Papers

In the context of renewable energies, wind energy appears as a sustainable alternative to address current environmental and energy challenges. This work studies the synchronization and stability of a network of wind turbines subjected to strong disturbances, by integrating a realistic modeling of wind variability by using the Ornstein-Uhlenbeck stochastic process. The dynamics of each wind turbine are described by a Kuramoto-type equation, while synchronization is analyzed through the time evolution of the phases. Stability is studied by analyzing the basin of attraction to the synchronous solution, namely the set of initial conditions leading to the stable synchronous state. Simulations carried out on various models ranging from an isolated wind turbine with constant power to an isolated wind turbine with variable wind power, reveal that the stability of the system is strongly influenced by inertia, damping, wind speed, wind fluctuation rate, correlation time, and coupling strength. Physically, these parameters control the balance between injected mechanical power, energy dissipation, grid-induced restoring forces, and the temporal structure of wind fluctuations, thereby determining the ability of the wind turbine to absorb perturbations and maintain synchronization under fluctuating wind conditions.

We establish three generalizations of the Küronya-Lozovanu jet separation criterion via Newton-Okounkov bodies: if an inverted standard simplex of size $n+k+\varepsilon$ is contained in all infinitesimal Newton-Okounkov bodies at $x$, then $K_X+D$ separates $k$-jets at $x$. We prove (1) a canonical-free version with a computable multiple $m(D)$; (2) a multipoint extension for simultaneous jet separation; and (3) a combination of both. Proofs use Trusiani's framework and Nadel vanishing. We conclude with explicit computations for a double cover of a product of elliptic curves.

In this paper, we view the collection of ideals of a commutative principal ideal ring from two perspectives: one as an ordered semigroup I(R) and the other as a category I_R . It is shown that I(R) is a regular ordered semigroup whereas I_R is a category with subobjects. Further we establish the correspondence between these structure

We investigate a reverse Faber-Krahn type inequality for the Robin Laplacian in a bounded smooth domain $\Omega \subset \mathbb{R}^N$ whose boundary has two connected components. We prove that a concentric spherical shell maximizes the first eigenvalue over a class of such domains under perimeter and volume constraints, and under an additional convexity assumption when $N \geq 3$. This result generalizes to a wider class, and extends to higher dimensions, the inequality of Hersch [20], whose approach was substantially based on a construction of the so-called effectless cut by Weinberger [35], so that we call it the Hersch-Weinberger inequality. Our method is based on the analysis of the gradient flow of the first eigenfunction and several approximation procedures, without relying on the effectless cut itself. The effectless cut being a complicated object related to the attractor of the gradient flow, we describe its most fundamental topological properties. In particular, we show that it does not necessarily have to be a hypersurface.

Neural networks have emerged as effective tools for solving ill-posed inverse problems. In many scientific applications, however, observational training data are insufficient, and learned inverse operators must instead be trained on synthetic data generated from the forward model. This requires specifying unknown parameters in the forward model and solving the model to generate synthetic observations. Typically, the unknown parameters are sampled from a prescribed probability distribution. Here, we show that this sampling strategy is not a neutral preprocessing step, but instead defines an implicit regularization operator. This result follows from the fact that the learned inverse operator minimizes empirical risk together with the classical result that conditional expectation minimizes mean-square error. We present theoretical results for the implicit regularization operator in both infinite- and finite-data settings, including Physics Informed Neural Networks (PINNs). These results are demonstrated numerically on three inverse problems of increasing complexity: a 1D linear Fredholm integral equation, a 1D nonlinear subsurface interface inversion, and a 2D nonlinear cross-well seismic traveltime tomography problem. Across all three problems, three distinct sources of regularization are identified in the learned operator: prior sampling, architectural, and physics-informed regularization. A mismatched sampling distribution is shown to degrade reconstruction quality in ways that neither more expressive architectures nor augmented physics residuals can fully correct. The results demonstrate that the sampling distribution should be chosen with the same care as a classical regularization functional and provide a practical framework for implementing more sophisticated regularization operators using neural networks.

