynomiality of the invariant eigenfunctions implies $\alpha=2$ without the drift regularity assumptions. The reduced equation is then Hermite or generalized Laguerre after normalization. The arguments use neither curvature bounds nor soliton equations.

Let $B^d$ denote the $d$-dimensional Euclidean ball of unit radius. What is the largest constant $f(B^d) \in [0,1]$ with the property that every finite collection $\mathcal{C}$ of unit balls in $\mathbb{R}^d$ admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $f(B^d)$ of the volume of $\mathcal{C}$? This problem was first raised by T. Radó in 1928, for axis-parallel squares in the plane; the author was motivated by a classical covering lemma in real analysis due to Vitali. The case of Euclidean balls was first considered by R. Rado in 1949. Until last year the best known estimates on $f(B^d)$ for unit balls where very far apart: \[ (1+\epsilon_d) 3^{-d} \leq f(B^d) \leq 2^{-d}, \] where $0<\epsilon_d=o_{d\rightarrow \infty}(1)$. Recently, the authors of this note observed that an exponential improvement on the upper bound follows from the Kabatiansky–Levenshtein spherical code bound, while the lower bound was improved by a linear factor by C.~Xie and G.~Ge (see arxiv:2608.09744). The current best estimates for large $d$ are \[ c \cdot d \cdot 3^{-d} \leq f(B^d) \leq 2.447^{-d}, \] where $c>0$ is an absolute constant. Here we offer the first exponential improvement of the lower bound in almost 80 years, which narrows the gap to: \[ 2.910^{-d} \leq f(B^d) \leq 2.447^{-d}. \] Our method is constructive and yields a polynomial time algorithm for finding a disjoint sub-collection realizing the estimate. Moreover the same technique gives similar exponentially improved lower bounds for all symmetric convex bodies satisfying a uniform convexity assumption, e.g., $\ell^p$-balls for all $p\in (1,\infty)$.

We construct an infinite family of directed strongly regular graphs (DSRGs) with intransitive full automorphism groups whose Weisfeiler–Leman closure are association schemes of rank 6. This is the smallest possible rank for an association scheme admitting a proper DSRG merging. We also exhibit rigid sporadic DSRGs with the same closure property, showing that combinatorial regularity, group-theoretic symmetry, and Weisfeiler–Leman regularity capture fundamentally different aspects of graph structure.

In this paper we obtain a complete description of all indecomposable characters (central positive-definite functions) of "block-diagonal" inductive limits of semidirect products $\mathbb{T}^N\rtimes\mathfrak{S}_{N}$ of the compact $N$-dimensional torus $\mathbb{T}^{N}$ with the symmetric group $\mathfrak{S}_{N}$. The classification is proved by means of a variant of the ergodic method of Vershik and Kerov, namely by studying the weak limits of characters of finite-dimensional subgroups. As an application of our results, we give another proof of the classification of indecomposable characters of the infinite unitary group with block diagonal embeddings.

A KZ-type connection in characteristic $p$ has $p$-curvature operators. These operators are commuting endomorphisms of the connection. We present a Bethe-ansatz-type construction of their eigenvectors and eigenvalues. The eigenvector is constructed as the value of a characteristic $p$ hypergeometric integral and extends to a flat eigensection over the Frobenius neighborhood of the corresponding Bethe point.

We prove that the Futaki invariant introduced by Futaki, Hattori, and Ornea vanishes identically on every compact locally conformally Kähler manifold with potential, thereby answering a question raised by Ornea and Verbitsky.

