Papers

In this work we investigate a relative notion of protomodularity. We say that a finitely complete category $\mathsf{C}$ satisfies the \defn{$\mathscr{M}$-version of protomodularity} when the change-of-base functors (concerning the fibration of points) are conservative with respect to a class of morphisms $\mathscr{M}$ of $\mathsf{C}$. If $\mathscr{M}$ is the class of all morphisms or the class of all monomorphisms of $\mathsf{C}$, then this notion is precisely that of an “ordinary” protomodular category. We analyse what conditions the class $\mathscr{M}$ should have to obtain the relative $\mathscr{M}$-versions of some well-known properties of protomodular categories, such as the Split Short Five Lemma (when $\mathsf{C}$ is also pointed) and their relation with Mal'tsev categories. We also give diverse examples of categories satisfying the $\mathscr{M}$-version of protomodularity for different choices of $\mathscr{M}$.

We study the gap between the integrated direction-wise kinetic minima and the minimum expected Dirichlet energy of a joint particle realization of a prescribed two-parameter family of conditional probability measures. Under regular coarea assumptions with compact connected fibres and a star-shaped parameter domain, we derive a curvature lower bound valid for every admissible law of Sobolev maps realizing the prescribed smooth positive conditional measures. Our method combines the weighted-Poisson characterization of local optimal-transport velocities with an exact energy-residual identity, a weak Stokes formula, and explicit transport constructions. The minimum excess vanishes exactly when the Lie-bracket curvature vanishes throughout the parameter domain, whereas at a nonzero-curvature centre it is of order $r^4$ as $r\to0$ on squares of half-side length $r$ with unnormalized area measure. In a two-phase torus model fitted to electricity-demand data, numerical tests illustrate this quartic order for explicit couplings, with radial transport approximately halving the ordered construction's excess on the tested small squares.

Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 4$ which defines a smooth hypersurface in $\mathbb{P}^{n-1}_{\mathbb{C}}$. For $n \geq 24 d^4 2^d$, we prove an asymptotic formula for the number of prime solutions to the equation $F(x_1, \ldots, x_n) = 0$, provided $F$ satisfies suitable local conditions.

We provide a description of the real variable Hardy spaces $H^p$, $p\in (0,1]$ associated to a regular polyhedral open convex cone, devised as real variable analogues of the holomorphic Hardy spaces on the corresponding tube domain.

Kernel methods are among the most powerful tools in machine learning and statistics, with a large number of successful applications. Their immense success stems from the flexible function class associated to each kernel—its reproducing kernel Hilbert space (RKHS)—which facilitates statistical analysis, as well as from their computational tractability and applicability to many domains. Multiple notions (such as characteristic, $L_p$-universal, and integrally strictly positive definite) capture the expressivity of kernels and their RKHSs and play a key role in understanding the statistical properties of kernel methods; these concepts and their relations are well-understood for bounded kernels. Even though unbounded kernels have received significant attention over the past decade (for instance, in the construction of kernel-based discrepancy and dependence measures such as the maximum mean discrepancy, the Hilbert-Schmidt independence criterion, and the kernel Stein discrepancy), surprisingly little is known about the relations of these notions in the unbounded case. In the present paper we tackle this severe bottleneck, establishing their relations under mild assumptions.

We prove global rigidity of smooth asymptotically flat stationary vacuum black holes with sufficiently small ADM angular momentum, under global regularity, bifurcation, and maximal-hypersurface assumptions, with a threshold depending on fixed geometric bounds. This answers the question of Alexakis, Ionescu, and Klainerman concerning replacement of their smallness assumption on the stationary field at the bifurcation sphere by smallness of the angular momentum at infinity. The main step establishes Kerr rigidity near infinity for arbitrary angular momentum. We transform the vacuum metric into an Einstein–Maxwell family and compare null focusing at its limiting end with the area–charge inequality on compact maximal hypersurfaces, constructed by continuation through the family. Equality determines both the transformed metric and its stationary field, from which the inverse transformation recovers Kerr. Two Kerr identities then give the horizon estimate required for the global conclusion.

