Papers

We study discrete Hamiltonian systems defined on cotangent bundles that are subjected to external forces, whose trajectories are determined by a discrete variational principle. We analyze the evolution of the canonical symplectic structure and, when a Lie group of symmetries is present, the corresponding evolution of the associated momenta. Given a continuous forced Hamiltonian system, we construct an exact discrete analogue whose order-$r$ approximations yield trajectories that approximate the continuous ones with accuracy of at least order $r$. We also give two methods to build approximate discrete systems. Combining these, we obtain a variational integrator: first approximate the exact discrete system and then solve the resulting algebraic equations of motion.

State estimation in power distribution networks often involves sparse, noisy, and heterogeneous measurements, making classical weighted least-squares methods unreliable in low-observability regimes. To meet this challenge we develop a physics-informed low-rank tensor completion method. The method encodes voltage phasor components, voltage magnitudes, and power injections over time in a low-rank tensor, while incorporating physics through convex residual penalties. For distribution network state estimation, these residuals are instantiated using linearized power-flow equations, yielding a convex optimization problem over a tensor gauge ball, equivalently represented as a polytope implicitly described via integer programming. We solve this problem using globally convergent blended conditional gradients, resulting in a general-purpose algorithmic framework that can be adapted across different domains and applications where low-rank structure and convex physics residuals are available. We also derive a deterministic error bound showing that, under a restricted stability condition, the estimation error is controlled by measurement noise and physics-model mismatch. Numerical experiments show that the proposed method substantially improves state recovery compared with purely data-driven tensor completion and weighted least-squares, especially in low-observability regimes with missing data.

We study the binomial channel with input alphabet $[0,1]$ and output alphabet ${0,\ldots,n}$. We investigate its capacity and the structure of the capacity-achieving input and output distributions. Since the output alphabet is finite whereas the input alphabet is continuous, different input distributions may induce the same output distribution; hence, uniqueness and support properties of optimal inputs do not follow from strict concavity arguments. We first establish structural properties of the capacity-achieving input distribution. In particular, we show that it is discrete, unique, symmetric around $1/2$, and contains the endpoints ${0,1}$ in its support. We also derive location constraints and bounds on the probability masses of support points, and improve the Witsenhausen-type upper bound on the support size from order $n$ to order $n/2$. We derive explicit nonasymptotic upper and lower bounds on the capacity $C(n)$. These bounds imply $C(n)=\frac{1}{2}\log(\frac{n\pi}{2e})+o(1).$ The lower bound is obtained by evaluating the mutual information at the reference input $X_r\sim \mathrm{Beta}(1/2,1/2)$, which induces a beta-binomial output distribution, while the upper bound follows from a minimax redundancy construction. Finally, we prove an improved lower bound on the support size of the capacity-achieving input distribution. We show that the beta-binomial output induced by $X_r$ is asymptotically optimal and close to the capacity-achieving output distribution in relative entropy and $\chi^2$ divergence. We also prove a finite-mixture approximation lower bound showing that the beta-binomial output cannot be approximated too accurately by binomial mixtures with few components. Combining these results yields a support-size lower bound of order $\Omega(\sqrt{n\log\log n})$, with explicit constants. Numerical results illustrate the capacity bounds and optimal input.

Integrating information across related tasks can improve estimation and prediction in transfer, multi-task, and federated learning, but contamination and heterogeneity make robust borrowing challenging. We study a contaminated multi-task empirical risk minimization (ERM) framework in which an $\epsilon$ fraction of $K$ tasks, each with sample size $n$, may be arbitrarily contaminated while the remaining tasks are heterogeneous. Our goal is to estimate both the global minimizer of the average risk and the clean task-specific minimizers, thereby combining robustness and personalization. In the Gaussian mean model, we show that several common paradigms, including adaptive and robust regularization around a shared center, global matrix regularization, decomposition-based regularization, and score-based outlier-task detection, all suffer from a worst-case contamination error of order $\epsilon\sqrt{d/n}$, which is suboptimal compared to the lower bound $\epsilon/\sqrt{n}$. This identifies a dimension-dependent barrier for these approaches. We then establish minimax lower bounds for a general heterogeneous ERM setting and propose a computationally efficient filtering-based robust multi-task gradient descent method. Under local strong convexity, smoothness, and sub-Gaussian gradient assumptions, the proposed method attains high-probability upper bounds matching the minimax rates up to logarithmic factors over a broad regime. In particular, it removes the extra $\sqrt{d}$ contamination dependence of many regularization-based methods and score-based outlier detection, while achieving personalization to local tasks under strong heterogeneity. Simulations and a real-data analysis demonstrate strong robustness and personalization relative to a broad range of benchmark methods.