This paper integrates the multivariate Arnoldi process into tangent space Hermite interpolation for manifold-valued data. A G-Arnoldi recurrence constructs a discrete orthogonal polynomial basis, improving numerical conditioning for high-degree approximation. The analysis uses a Markov inequality with a geometry-dependent exponent r; for the convex parameter domains considered here, r = 2 is available. Under suitable regularity and sampling assumptions, we establish C^0 and C^1 error bounds and analyze their propagation through the exponential map. When only function values are sampled, these bounds are O(sqrt(M) n^{-m}) and O(sqrt(M) n^{2-m}), respectively. When first derivatives are also sampled, the C^1 bound improves to O(sqrt(M(d+1)) n^{1-m}). Here n is the polynomial degree, d the parameter dimension, m the target regularity, and M the number of sampling points. For sampling sequences satisfying the fill distance condition with M \asymp n^{2d}, the corresponding bounds become O(n^{d-m}), O(n^{d+2-m}), and O(n^{d+1-m}), yielding superalgebraic convergence for targets admitting smooth extensions. Numerical experiments on SO(3) and S^2 compare the accuracy and computational cost of the Arnoldi-enhanced method with THI+Kriging, with supplementary experiments demonstrating approximation without derivative samples.

Kernel Stein discrepancy (KSD) is among the most popular goodness-of-fit (GoF) measures on general domains with a large number of successful deployments. One of the main applications of KSD is in constructing powerful GoF tests. However, tests relying on the classical U-/V-statistic-based KSD estimators have two major drawbacks. (i) Their runtime scales quadratically in the number of samples. (ii) Their asymptotic null distribution is computationally intractable in most cases, typically handled by bootstrapping. While it is known that the Nyström method permits accelerating KSD estimation with no loss of statistical accuracy under mild conditions, to the best of our knowledge, the fundamental question of its impact on bootstrap-based GoF testing is open; resolving this question is the focus of the current paper. In particular, we prove that the key properties of the quadratic-time bootstrapped KSD-based GoF test (asymptotic level and local consistency) are preserved by its Nyström acceleration. We numerically demonstrate the efficiency of the accelerated KSD estimator and bootstrap in the context of GoF testing of spherical and functional data. Our numerical results show that the Nyström-accelerated method performs statistically on-par with the quadratic-time approach, while requiring substantially smaller runtime.

Let $K$ be a number field, and $\varphi_{1},\ldots,\varphi_{g}\in K(t)$ be finitely many rational maps, each of degree at least $2$. We first show that for generic finite sets $\mathcal{A}_{1},\ldots,\mathcal{A}_{g}$ consisting entirely of points that are not $\varphi_{i}$-periodic, there exists a set of primes $\mathfrak p$ of $K$ of positive density such that for each $\mathcal{A}_{i}$ and every $\alpha\in\mathcal{A}_i$, $\alpha$ is not $\varphi_i$-periodic modulo $\mathfrak p$. The notion of genericity used here is defined in terms of the associated arboreal fields and is sharper than those previously used in the literature. Leveraging our proof in the generic case, we then show that the same conclusion holds for most expected cases of non-generic sets $\mathcal{A}_{i}$. Finally, we apply our result to confirm the dynamical Mordell–Lang conjecture for coordinate-wise actions of a class of maps that includes rational maps that are generic in this sense.

In this paper, we consider the following mixed local nonlocal Brezis-Nirenberg problem \begin{equation}\label{crit_pro_abstract}\tag{$\mathcal{P}_{2^*}$} -\Delta u+(-\Delta)^s u=\lambda |u|^{p-2}u+|u|^{2^*-2}u \text{ in } \Omega,\quad u=0 \text{ in } \mathbb{R}^N \setminus \Omega, \end{equation} where $\Omega\subset\mathbb{R}^N$ is a smooth bounded domain, $N\geq3$, $s\in(0,1)$, $\lambda>0$, and $2\leq p<2^*=\frac{2N}{N-2}$. We establish a compactness result for the following class of subcritical/critical problems \begin{equation}\label{sub_pro_abstract}\tag{$\mathcal{P}_{p_n}$} -\Delta u+(-\Delta)^s u=\lambda |u|^{p-2}u+|u|^{p_n-2}u \text{ in } \Omega,\quad u=0 \text{ in } \mathbb{R}^N \setminus \Omega, \end{equation} where $p_n \in (p,2^* ]$ and $p_n\to 2^*$. Specifically, for $p \in (2+\frac{4s}{N-2},2^*)$ when $N>6-4s$, and for $p \in (2^*-1,2^*)$ when $N\leq6-4s$, we prove that any bounded sequence of solutions $\{u_n\}$ to \eqref{sub_pro_abstract} is relatively compact in the energy space, and converges strongly to a nontrivial solution to \eqref{crit_pro_abstract}. To the best of our knowledge, this is the first paper to address this type of compactness result for a non-homogeneous operator. Due to the presence of the non-homogeneous operator, proving the compactness result requires several delicate new and novel estimates, which we believe will be of independent interest for further studies of related problems. As an application of this compactness result, under the same ranges of $N$ and $p$, we prove that \eqref{crit_pro_abstract} admits infinitely many sign-changing solutions.