We study commutativity and zero products of slant Toeplitz operators on the Bergman space of the unit disk. For bounded polyanalytic symbols and for bounded harmonic symbols with boundary values in $W^{1,2}(\mathbb T)$, we prove that two such operators commute if and only if their symbols are linearly dependent over $\mathbb C$, with both symbols belonging to the same class. In the polyanalytic case, no boundedness assumption is imposed on the individual analytic components. We also show that every finite-rank commutator in this class is zero and that every finite product associated with nonzero polyanalytic symbols has infinite rank. Finally, for $s\ge2$, we prove that a product of $s$ slant Toeplitz operators with bounded harmonic symbols whose boundary values belong to $W^{s-1,2}(\mathbb T)$ is zero if and only if at least one symbol vanishes identically.

Let $f_n:\{0,1\}^n\to\{0,1\}$ be Boolean functions, and let $C_n$ count their changes in a unit time interval under stationary dynamics in which each coordinate is independently resampled at rate one from the Bernoulli$(p_n)$ distribution. We prove that, if $f_n$ is transitive and $n\min\{p_n,1-p_n\}^{r}\to\infty$ for every $r>0$, then tightness of $(C_n)$ implies $\mathrm{Var}(f_n)\to0$. This settles Forsström's conjecture in the $p_n\le 1/2$ regime and extends it symmetrically to arbitrary biases. The proof uses a mixed jump sum whose second moment is bounded by the generator of a low degree Fourier projection.

We establish an $L^p$–$L^q$-type Carleman estimate for the Schrödinger operator $H_{A,V}$ with scaling-critical electromagnetic singular potentials on a conical singular space $(X,g)$ of dimension $d\geq3$, where the metric is $g=dr^2+r^2h$ and $X=C(Y)=(0,\infty)_r\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$. More precisely, we prove $$ \|e^{\tau\phi}u\|_{L^{2d/(d-2)}(X)}\le C\,\|e^{\tau\phi}H_{A,V}u\|_{L^{2d/(d+2)}(X)} $$ with the quadratic weight $\phi(r)=\tfrac14(\log r)^2$, uniformly in $\tau$. As an application of this Carleman estimate, we prove the unique continuation property for this electromagnetic Schrödinger operator when the cross-section is $Y=\mathbb{S}^{d-1}$.

We prove an Ax-Lindemann theorem for the rank one Riemann-Hilbert correspondence on a smooth projective complex algebraic curve and recover a result of Simpson.

We develop a continuous-time deterministic policy gradient framework for reinforcement learning with reflected state dynamics, where the state process is governed by a controlled reflected stochastic differential equation on a hypercube. Under suitable regularity assumptions, we establish the connection between the value function and the Neumann Bellman equation, introduce an advantage-rate function that yields a deterministic policy gradient formula, and prove the martingale characterization theorem. Motivated by these theoretical results, we propose a continuous-time deep deterministic policy gradient algorithm for reflected stochastic systems, in which the Neumann boundary condition is imposed via either soft penalization or hard architectural constraint. We further quantify the discrepancy between the ideal continuous-time dynamics and the discretely sampled exploratory dynamics executed in practice, showing that the error decays as the time grid is refined and exploration noise vanishes. Our experiments on reservoir control problems illustrate the effectiveness of the RL framework, highlighting that boundary-aware methods substantially reduce Neumann boundary residuals and enhance learning stability.

Let $\mathcal{G}$ be a finite connected simple graph with at least two vertices. We construct a finite regular hypertope $\Gamma$ whose diagram is the graph $\mathcal{G}$ with every edge labelled by $4$, and for which $Cor(\Gamma)/Aut(\Gamma) \cong Aut(\mathcal{G})$. This shows that any possible group of diagram symmetries can be realized by the correlations of a hypertope. The construction uses a group generated by involutions, with nilpotency class two, exponent four and commutator relations dictated by the defining graph.