The classical Néron-Ogg-Shafarevich criterion characterises good reduction of an elliptic curve $E$ over a $p$-adic field via the action of inertia on the $\ell$-adic Tate module. However, the action of inertia on $E[\ell]$ is not sufficient to distinguish between potentially good and multiplicative reduction, and the action on $T_{\ell}(E)$ is not sufficient to determine the Kodaira type. We remedy this situation by endowing $E[\ell]$ with a distance function that records the $p$-adic distances between the $x$-coordinates of the points. We show that, equipped with this additional structure, $E[\ell]$ determines the Kodaira type of the elliptic curve. In the case of residue characteristic $2$, we assume that $E$ does not have potentially good reduction of type $I_n^*$.

We prove that the symmetry Lie algebra of a parametrized variety can be determined directly from the parametrization, without computing the vanishing ideal of the variety. We derive a practical polynomial-time Monte Carlo algorithm for computing the symmetry Lie algebra of a parametrized variety. We discuss applications to testing the binomiality of the ideal of a parametrized variety after changing coordinates, and test this property on varieties arising from staged tree models and colored Gaussian graphical models. Finally, we discuss symmetries and binomiality after changing coordinates for rational curves and give a characterization of the symmetries of many secant varieties.

We prove a Bernstein–Ganzburg type limit relation \[ \lim_{n\to\infty} \Bigl(\frac{n}{\sigma}\Bigr)^{(2a+1)/p}E_{n,\sigma}(f)_{p,a,b} =A_{\sigma}(f)_{p,a}, \] where $E_{n,\sigma}(f)_{p,a,b}$ is the error of best approximation of $f(nt/\sigma)$ by trigonometric polynomials of degree at most $n$ in $L^{p}((-\pi,\pi],|2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}\,dt)$, and $A_{\sigma}(f)_{p,a}$ is the error of best approximation of $f$ by entire functions of exponential type at most $\sigma$ in $L^{p}(\mathbb{R},|x|^{2a}\,dx)$. For $a=b=0$, this result was obtained by M.~I.~Ganzburg. The proof uses ideas from the Bernstein–Ganzburg limit theorems and a localization method with the Fejér kernel from the proof of the limit relation for Nikol'skii constants. As an application, using known results for polynomial approximation, we compute the exact value of $A_{\pi}(\mathbf{1}_{(-1,1)})_{1,a}$ for $a=0$ and $a=1/2$.

Solvable time-dependent systems provide important settings for studying non-equilibrium physics, where exact results are rare. They are also useful for benchmarking quantum simulations, which can directly probe real-time dynamics beyond the reach of conventional numerical approaches. In this paper, we show that the four-dimensional Chern-Simons theory offers a natural and unifying framework for constructing such systems. Focusing on classically integrable field theories, we consider a generalization of the four-dimensional Chern-Simons theory in which the usual holomorphic one-form is replaced by a more general, spacetime-dependent one-form. This yields a systematic procedure for generating time-dependent integrable field theories and establishes a universal relation: for every theory obtained in this way, the allowed time dependence coincides with the one-loop renormalization group flow. Despite the explicit time dependence, these theories retain Lax integrability and remain solvable by inverse scattering methods. Our construction applies to both ultralocal and non-ultralocal theories and extends previously known time-dependent sigma models to a much broader class of integrable systems. It also admits rewriting as dilaton gravity coupled to matter, producing a large family of classically integrable dilaton gravity theories in two dimensions. We also comment on connections to time-dependent integrable models studied recently in condensed matter physics and non-autonomous integrable systems arising from dimensionally-reduced Einstein gravity.