Safety evaluation in sampling-based stochastic model predictive control often requires numerical estimation of exit functionals. The approximation of first-exit times and exit indicators is therefore a key numerical bottleneck, and discretization error in these quantities directly affects the resulting controller. This paper studies how existing higher-order methods for strong approximation of exit times can be brought into safe control. Two cases are highlighted. For general noncommutative dynamics, an adaptive order-1 Milstein discretization is used together with Lévy-area simulation via Wiktorsson's method. For commutative dynamics, an adaptive order-1.5 construction achieves a stronger exit-time rate. Under a local anti-concentration condition on the exit-time law, we show that strong exit-time approximation transfers to strong approximation of the failure indicator. The methods are then studied in the context of chance-constrained path integral control, which provides an exact continuous-time representation of safety through exit events. Numerical experiments compare the two cases in terms of strong exit-time error, failure-indicator error, and closed-loop constraint satisfaction, showing improvement over Euler-Maruyama and thereby enabling existing and future techniques whose applicability depends on improved strong approximation.

We study ergodic averages over Gaussian integers in shrinking angular sectors, which impose an increasingly restrictive directional relation between the real and imaginary coordinates. In uniquely ergodic systems, averages of continuous functions along $\Omega$ converge to the invariant integral uniformly in the base point and sector location for sectors whose widths shrink subpolynomially in the norm cutoff. For finitely generated multiplicative actions, we prove a quantitative sector–disk comparison in an explicit logarithmic shrinking range, with a power-saving error in $\log N$, uniformly in the base point and sector location. The exponents depend only on the number of distinct prime transformations, and the result requires neither ergodicity nor angular-distribution hypotheses on the sets of primes inducing the individual transformations. Strong unique ergodicity then yields convergence to the invariant integral at each fixed base point. Consequences include asymptotic independence of relative angular position and $T^{\Omega(n)} x$, Gaussian Liouville cancellation, and joint equidistribution modulo integers of prime-factor counts associated with two prime classes having divergent reciprocal-norm sums.

An effective strategy for constructing compact representations of integral coefficients in scattering amplitudes is to make their singularity structure manifest. This is naturally achieved by treating them as elements of fraction fields of polynomial quotient rings, where poles correspond to algebraic vanishing loci. Simplification via partial fraction decomposition relies on exploiting the structure of the numerators and is governed, at least in a first approximation, by their behaviour on irreducible codimension-two varieties. These are identifiable through primary decompositions of the associated ideals. Despite the availability of general algorithms, primary decompositions remain computationally challenging and often require tailored strategies. In this work, we extend an approach based on ideal saturation, identify an issue related to the breaking of Lorentz covariance in the choice of ideal generators, and present a solution based on fitting ansätze in special kinematic limits. We also provide substantial improvements to an algorithm for the generation of numerical points in proximity of given varieties and to one for testing primality of unmixed ideals, both of which rely on a semi-numerical procedure for computing independent sets and variety dimensions. Finally, we present new codimension-two primary decompositions for five-point massless amplitudes at higher powers in the dimensional regulator, and for finite remainders with five-point one-mass kinematics, where a new spurious singular locus appears.

We present a gluing construction for a class of 6th order geometric non-linear PDE's on a Kähler manifold or orbifold, arising in the study of Bismut Hermitian Einstein complex $3$-folds, the supersymmetric ${\rm AdS}_3 \times Y_7$ and ${\rm AdS}_2 \times Y_9$ solutions of type IIB and $D=11$ supergravity models in physics, and in Dervan's Z-critical Kähler metrics. As applications, we obtain a desingularization theorem for orbifolds with crepant singularities admitting a solution of the PDE, as well as a blowing-up of points theorem for smooth complex $3$-folds admitting such solutions. These results yield new examples of regular Bismut Hermitian Einstein structures on the smooth manifolds $3(S^2\times S^4)\sharp 4(S^3\times S^3)$, $5(S^2\times S^4)\sharp 6(S^3\times S^3)$ and $S^1\times 9(S^2\times S^3)$, as well as infinite topological types of smooth compact $7$-dimensional manifolds $Y_7$, providing new solutions to the supersymmetric ${\rm AdS}_3 \times Y_7$ type IIB and ${\rm AdS}_2 \times T^2 \times Y^7$ type $D=11$ supergravity models.