In this paper, we investigate the eigenvalue problem for a non-local dispersal operator defined on a bounded spatial domain with Neumann-type boundary conditions. Unlike the classical Laplacian, the non-local operator lacks compactness, which gives rise to an essential (continuous) spectrum and severely complicates the study of its discrete eigenvalues. The main contribution of this work is the rigorous variational construction of a finite or infinite sequence of non-trivial eigenfunctions corresponding to isolated eigenvalues located strictly above the continuous spectrum. Since the existence of these eigenvalues is not generally guaranteed due to the potential collapse of the spectral gap, we establish explicit sufficient geometric conditions linking the domain's size and geometry with the variance of the dispersal kernel that assure the emergence of the principal eigenvalue. Subsequent eigenpairs are constructed inductively via finite-dimensional Galerkin approximations. Utilizing a decomposition of the operator into compact and multiplicative part, we prove the strong convergence of the minimizing sequences.

Let $G$ be a graph and $r \ge 1$. A vertex subset is $r$-independent if every connected component of its induced subgraph has size at most $r$. The family of all such subsets forms a simplicial complex, the $r$-independence complex $\Ind_r(G)$, generalizing the classical independence complex. Recent work has focused on shellability and vertex decomposability of these complexes. For chordal graphs, $\Ind_r(G)$ has the homotopy type of a wedge of spheres for all $r$, and some chordal subfamilies are known where these complexes are not even sequentially Cohen-Macaulay. Thus, determining chordal graph classes and values of $r$ for which $\Ind_r(G)$ is sequentially Cohen-Macaulay, shellable, or vertex decomposable remains an active area. Existing methods, based on chordal hypergraphs or special graph properties, do not extend to arbitrary chordal graphs. In this paper, we show that for every tree $T$ and every integer $r \ge 1$, the complex $\Ind_r(T)$ is vertex decomposable, resolving a conjecture \cite[Conjecture 3.15]{PD23chordal} of Abdelmalek et al. Our approach gives a structural description of shedding vertices via rooted subtrees and uses it to prove vertex decomposability recursively.

Computing many eigenpairs of the Schrödinger operator presents a computational bottleneck in large-scale quantum simulations due to the global communication overhead of explicit orthogonalization. To address this issue, we propose a quasi-orthogonal evolution model utilizing inverse operators and develop a corresponding discrete numerical scheme. Instead of forcing explicit orthogonalization, the proposed framework confines the numerical approximations within a quasi-Stiefel set, ensuring the iterates maintain full column rank without requiring $\left\langle U, U \right\rangle=I_N$. Moreover, the method naturally absorbs orthogonality errors and asymptotically converges to the exact eigenfunctions, even when initialized with non-orthogonal random data. The scheme guarantees monotonic dissipation of the target energy functional, with exponential convergence rates rigorously established for the discrete energy, gradient, and eigenfunction approximations. Furthermore, infinite-dimensional analysis proves that the admissible time step size is independent of the spatial discretization. This property overcomes the mesh-dependent stability constraints typical of conventional explicit or semi-implicit schemes, permitting larger time increments to accelerate global convergence. Numerical experiments validate the theoretical findings.

This paper introduces the Quantum Covariance Embedding, which embeds Positive Operator-Valued Measures into a tensor product of a Reproducing Kernel Hilbert Space and the quantum state space via a tensorized Bochner integral. This construction induces the Quantum Maximum Discrepancy that metrizes the space of quantum measurements. Applying this framework to Quantum State Tomography, we reformulate density estimation as a tensorized kernel regression, enabling optimal inference without the basis-dependent sparsity constraints that restrict existing methods. We develop a unified geometric design theory for quantum Gram superoperators, establishing that Unitary Designs are strictly E-optimal experimental designs and thus statistically superior to Pauli observables. For general structure-free estimation, we derive the exact minimax lower bound and prove that our tensorized estimators achieve this optimal rate. Furthermore, we introduce the QUAntum Regression with Kernels (QUARK) estimator to accommodate the spectral geometry of physical implementations, deriving central limit theorem and concentration inequalities. To facilitate practical estimation, we establish the exactness of trace-preserving projections and demonstrate efficient estimation under mutually unbiased bases via the fast Walsh-Hadamard transform.