This article deals with combinatorial identities with two complex parameters. Starting with a fundamental lemma, we derive various polynomial identities, combinatorial sums and related results. For example, we generalize a polynomial identity of Carlitz involving central binomial coefficients and present a second identity of the same nature. Special cases of our findings lead to sums involving Catalan numbers, harmonic numbers, and Fibonacci numbers.

Variational quantum algorithms (VQAs) are a leading approach to near-term quantum computation, but their utility is limited by barren plateaus and other pathologies in their loss landscapes. Existing landscape theories based on dynamical Lie algebras, Jordan-algebraic Wishart systems, approximate t-designs, and Haar-random circuits are foundational, but they often neglect the finite-depth geometry of realistic ansätze and are therefore ill-suited to the shallow-depth regime, where VQAs are poor approximators of 2-designs and trainability is most feasible. This work introduces Krylov algebras, algebraic structures induced by the Krylov span of a finite generator set acting on one or more seed vectors, as a framework for VQA landscape theory. We show that VQA reachable manifolds can be approximated in a numerically robust, geometrically faithful fashion by Krylov-Lie algebras and groups, and that these structures induce canonical invariant measures for computing expectation values and variances under general sampling measures. In particular, we derive weighted non-Haar variance formulas that recover the usual Lie-algebraic Haar formulas as a special case while isolating non-Haar effects into explicit correction terms. We also show that the common heuristic that sufficiently deep circuit ensembles must converge to Haar fails in general without additional hypotheses, identify concrete obstructions to naive Haar convergence, and recover convergence under natural necessary and sufficient ergodic conditions. Lastly, our formulas further imply that non-Haar contributions to landscape statistics may mitigate barren plateaus by reweighting the visible sectors of the loss landscape, suggesting that VQAs may be more trainable than recent literature has posited.

A 2-partition of a digraph is a partition of its vertex set into two nonempty parts. Degree-constrained 2-partition problems are generally computationally difficult, even when the prescribed properties are expressed only in terms of minimum indegree, minimum outdegree, or minimum semidegree. Bang-Jensen and Christiansen~\cite{B-C} conjectured that the minimum-degree partition problems would be polynomial-time solvable on semicomplete digraphs when the degree thresholds are fixed, and Bang-Jensen and Gutin~\cite{B-G-Classes} posed the related Problems~2.8.15 and~2.8.16. We resolve this conjecture. More precisely, for every fixed pair of integers $k_1,k_2\ge 2$, we give deterministic polynomial-time algorithms that decide whether a given semicomplete digraph admits a $(\delta^+\geq k_1,\delta^-\geq k_2)$-partition, a $(\delta^+\geq k_1,\delta^0\geq k_2)$-partition, or a $(\delta^0\geq k_1,\delta^0\geq k_2)$-partition, and construct such a partition whenever one exists. Here, $\delta ^+,\delta ^-,\delta ^0$ represent the minimum out-, in-, semi-degree, respectively. The algorithms use small degree certificates, minimal cores, closure and protective-set arguments, and deterministic universal colorings with monotone recoloring, which develop a new method in partition algorithm construction.

This paper develops a probabilistic sign rule for quotients of functions represented by positive series or integrals. For a function in this class, normalising the summand function in the series case or the integrand function in the integral case induces a probability law under which parameter log-derivatives of the function are expressed as moments of kernels, the log-derivatives of the same summand or integrand function with respect to the same parameters. The resulting moment identities reduce quotient monotonicity, log-supermodularity, and log-convexity to sign criteria based on kernel monotonicity, stochastic ordering of the induced laws, and covariance or variance identities. The criteria are applied to generalised hypergeometric, Stieltjes-transform, and Prabhakar quotients, yielding new Turán inequalities, two-sided Stieltjes bounds, and a local failure threshold for a monotonicity conjecture for the zero-balanced Gauss function.

A poset $P$ is said to satisfy the finite antichain condition, or FAC for short, if it has no infinite antichain. Such posets exhibit rich and complex structure, and it was conjectured by Aharoni and Korman in 1992 that any FAC poset $P$ possesses a chain $C$ and a partition into antichains such that $C$ meets every antichain of the partition. While this conjecture is now known to be false, in this paper we prove that the conjecture does hold true for a broad class of posets. In particular, we prove that the Aharoni–Korman conjecture holds for countable posets containing no saturated chain $D$ such that either $D$ or its reverse $D^*$ is of the form $\bigoplus_{x\in\omega} D_x$, where each $D_x$ is infinite and co-wellfounded. In pursuit of this goal, we prove several structural results, the foremost of which demonstrates how a countable FAC poset may be broken up into a collection of scattered posets which reflect the structure of the poset as a whole.