We introduce and develop Functional Dimensional Regularization (FDR), a novel functional renormalization group (RG) framework that extends the principles of dimensional regularization beyond the perturbative $\varepsilon$-expansion. The key insight is to define the FDR beta functions, valid in any continuous dimension $d$ and characterized by threshold functions, as a sum over all critical dimensions of the corresponding scheme-independent beta functions computed in DR. We demonstrate that this formalism exhibits all the hallmarks of a fully-fledged functional RG. Within the local potential approximation, we perform a general fixed point analysis, recovering the $\varepsilon$-expansion for all multi-critical models and establishing a direct connection to functional scaling solutions, spike plots, and eigen-perturbation spectra. Extending the framework to the second order of the derivative expansion, we compute the critical exponents for the Ising universality class in $d=3$, providing estimates for the anomalous dimension $\eta$ and RG spectrum that converge rapidly and show favorable agreement with state-of-the-art results. We present a detailed comparison with non-perturbative and proper-time RG approaches, highlighting structural similarities and key differences. In particular, we introduce a “distillation” procedure that allows one to systematically derive the FDR flow from any other one-loop-exact RG scheme. Our results establish FDR as a self-contained and competitive RG framework that combines the technical simplicity of dimensional regularization with the versatility of the functional approach.

For systems of rough stochastic differential equations in the sense of Friz–Hocquet–Lê, with a fixed deterministic rough driver, we prove two results. The first is weak existence with unbounded coefficients. The Itô data is assumed continuous and of linear growth, with no Hölder or Lipschitz continuity and no ellipticity. The rough vector field is a controlled pair of linear growth with bounded derivatives, subject either to a norm-curvature decay or to diagonality with coordinatewise curvature decay; in the diagonal case a coordinatewise upper bound on the diffusion matrix is added. The existing results ask the Itô data and the rough field to be bounded. We remove that boundedness, and ask in exchange for a curvature hypothesis on the rough field, bounds on its derivatives, and a narrower index configuration. The conclusion is weaker on two counts: the bounds that define a solution are imposed in the mean rather than uniformly on the probability space, and the solution's moments stop at the order the initial law carries. The second is the Yamada–Watanabe implication for rough equations: weak existence, together with pathwise uniqueness, yields a strong solution and joint uniqueness in law. A solution here is not defined by a pathwise identity but by two bounds on a remainder, one of them conditional, and an arbitrary enlargement of the filtration need not preserve that one. The enlargements the Yamada–Watanabe construction produces are immersions, however, and along an immersion both bounds transfer unchanged.

Previous studies of network dynamics have suggested that heterogeneity among nodes inhibits stability, at odds with the ubiquity of inherently heterogeneous natural and engineered systems. Here, we show that this conclusion arises from model reductions introduced for mathematical tractability and breaks down when nodal dynamics are higher-dimensional, yielding non-Hermitian Jacobians. In such systems, including neural, power-grid, and material networks, nodal heterogeneity can instead enhance stability, even when parameters are randomly disordered. Non-Hermiticity also underlies the stabilizing effects of network heterogeneity, which can arise even in one-dimensional nodal dynamics through nonreciprocal interactions, as shown for ecological networks. Our framework reveals disorder not as a liability but as a general resource for stabilizing complex systems.

We present bulk-primary operator-product coefficients for every Virasoro minimal model with an $E_6$, $E_7$, or $E_8$ modular invariant. Dividing by a universal product of chiral generalized-minimal-model OPE coefficients reduces the conformal bootstrap problem to a finite algebraic problem governed by root-of-unity quantum-group $6j$ symbols. The exceptional factors of the OPE coefficients can be chosen independent of the second Kac indices $s$, and their dependence on the central charge reduces to finitely many reference cases and explicit phases. A finite computer-assisted calculation yields candidate algebraic $s=1$ seed data and high-precision numerical evidence for existence and uniqueness modulo primary-field signs. We prove that exact existence and uniqueness for these seeds imply the corresponding result for all Kac labels within the genus-zero bulk locality and crossing equations; the finite seed step remains uncertified. The three families require $18$, $32$, and $107$ representative reduced OPE coefficients for $E_6$, $E_7$, and $E_8$, respectively. We provide reconstruction conventions and coefficient tables for both unitary and nonunitary models. Our calculation assumes only Virasoro symmetry, bulk locality (permutation symmetry) and OPE associativity (crossing equations).