We prove that if $(M^m, h)$ is a Yamabe metric, then the product metric $h + g_{\mathrm{flat}}$ on $M^m \times T^{n-m}$ is also a Yamabe metric whenever the flat torus $T^{n-m}$ is sufficiently small. This generalizes earlier results for $S^1 \times S^{n-1}$. Our method extends to the study of Type~I and Type~II Yamabe constants, $Q$-curvature problems, and an isoperimetric-ratio type problem.

Deploying metasurfaces (MTSs) to eliminate wireless blind spots requires jointly determining the physical placement of MTSs and the meta-atom phase shifts. Existing methods typically rely on explicit channel estimation, which incurs prohibitive overhead and is often intractable in real-world networks. To sidestep this bottleneck, we propose RFZero, a channel-state-information (CSI)-free deployment paradigm. Instead of estimating channels, RFZero extracts macro-environmental features from visual photos to guide MTS placement, and leverages reference signal received power (RSRP) feedback for dynamic phase-shift optimization. Most importantly, RFZero operates independently of base stations, thereby enabling seamless plug-and-play implementation. Real-world field tests confirm that RFZero completely eliminates all blind spots in a $100\text{ m}^2$ indoor area using just a pair of $1.5\text{ m}\times 0.9\text{ m}$ MTSs.

(A) It is known that among the currently unresolved cases of the axially symmetric Navier-Stokes equations (ASNS), the most relatively tractable one is where the fluid passes the exterior of a cone. In this paper, we investigate this case with Navier total-slip boundary condition. We show that there exists an absolute constant $C_* > 0$ such that if \[ \sup_{x\in D}r|v_{0,\theta}|\leq C_* \quad\text{and}\quad \int_{D} r v_{0,\theta}(x) \mathrm{d} x = 0, \] then there exists a unique global bounded strong solution with finite energy. Note that, for the initial velocity, there is neither a size restriction on other components, nor a parity assumption. There are four key ingredients in the proof. (1) Three new good unknowns are introduced, and a self-closed energy estimate for them is derived. (2) An elliptic estimate for pressure is established to control boundary terms arising from the boundary condition. (3) A De Giorgi iteration scheme is applied to establish the boundedness of $rv_\theta$. (4) A new anisotropic Hardy's inequality is derived for weighted mean-zero functions to overcome the lack of parity of $\boldsymbol{v}$. (B) Based on (A), we introduce and prove the so-called controlled regularity for the above problem, i.e. for suitable initial data without any smallness assumption, there exists an external force supported away from the axis of symmetry such that the corresponding problem admits a global strong solution. This seems to add a little weight to the regularity scenario for ASNS, since the force is supported away from the axis which is the only place regularity may break down. We also prove that if there exists a solution that blows up in finite time, an unstable blow-up solution must exist.

For a full-rank integral lattice $\mathcal{L}\subset\mathbb{R}^n$, Regev and Stephens-Davidowitz proved that \[N_{=k}(\mathcal{L}):=|\{y\in\mathcal{L}:\lVert y\rVert^2=k\}|\le 2\binom{n+2k-2}{2k-1}.\] We classify the equality cases. For $n\ge2$, equality holds if and only if either $k=1$ and $\mathcal{L}\cong\mathbb{Z}^n$, or $n=8$, $k=2$, and $\mathcal{L}\cong E_8$. For $n=1$, equality holds exactly when $\mathcal{L}$ represents $k$. The proof shows that equality is rigid. Saturation of the shell bound forces the normalized norm-$k$ shell to be an antipodal tight spherical $(4k-1)$-design. The associated Delsarte–Goethals–Seidel annihilator polynomial gives an arithmetic root condition, which isolates $E_8$ at $k=2$, rules out $k=3$, and combines with the Bannai–Damerell/Bannai theorem and an elementary circle argument to exclude all remaining cases in dimension at least $2$.

We extend the notion of illumination bodies to Riemannian spaces of constant curvature and to projective Finsler geometries. We prove that the derivative of their volume defines a notion of surface area for convex bodies in these settings, generalizing the affine surface area in Euclidean space. The proof is based on a general result on the derivative of weighted volumes of weighted illumination bodies in Euclidean space. In the appendix, we give some explicit examples for non-Euclidean illumination bodies.