An analytic subring of $\mathbb{R}$ has Hausdorff dimension $0$ or is equal to $\mathbb{R}$, by a theorem of Edgar–Miller and Bourgain. We show that this fails for coanalytic subfields in the constructible universe: if $V=L$, then for every $\alpha\in[0,1]$ there is a proper coanalytic subfield of $\mathbb{R}$ of Hausdorff dimension $\alpha$. In fact there is a single parameter-free $\Pi^1_1$ set $G\subseteq[0,1]\times\mathbb{R}$ whose sections $A(\alpha)$ are such fields, are strictly increasing in $\alpha$, and satisfy $A(\alpha)=\bigcup_{r<\alpha}A(r)$ for $\alpha>0$.

We study the $\ell^2$-Betti numbers of graph products of cyclic groups. We relate these invariants to the $\ell^2$-Betti numbers of right-angled Artin groups and obtain explicit formulas in several cases determined by the structure of the finite-order vertices in the defining graph. In particular, we consider the cases where these vertices form a clique or have identical links, and derive a formula for the first $\ell^2$-Betti number when the defining graph has exactly two finite-order vertices. We also compute the BNSR-invariants of graph products of cyclic groups and the first $\ell^2$-Betti number of their automorphism groups when the defining graph has at most two finite-order vertices.

A combination of recent results of Canning–Larson–Payne–Willwacher and Payne–Willwacher is that the motivic Euler characteristic of the moduli space $M_{g,n}$ of nonsingular curves with $n$ markings is polynomial in the Tate motive if and only if $g=0$ or $3g+2n<25$. In the cases when $g\geq 1$ and $3g+2n<25$, there are $32$ pairs $(g,n)$ where the motivic Euler characteristic has been computed. We compute the motivic Euler characteristic for $9$ more pairs, leaving only $6$ pairs where the motivic Euler characteristic is polynomial but unknown. In genus $4$, we moreover compute the answer when $n=7$, i.e.~the first non-polynomial case.

We prove an $L^4$ Strichartz estimate for the dispersion $n_1^2+\alpha n_2^2$ on $\mathbb T^2$, for every $\alpha>0$. The loss is controlled by the Shannon entropy of the normalized squared Fourier coefficients and the cardinality of their support $S$. In particular, we obtain a norm loss of $(\log\#S)^{1/2}$ on fixed time intervals, with constants locally uniform in $\alpha$. This provides an irrational-dispersion counterpart of the cardinality estimate of Herr and Kwak [Forum Math. Pi (2024)] and improves the $N^\varepsilon$ loss of Bourgain and Demeter [Ann. of Math. (2015)] to $(\log N)^{1/2}$ on frequency boxes. The proof combines weighted incidence estimates, smooth localization, and an entropy recursion.

We study the stability of optimal transport for the squared Carnot–Carathéodory distance. We prove the sharp $W_1^{1/4}$ stability estimate for optimal transport maps on complete Riemannian and fat sub-Riemannian manifolds, assuming bounded source support and an upper density bound. On John domains with two-sided density bounds, we obtain logarithmic $L^2$ stability of Kantorovich potentials on complete smooth sub-Riemannian manifolds that are equiregular near the source, with arbitrary compact targets. For potentials, we obtain every Hölder exponent $0<\alpha<1/2$ on two-generating structures, and the exponent $1/2$ in the Riemannian and fat settings and on all step-two Carnot groups. Three-atom examples show that the map and potential exponents with respect to $W_1$ cannot exceed $1/4$ and $1/2$, respectively, on any smooth sub-Riemannian manifold of horizontal rank at least two. We also construct smooth transport families whose target variation of order $t$ in $W_1$ produces map variation of order $t^{1/s}$ in an equiregular region of step $s$. The map estimates follow from coercivity of the duality defect, while the potential estimates follow from quantitative concavity of the Kantorovich functional.