Asymmetric orbifolds provide concrete examples of non-geometric constructions dubbed T-folds. All Z2 point groups of six dimensional asymmetric order-two orbifolds are identified. The order-two T-fold configurations on the SO(12) lattice are classified at the fermionic point of type II string theories for all Z2 point groups. The spectra of the models on these configurations are presented parametrically, in terms of certain generalised GSO phases. The minimal effective Hodge numbers were found to be (h_{11},h_{12})=(1,1) for six T-fold configurations. Asymmetric generalisations of the mirror symmetry map are conjectured. The orientifoldable configurations using the basic worldsheet parity were identified within the classification. Twisted sectors corresponding to pure asymmetric twists may contain Rarita-Schwinger multiplets with spin-3/2 states. Throughout the paper, generalised GSO projections are chosen to preserve the maximal amount of supersymmetry possible. Relaxing this, the order-two point groups were identified for which non–supersymmetric T-folds can be constructed. Since some of them may enhance to N=1 or, even, N=2 supergravities, we argue that appearance of spin-3/2 states necessarily implies that the spectrum has reorder itself in a supersymmetric fashion and hence that the vacuum energy vanishes.

We investigate the associated vertex operator algebras and Higgs branches of two closely related families of generalised Argyres–Douglas theories, arising from the twisted $A_{2N}$ and twisted $D_{N+1}$ series, respectively. We propose that the $A$-series theories are realised as nilpotent Higgsings of the $D$-series, and we assemble a variety of pieces of evidence for this proposal. A consequence is that the associated vertex operator algebras of the $A$-series are finite extensions of affine Kac–Moody vertex algebras at (non-boundary) admissible levels, rather than those Kac–Moody algebras themselves, and that their Higgs branches are in many cases double covers of nilpotent orbit closures. The $\mathbb{Z}_2$ gaugings of the $A$-series theories then furnish examples of unitary SCFTs whose associated vertex operator algebras are precisely affine Kac–Moody algebras at non-boundary admissible levels. We propose that these vertex algebras are equipped with a non-standard $\mathfrak{R}$-filtration, and comment on compatibility with graded unitarity.

We introduce collision rank and show that groups of finite collision rank are good in the sense of topological $4$-manifold theory. Finitely generated groups of collision rank two are shown to exist outside the class of subexponentially amenable groups. Such examples are given by commutator subgroups of topological full groups of Sturmian systems, and include continuum many pairwise nonisomorphic infinite simple amenable groups.

Ferrero and Lesniak in 2018 found the maximum numbers of edges in $r$-partite non-Hamiltonian graphs. Recently we found the maximum numbers of edges and $t$-cliques in $K_{r+1}$-free graphs (1) that are not Hamiltonian or (2) that satisfy a condition on low-degree vertices related to Pósa's theorem. Applying theorem (2), here we extend theorem (1) from Hamiltonicity to other properties. We determine the maximum numbers of edges and $t$-cliques in $K_{r+1}$-free graphs that avoid one of the following properties: traceability, Hamiltonian-connectedness, $k$-path Hamiltonicity, $k$-Hamiltonicity, $k$-Hamiltonian-connectedness, and $k$-connectedness. We find all extremal graphs having the maximum numbers of edges. On the way, we prove upper bounds on the numbers of edges and $t$-cliques in $K_{r+1}$-free graphs that avoid an arbitrary stable property that holds for sufficiently large complete graphs.