We study the long-time stability of global self-similar solutions to two energy supercritical nonlinear wave equations, namely, the cubic nonlinear wave equation in $6$ dimensions and the corotational wave maps equation in $4$ dimensions. We prove the stability of self-similar solutions under perturbations that are small in the critical Sobolev spaces. The proof is based on Strichartz estimates for wave equations with potentials in similarity variables.

We study the spectral Barron regularity of Coulombic electronic eigenfunctions after extraction of a cut-off Jastrow factor. Let \(H=-\Delta+V\) be an \(N\)-electron Coulomb Hamiltonian with clamped nuclei, and let \(\psi\) be an eigenfunction associated with a discrete eigenvalue below the bottom of the essential spectrum. For the cut-off Jastrow factor \(F_{\rm cut}\) of Fournais–Hoffmann-Ostenhof–Hoffmann-Ostenhof–Sørensen, we set \[ \phi=e^{-F_{\rm cut}}\psi . \] Whereas the original wave function satisfies the sharp global threshold \(\psi\in \mathcal B_{\rm sp}^s(\mathbb R^{3N})\) for every \(0\leq s<1\), we prove that the Jastrow quotient gains one full order: \[ \phi\in \mathcal B_{\rm sp}^s(\mathbb R^{3N}) \qquad \text{for every } 0\le s<2 . \] The endpoint \(s=2\) is shown to be natural through an explicit hydrogen-like eigenfunction. The many-body proof is a global Fourier-side resolvent argument. After conjugation by the cut-off Jastrow factor, the Coulomb singularities are converted into localized angular coefficient blocks with admissible Fourier-control measures. Low frequencies are controlled by the a priori \(H^1\)-bound, while high frequencies are recovered by a Neumann fixed-point argument using the resolvent multiplier and annular estimates for the coefficient measures.

Olivier's 1827 theorem provides a rate of convergence to zero of the general term of a decreasing summable sequence of positive reals. We derive stochastic extensions of this result in the context of almost supermartingales. The results are applied to the analysis of stochastic iterative processes.

The Erdős–Gallai theorem states that every graph of average degree $d$ contains a cycle of length at least $d$. We prove the following robust extension of the Erdős–Gallai theorem: For every $c>0$ there exists $K$ such that for all $d\geq K$, $p\geq K/d$ and every graph $G$ with average degree $d$, the random graph $G_p$ obtained by independently sampling each edge of $G$ with probability $p$ contains a cycle of length at least $(1-c)d$ asymptotically almost surely as $|V(G)|\to \infty$. With related methods, we prove the following hyperstability version of the Erdős–Gallai theorem: any graph $G$ without a cycle of length at least $d$ is at most $c d|V(G)|$ edge deletions away from a graph all of whose connected components have a vertex-cover of size at most $d$. At the core of our argument lies a very general structure theorem about graphs that originates from results of Pokrovskiy concerning the hyperstability of bounded-degree trees.

We study the mixing times of stochastic spin systems corresponding to nearest-neighbour Glauber dynamics on dynamical percolation, defined on $d$-dimensional torus of side-length $N$. In this model, the status of each edge (open or closed) updates independently at rate $\lambda>0$, according to $\mathrm{Ber}(p)$ samples. Simultaneously, the spin of each site updates at rate $1$ according to Glauber dynamics on the environment restricted to open edges. We show that for a relatively general class of nearest-neighbour systems, as long as $p<p_c(d)$, for any temperature, if $\lambda$ is sufficiently small, the mixing time is of order $\frac{\log N}{\lambda}$. This Markov chain is non-reversible, and the proof is obtained by developing a particular coupling that couples together local configurations whenever the environment behaves well.