Structured recovery phenomena, such as restricted isometry properties in compressed sensing, have shown that high-dimensional objects can often be reconstructed from remarkably low-dimensional linear measurements. This work develops an analogous recovery framework for low-rank signed measures on $\mathbb{R}^2$, defined here as measures that can be expressed as finite sums of product measures with one-dimensional factors. The framework is based on linear operators, termed "silhouette operators," that map a measure to a fixed finite collection of one-dimensional linear pushforwards. The main results show that a suitably chosen collection of $2k$ projected marginals suffices to identify every compactly supported rank-$\le k$ signed measure, that this number is optimal, and that the projection directions cannot be chosen arbitrarily. The framework is also extended to higher-dimensional sums of product measures by establishing sufficient conditions under which collections of pairwise marginals identify the full model. Building on this framework, a computationally efficient estimator, termed "silhouette mixture estimation" (SME), is introduced for constructing a low-rank empirical measure from data by matching its one-dimensional projected marginals to the corresponding empirical marginals in Wasserstein distance. When combined with one-dimensional density estimators, SME yields an efficient nonparametric density estimator that performs strongly relative to a range of parametric, nonparametric, and deep-learning baselines in settings of moderate dimension and sample size.

We study linear Lagrange interpolation on exact geodesic spherical triangles obtained by radial projection of chordal triangles. Factorising the element map into an affine map and a radial correction separates chordal anisotropy from radial distortion. We derive exact identities for the surface measure, Dirichlet energy, and directional derivatives, and use them to transfer planar anisotropic interpolation estimates to the sphere. The resulting local error estimates retain the two chordal directional length scales and keep the geometric factors explicit, with constants independent of the element size, shape, and sphere radius. In the gradient estimate, the maximum-angle factor multiplies only the longer directional scale. Four explicit triangle families quantify the loss caused by replacing these scales with the diameter and establish the sharp maximum-angle dependence. They also show that, within the same two-term directional bound with a constant independent of the element, the maximum-angle factor cannot be moved to the shorter scale or have its exponent reduced below one. The fourth family proves the optimality of the square-root radial-transfer factor while retaining the same transported directional scale. The local estimates extend to compatible spherical triangulations. As an application, they yield an anisotropic energy-error estimate for the conforming approximation of the mean-zero Laplace–Beltrami problem with exact integration.

A KMS-symmetric quantum Markov semigroup is completely determined by the quadratic form in induces on the GNS Hilbert space. The quadratic forms that arise in this way are called quantum Dirichlet forms. In this article we give a complete description of bounded quantum Dirichlet forms in terms of derivations. In particular, we exhibit an explicit characterization of the bounded quantum Dirichlet forms on type I factors. We also show that any unbounded quantum Dirichlet form can be represented in terms of derivations. The key technical insight is that the modular group always restricts to a strongly continuous group on the domain of a quantum Dirichlet form such that the square root of the analytic generator is accretive with respect to the form norm.

In [6], we introduced the notion of a Hermitian-relative curve, which is a plane curve defined by $(x^{\sqrt{q}}, y^{\sqrt{q}}, z^{\sqrt{q}})A (x,y,z)^t =0$ with $A \in GL(3, \mathbb{F}_q)$. After investigating their basic properties, we classified those curves with two or more rational inflexions. In this paper, we first complete the classification of curves whose rational points are all inflexions by demonstrating the existence of curves with exactly one rational point. As a continuation, we then classify Hermitian-relative curves that possess at least one rational point which is not an inflexion. We categorize these curves according to the ordered pair consisting of the number of rational points and the number of inflexions. Finally, we enumerate all possible pairs and prove that each pair is realized by a suitable curve.