We study the Gromov–Sormani MinA scalar curvature compactness conjecture for warped product metrics on $\mathbb{S}^2\times\mathbb{S}^1$ of the form introduced by Kazaras-Xu in \cite{KazarasXu2023} as follows: \[ g_i=\varphi_i^{-2}h_i+\varphi_i^2d\xi^2, \qquad h_i=dr^2+u_i^2(r)d\theta^2. \] Assuming nonnegative scalar curvature, a uniform volume upper bound, and a positive lower bound for the areas of closed minimal surfaces, we prove a uniform diameter bound for the base surfaces $(\mathbb{S}^2,h_i)$. Based on this key estimate, we further obtain compactness of the base warping functions $u_i$ and local and global estimates for the fiber warping functions $\varphi_i$. After passing to a subsequence, the metrics converge in $L^p$, for every finite $p$, to a limit metric $g_\infty$. %on the regular region. We also obtain Gromov–Hausdorff and Sormani–Wenger intrinsic flat subconvergence, and prove that $g_\infty$ has nonnegative scalar curvature in the distributional sense of Lee–LeFloch. Thus the Gromov–Sormani scalar curvature compactness conjecture is verified for this warped product class. Finally, we construct a $C^{1,\alpha}$ example illustrating the subtlety of volume-limit tests for nonnegative scalar curvature in low regularity.

We show that SU(1,1) NLFT can diverge pointwise for square-summable coefficients. As a consequence, we prove that the classical pointwise asymptotics of polynomials orthogonal on the unit circle can fail for measures in the Szegö class. We also discuss some special cases when the pointwise convergence holds.

Existing work on population dynamics inference often focuses on flows arising from vector fields that are the gradients of scalar potentials. Among all admissible flows that are compatible with the population dynamics, gradient flows are optimal in a specific sense: they minimize kinetic energy. The selection of fields based on different criteria corresponds to a gauge freedom when determining population dynamics, which we leverage in this work. We propose Non-Gradient Inference Flows (NGIF), an algorithm to infer non-gradient population dynamics using a weak formulation of the continuity equation. This allows us to parameterize general vector fields and choose other selection criteria beyond minimal kinetic energy. We demonstrate on a variety of low- and high-dimensional physics problems that this more general approach improves distributional accuracy over gradient-restricted baselines and better captures non-potential transport.

In this paper, we mainly establish the uncertainty principle (UP) for a function and its quaternion Fractional Fourier transform (QFrFT), as well as the UP for two QFrFTs. Using the polar representation of quaternion-valued signals, we give the UP for QFrFT in both the spatial and directional domains, providing a more precise condition for equality, example is given to verify the results. Furthermore, we extend the time-frequency UP to a frequency-frequency setting.

Multilevel programming is the standard framework for modeling hierarchical decision-making. In this paper, we characterize the computational complexity of deciding the existence of feasible and optimal solutions, as well as computing the optimal objective value in multilevel linear programming (LP). Our analysis considers various combinations of modeling assumptions, including the presence or absence of linking (coupling) constraints and whether all variables are bounded. In particular, we show the feasibility problem of $k$-level LP is $\Sigma^{p}_{k-1}$-complete for $k \ge 2$. Without linking constraints and unbounded variables, it is polynomial-time solvable for $k \le 4$ but becomes $\Sigma^{p}_{k-1}$-complete for $k \ge 5$, indicating a sharp jump in computational complexity assuming the polynomial hierarchy does not collapse. Combined with other results, one major implication is that no polynomial-time Turing machine can transform a bilevel LP instance with linking constraints into one without linking constraints while preserving feasibility unless P $=$ NP. In contrast, such machines exist for all $k \ge 5$. We observe similar phenomena with the decision of the existence of an optimal solution. In the bilevel case, feasibility and boundedness fully characterize the existence of an optimal solution, implying that the problem is DP-complete. However, these conditions are insufficient for $k \ge 3$ and the problem for $k \ge 3$ is $\Delta^{p}_k$-complete. Similar to the feasibility problem, the problem becomes polynomially solvable for $k=2,3$ without linking constraints and unbounded variables. However, the problem is $\Delta^{p}_k$-complete for $k \ge 4$, even with these simplifying assumptions. The computation of the optimal objective value is F$\Delta^{p}_k$-complete for any $k \ge 2$, even without linking constraints and unbounded variables.