Let $(a_n)_{n\geq 0}$ be a sequence of numbers. We derive an expression for the Euler transform of a general binomial sum involving $a_n$ and weighted by the floor function. To demonstrate the usefulness of our approach, several examples are discussed. We rediscover some known identities and prove several new. In particular, we derive some new identities for weighted binomial sums involving harmonic numbers and central binomial coefficients. We also present new closed-forms for weighted series involving the Riemann zeta function.

We introduce causal graph rewriting, a model of computation in which local rules are applied on directed acyclic graphs in an asynchronous manner. The non-determinism arising from asynchrony is disciplined by the oriented edges, which must be understood as both computational dependencies and locality constraints—and are themselves subject to the rewriting. We illustrate the model through two examples: a particle system, and a time-dilation example—reminiscent of general relativity. We study the well-definedness and properties of induced subgraphs and graph composition, which isolate and recombine the region affected by a rewrite. We then study locality with respect to these constructions, showing how a local rewrite preserves positions, borders, and context. Our main result concerns sequential composition: locality extends from single rule applications to arbitrary valid sequences, as any local rule is automatically $*$-local and $*$-extensive. We also formalise and prove the simulation of any one-dimensional cellular automaton.

For every odd $N$, we construct a Haagerup-Izumi (HI) category for $\mathbb Z/N$, using Barnes double-sine functions. Equivariantization gives near-group categories of type $((\mathbb Z/N)^2,N^2)$ for every odd $N$. We classify connected separable algebras up to Morita equivalence in such HI categories and describe their dual categories. These duals are often HI for noncyclic groups with nontrivial pointed associators. We prove the Evans-Gannon conjecture that the modular data of the center is always a smashed sum involving a finite metric group of order $N^2+4$ and recover this group and its quadratic form directly from the data of the HI category. We also give the HI equations and reconstruction in arbitrary characteristic and prove a Frobenius symmetry in characteristic two. Using this, we obtain the complete list of (untwisted, subfactor type) HI categories over $\mathbb{C}$ for odd $N\le 43$. Other applications include finding three pairwise non-Morita-equivalent $\mathbb Z/15$ categories whose centers have the same modular data. We show that the center of a $\mathbb Z/5$ HI category cannot be defined (i.e., admits no split ribbon form) over a cyclotomic field, disproving a conjecture of Davidovich-Hagge-Wang.

For every nonzero real number $a,$ we prove that the integral Hankel operator on $L^2(\mathbb R_{+})$ with kernel $k(t)=t^{-1+ia}$ is noncompact quasinilpotent. This answers in the affirmative a question by Peller \cite{Peller1998} about the existence of noncompact quasinilpotent Hankel operators. Moreover, we obtain the corresponding Hankel operators on the Hardy space with symbols $\left((1-z)/(1+z)\right)^{ia}.$

Reverse mathematics is a branch of mathematical logic dedicated to determining the minimal set existence principles necessary and sufficient to derive ordinary mathematical theorems about concrete structures like the real line. Since the mid-1970s, reverse mathematics has developed a systematic classification of the strength of theorems in areas of mathematics ranging from real and complex analysis to infinitary combinatorics. This essay will place reverse mathematics in its historical and philosophical context, and reveal its relevance to central issues in the philosophy of mathematics, from the foundational programmes of Hilbert and Brouwer to contemporary debates about realism, determinacy, and applicability of mathematics. In doing so, it will discuss the role of computability theory in measuring the strength of set existence principles, as well as related questions about idealisation when these principles are applied in the physical sciences and in philosophy.

We show that every lattice weak Minkowski summand of a smooth polytope combinatorially isomorphic to a product of simplices has a quadratic triangulation. This is achieved by identifying this class of polytopes with the class of Nakajima polytopes. We combine these results to give a new proof that two lattice weak Minkowski summands of the same such smooth polytope are relatively IDP. Along the way, we prove a structure theorem for simple polytopes whose 2-faces are triangles and trapezoids.