We present a novel dual approach to characterizing the solution to optimal stopping problems for strong Feller processes on locally compact, separable metric spaces. Our characterization builds upon reinterpreting a standard result in optimal stopping theory as a dual formulation, which yields computationally tractable algorithms. Our framework is applicable to a broad class of Markov processes. First, in the one-dimensional case, while previous studies have managed to comprehensively solve only diffusion and one-sided jump Lévy processes over an infinite time horizon, our proposed method successfully extends to both infinite- and finite-horizon scenarios across all these processes, including two-sided jump Lévy processes. In addition, for multi-dimensional cases, our framework is applicable to diffusions over an infinite horizon with certain structural features such as convex stopping regions. To illustrate this versatility, we explicitly solve several challenging problems including a case where the optimal stopping region consists of two disjoint connected components – a structural feature that remains unaddressed in the literature.

We study two graph pebbling parameters, the stacking number and the clearing number, through the Almost Stacked Hypothesis (ASH). This hypothesis asserts that these thresholds can be determined by testing only configurations in which at most one vertex carries more than one pebble. We prove that every almost stacked configuration of size $2^{n+1}-1$ on $C_{2n}$ is stackable and that every almost stacked configuration of size $3\cdot 2^n-2$ on $C_{2n+1}$ is clearable. Together with the known lower bounds, these results show that ASH implies $\operatorname{stack}(C_{2n})=2^{n+1}-1$ and $\operatorname{clear}(C_{2n+1})=3\cdot 2^n-2$. For a finite tree $T$, we introduce an explicit invariant $\operatorname{estim}(T)$. We prove unconditionally that $\operatorname{stack}(T)\geq\operatorname{estim}(T)$ and prove the reverse inequality under ASH. Consequently, ASH yields $\operatorname{stack}(T)=\operatorname{estim}(T)$, and we conjecture that this equality holds unconditionally. Finally, we study perfectly pebblable graphs: finite connected non-bipartite graphs whose clearing number has the minimum possible value $\operatorname{clear}(G)=|V(G)|+1$. Every complete graph with at least three vertices is perfectly pebblable, which might suggest that perfect pebblability requires high edge density. Assuming ASH, however, we show that this is not the case. We give a sufficient criterion involving strong edge-triangulation and Hamiltonian-path and path-cover conditions in vertex-deleted subgraphs and use it to construct two explicit infinite families of perfectly pebblable graphs with edge density tending to zero, one of which has only a linear number of edges.

Let $w(G)$ denote the number of distinct vertex degrees of a finite simple graph $G$. Melnikov asked for a lower bound on the chromatic number $\chi(G)$ in terms of $|V(G)|$ and $w(G)$, and conjectured a strict bound of this type. We resolve Melnikov's valency-variety problem by proving that every graph $G$ with at least two vertices satisfies \begin{equation*} \chi(G)\ge\left\lceil1+\frac{2w(G)(w(G)-1)}{4w(G)(|V(G)|-w(G))+(|V(G)|-w(G)-1)^2}\right\rceil, \end{equation*} and consequently \begin{equation*} \chi(G)\ge\left\lceil\frac{\lfloor w(G)/2\rfloor}{|V(G)|-w(G)}\right\rceil. \end{equation*} Finally, we also construct an explicit infinite family of graphs attaining equality in both bounds. In particular, these examples show that Melnikov's proposed strict inequality is false and that the bounds above are best possible.

Let $f(d)$ be the smallest integer such that every finite connected bridgeless simple graph of diameter $d$ admits a strong orientation of diameter at most $f(d)$. Chvátal and Thomassen (1978) proved $\lceil d^2/2\rceil+d\le f(d)\le2d^2+2d$. In this paper, we prove that $\lceil d^2/2\rceil+d\le f(d)\le\lceil d^2/2\rceil+d+18$ for every $d\ge2$, which shows that $f(d)=\lceil d^2/2\rceil+d+O(1)$ and determines both the quadratic and linear terms up to a bounded additive error. The key method of our proof is to construct a central subgraph $H$ that admits a strong orientation of diameter $O(d)$ and is within distance $\lfloor d/2\rfloor$ of every vertex outside it. We obtain the sharp linear coefficient by jointly estimating outside paths and their connecting paths in $H$ at the actual attachment vertices, rebuilding $H$ when necessary.