We prove regularity of a fully nonlinear equation that arises from the study of geodesics in the space of mixed volume forms on Hermitian manifolds admitting a balanced metric. Under conditions for ellipticity, we prove that this degenerate equation has a $C^{1,1}$ solution on Hermitian manifolds. We derive uniform Laplacian estimates for the perturbed equation, and also construct explicit subsolutions. In particular, this shows the existence of a unique $C^{1,1}$ solution to the Donaldson equation on Hermitian manifolds.

The Clifford torus in a Berger sphere with parameter $\tau$ is a critical point of the Willmore functional for every $\tau>0$, yielding a smooth path of Willmore surfaces. By estimating the Morse index along this path, we apply bifurcation theory to produce new symmetric Willmore tori emerging from the Clifford torus.

Let $G$ be a real connected semisimple Lie group with trivial center, no non-trivial compact factors, and all simple factors of real rank at least two. Let $\Gamma<G$ be an irreducible lattice and $P<G$ be a minimal parabolic subgroup. Amrutam–Hartman conjectured that every $\Gamma$-invariant von Neumann subalgebra $M\subset L^\infty(G/P,\nu_P)\rtimes \Gamma$ is of the form $L^\infty(G/Q,\nu_Q)\rtimes\Lambda$. We prove this classification whenever either $M\cap L^\infty(G/P,\nu_P)\neq\mathbb{C}1$ or $M\cap L(\Gamma)\neq\mathbb{C}1$, thereby reducing the Amrutam–Hartman conjecture to the extreme case $M\cap L^\infty(G/P,\nu_P)=M\cap L(\Gamma)=\mathbb{C}1$.

We consider a very general Rosenzweig-Porter-type model, $H=H_0+\lambda W$, where $H_0$ is an arbitrary Hermitian matrix and $W$ is a standard Wigner matrix. We precisely trace the localization properties of the eigenvectors and the eigenstate thermalisation hypothesis (ETH) as the coupling constant $\lambda$ interpolates between the trivial $\lambda=0$ case and the fully mean field regime of large $\lambda$. Our results hold uniformly in $H_0$ and $\lambda$, substantially generalising all previous local laws on deformed Wigner matrices even in the mean field regime. Our proof precisely captures the deterministic approximation to the resolvent which exhibits a strongly inhomogeneous structure. As a byproduct, we conclude the emergence of a mobility edge and study the phenomenon of re-entrant localization.

We study stabilizer state testing and learning with limited coherent quantum memory. Here an algorithm sequentially receives copies of an unknown $n$-qubit state, but may keep only $k$ qubits of coherent quantum memory between measurements. With unrestricted memory, seminal work of Gross, Nezami and Walter showed how to test $n$-qubit stabilizer states using $6$ copies, which is dimension independent, unlike the learning complexity of $\Theta(n)$. We show that this testing-vs-learning separation is lost under memory constraints. More concretely we show that (1) The sample complexity of testing stabilizer states in the $k$-qubit memory framework is $\Theta(n-k)$. Our upper bound goes via a novel connection to the hidden shift problem and the lower bound is proven using a novel approach to average case bounds on likelihood ratios via combinatorics of the stochastic orthogonal group. (2) The sample complexity of learning stabilizer states with $k$ qubits of memory, in the non-adaptive framework, is $\Theta(n^2/k)$. As a further application of our techniques, we prove an exponential lower bound for purity testing even when the memory may be left coherent throughout the protocol. Our main results identify coherent quantum memory as the resource enabling the usual separation between stabilizer testing and learning. In particular, even with $k=0.99n$ qubits of memory, there is no constant-copy stabilizer tester; furthermore for $k=cn$ qubits of memory (for $0< c < 1$), stabilizer testing is as hard as learning, with both requiring $\Theta(n)$ copies.

We exhibit a formula relating the correlation between Wilson loop variables in Potts lattice gauge theory to a topological quantity in the plaquette random cluster model. As applications we prove exponential decay of correlations between slowly growing Wilson loop variables for Ising lattice gauge theory on $\mathbb{Z}^3$ at all but the critical temperature, we demonstrate that the correlation length is finite at sufficiently high or low temperatures in any dimension, and we show that the correlation length of the model on $\mathbb{Z}^4$ with free boundary conditions equals that of the dual model with constant boundary conditions when $q$ is prime.

We study the Sylow permutation character of the symmetric group at the prime 2 and prove some new bounds on the number of parts of partitions corresponding to its constituents.

A conjecture posed by Ohsugi and Tsuchiya (2019) postulates that the Ehrhart $h^*$-polynomials of symmetric edge polytopes are $\gamma$-positive. We disprove this conjecture by exhibiting an infinite family of counterexamples. The smallest example provided by our construction is a $36$-dimensional symmetric edge polytope.

A fundamental component of homotopy type theory, a synthetic theory of $\infty$-groupoids, is Voevodsky's univalence axiom. Univalence characterizes the identity types in the universal fibration, a classifier for small type families: identity types in the universe are equivalent to types of equivalences. The directed univalence axiom plays a similar foundational role in simplicial type theory, a synthetic theory of $\infty$-categories. In its original form, which does not include universes or directed univalence, the simplicial type theory has semantics in categories of simplicial objects in an $\infty$-topos, with synthetic $\infty$-categories corresponding to internal $\infty$-categories. We verify that directed univalence holds in this semantic setting, constructing an equivalence between hom types in the universal left fibration and function types. In fact, we verify a higher version of this result, constructing an equivalence between homotopy coherent composites in the universal left fibration and composable sequences of functions between types. Using the technique of weighted limits, we reduce this theorem for simplicial objects in an arbitrary $\infty$-topos to calculations "on the left" with simplicial sets.

We study characterisations of strong $\Delta$-matroids, compiling a list of five equivalent descriptions. We show a variant of Wenzel's exchange property and the hyperplane exchange property of Borovik-Gelfand-White are equivalent. We also introduce two novel characterisations in terms of 'peerless' and 'isolated' antipodes within the system of feasible sets, banning certain configurations of antipodes either globally or locally. As a corollary, we obtain new 'local' exchange axioms for matroids and $\Delta$-matroids. We give algebraic motivation for these new characterisations by introducing the peerless antipode equations, tropical equations that govern whether a $\Delta$-matroid has no peerless antipodes. We show that these arise as the tropicalisation of a specific basis of quadratics cutting out the orthogonal Grassmannian.

We consider a quasilinear Allen–Cahn system which arises when the gradient energy term in the Ginzburg–Landau energy also contains zero order terms. Such systems offer significant advantages in applications, since surface tensions and mobilities can be easily calibrated. The analysis for these systems is highly challenging, partly due to the fact that the gradient term in the energy is non-convex and since gradient terms appear quadratically in the weak formulation. This explains why an existence theory has been lacking for nearly thirty years. In this paper, we give the first existence and uniqueness results for such systems. Firstly, we prove existence and uniqueness of local-in-time strong solutions using the theory of maximal regularity. Here, non-standard techniques have to be applied due to the fact that linear constraints on the solution are involved and due to nonlinear boundary conditions. Secondly, using a minimizing movement approach we show the existence of global-in-time weak solutions. Here, the main difficulty arises from the fact that the underlying energy is not $\lambda$-convex. We overcome this issue by proving higher integrability of the gradient of the solution, first showing that solutions are bounded and then using an approach by Giaquinta and Modica. This finally allows us to pass to the limit in the time-discrete approximation. Using the de Giorgi interpolation technique, we are also able to show a sharp energy decay property despite the lack of convexity of the energy.

Let $\{\eta^i\}_{i=1}^2$ be a an orthonormal coframe on a domain $U$ on a smooth surface $(\Sigma,g)$. When $\eta^i$ is smooth, it is well-known that there is a unique connection 1-form $\omega$ verifying Cartan's first structural equations $d\eta^i = (*\eta^i) \wedge \omega$, and Cartan's second structural equation $d\omega = K_g dvol_g$. We prove that this statement remains valid when the frame is $C^0 \cap H^{\frac12}$, where the structural equations are understood in the sense of distributions. From this, we deduce that the Gauss equation $\mathrm{Det}\, D^2 f = K_g (1+|Df|^2)^2$ holds for every graphical representation $f$ of an isometric embedding of regularity $C^1 \cap W^{1+\frac23,3}$ or $c^{1,\frac12} \cap BV^2$. As an application, we prove regularity and convexity results for isometric embeddings of closed surfaces and convex caps with $K_g \geq 0$.