We develop and analyze a locking-free virtual element method for the three-dimensional quasi-static electroporoelasticity equations. The pressure, electric, magnetic, and displacement fields are discretized by the lowest-order nodal, edge, face, and Stokes-like virtual element spaces, respectively, so that the scheme works on general polyhedral meshes, while the temporal discretization is carried out by a backward Euler scheme. The locking phenomenon is overcome by defining the Stokes-like space on a refined mesh $\widetilde{\mathcal{T}}_h$, on which we can construct an interpolation operator $\mathrm{I}_s$ satisfying $\div(\mathrm{I}_s\bm v)|_K=\Pi^K_0\div\bm v$, which renders the error estimates independent of the Lamé constant $\lambda$. Moreover, since the exact sequence $\curl\bm V^e_h=\bm V^f_{h,0}$ holds, the magnetic field is updated explicitly, reducing the four-field system to a three-field one and lowering the computational cost. Numerical experiments are provided to demonstrate the performance of the proposed method.

We prove quantitative stability estimates for quadratic optimal transport maps by regularizing a Brenier potential and estimating the error in the duality gap. The source has an $L^p$ density, $p>1$, relative to a probability measure with a BV density. When one target lies in a fixed ball, we obtain the $W_2$ Hölder exponent $(p-1)/(4p-2)$. The constant depends on the dimension through three explicit source quantities: the variance, the total variation of the reference density and the $L^p$ norm of the weight. The proof uses a second-order bound on the averaged regularized Fenchel gap. By restricting the slopes of the potential, we also treat targets with a bounded moment of any order greater than two. In both estimates, the condition is imposed on one target, while the other can be any probability measure with finite second moment. For Gaussian and isotropic log-concave sources, the constants grow polynomially with the dimension. For $L^p$ changes of Gaussian measure, the factor in the bounded-target estimate is $d^{(p-1)/(4p-2)}$. For each $p$, a fixed source in two dimensions proves optimality of both stability exponents in this source class. A Gaussian construction provides polynomial lower bounds on the dimension dependence.

Using the theory of symmetric polynomials, we prove that the spectrum of every Toeplitz operator with an analytic polynomial symbol on the harmonic Bergman space can be represented as the union of a closed curve and a finite set of at most $(n-1)2^{n-1}$ isolated eigenvalues, where $n$ is the degree of the symbol. This implies that Weyl's theorem holds for such operators.

We determine the sharp $L_p$ Brunn–Minkowski range for planar dual quermassintegrals on origin-symmetric convex bodies: $0<q\leq 2$ for $0\leq p<1$, and $0<q\leq4$ for $p=1$. The main new results are the Minkowski inequality for $2<q<4$, proved using a weighted spectral inequality on the projective circle, and parallelogram counterexamples for $0<p<1$ and $q>2$. Together with the known logarithmic and endpoint results, we obtain the complete equality classification: only positive dilates occur, except at $(p,q)=(0,2)$, where parallelograms with corresponding sides parallel also give equality.

Differential-algebraic equation (DAE) systems arise in power networks, multibody mechanics, and chemical processes, where algebraic constraints couple dynamic and algebraic states. Standard control barrier function (CBF) methods, designed for ordinary differential equations, fail for DAE systems because the algebraic constraints impose compatibility requirements that may render safety filters infeasible. This paper develops a safety-critical control framework for semi-explicit DAE systems using CBFs that consider compatibility with the algebraic constraints, termed DAE-aware CBFs. We construct projected vector fields on the constraint manifold to account for the hidden dynamics, while ensuring compatibility conditions that keep the algebraic constraints satisfied. We then present sum of squares (SOS) verification conditions that certify both correctness and feasibility of the resulting DAE-aware CBF. The framework is validated on power system case studies.