In this paper, we prove that weak solutions to the 2D viscous and resistive magnetohydrodynamic (MHD) equations are non-unique in $L^2_t L^p(\mathbb{R}^2) \cap L^1_t W^{1,p}(\mathbb{R}^2)$ for given any $1\le p<\infty$, showing the sharpness of the Ladyzhenskaya–Prodi–Serrin condition at the endpoint $(2,\infty)$ and the solutions live on the borderline of the Beale–Kato–Majda criterion. To the best of our knowledge, this is the first non-uniqueness result for the 2D viscous and resistive MHD system. As byproducts, we also obtain non-uniqueness for the Navier–Stokes equations in $L^2_t L^p$ with $1\le p<\infty$, and for the MHD system with large $\mathrm{BMO}^{-1}$ initial data.

Autonomous driving stacks must pick one trajectory from a multi-modal candidate set; choosing by model confidence ignores safety, traffic-law, and comfort constraints. We present RECTOR (Rule-Enforced Constrained Trajectory Orchestrator), a post-generation reranking layer that scores candidates against a tiered rulebook (Safety~$\succ$~Legal~$\succ$~Road~$\succ$~Comfort) via differentiable proxies and a scene-conditioned applicability mechanism, then selects with a deterministic $\varepsilon$-lexicographic rule that preserves cross-tier priority by construction – without retraining the predictor. On the Waymo Open Motion Dataset validation\_interactive split (43{,}219 augmented instances, $K{=}6$), under Protocol~B (28-rule proxy catalog, oracle applicability) rule-aware selection cuts Safety+Legal violations from 28.58% to 20.42% and Total from 40.32% to 32.41% versus confidence-only on the same candidates. A uniform-weight weighted-sum baseline matches binary compliance on this benchmark – the empirical lift comes from rule-aware ranking, while the lexicographic guarantee is the structural differentiator no weight calibration can replicate. Under adversarial confidence corruption, confidence-only selection fails in 100% of scenarios while both rule-aware selectors reject the injected mode in $\sim$96%. All figures are proxy-evaluator results (not a safety certificate), open-loop, 5\,s horizon, U.S.\ rules, validation split.

Particle Swarm Optimization (PSO) frequently suffers from premature convergence. This paper introduces a family of problem-informed diversity-enhancing strategies that manipulate the swarm's social and cognitive components. These include opposing-best strategies that repel particles from optimal regions, negative learning strategies that guide exploration toward poor solutions, and reverse learning strategies that push particles away from inferior regions. These socio-cognitive mechanisms are evaluated against an analogous suite of problem-unaware, explicit randomization strategies that inject randomness either into velocity update components or directly into position updates. The results reveal that the effectiveness of diversity enhancement is determined primarily by how it is embedded within the swarm dynamics, rather than by the mere presence of extraneous problem-informed guidance. Particularly, random perturbations introduced at the velocity-update level consistently outperform those applied directly to particle positions.

We improve upon the Johnson-type bounds of Hayashi–Yasunaga and Liu–Tjuawinata–Xing for insertion–deletion codes by encoding each local list into a binary constant-weight code. The resulting local list-size bound is tight over sufficiently large alphabets. Combining this bound with an averaging argument and the constant-weight McEliece–Rodemich–Rumsey–Welch bound yields an asymptotic rate bound that strictly improves Yasunaga's Elias-type bound throughout the nontrivial range.

For a graph $G$, let $\tau(G)$ denote the number of spanning trees. We show that for every fixed $0 < c < 1/4$, the number of distinct values of $\tau(G)$, as $G$ ranges over simple graphs on $n$ vertices, is at least $\exp(c n \log n)$ for all sufficiently large $n$. This is optimal up to the choice of the constant $c$ and resolves a conjecture of Chan-Kontorovich-Pak regarding a problem of Sedláček from the late 1960s.

A lot is known about the moduli space of parabolic bundles over curves of genus $g\geq 2$, but the lower genus cases are notably different. The goal of this article is to study the geometry of the moduli space of semistable parabolic bundles of rank $3$ with trivial determinant and one marked point on an elliptic curve. We show that this moduli space is rational, give an explicit description of its geometry, its group of automorphisms and also prove a Torelli-type result.

We define a complete metric structure on the family $\text{PL}_q^p(\mathbb{R}^n)$ of probability measures with densities in $L^p(\mathbb{R}^n)$ and finite $q$-moments. We establish the existence of generalized minimizing movements for the isoperimetric ratio and characterize absolutely continuous curves in this space through weak solutions of the continuity equation with velocity fields satisfying a first-order integral condition. We also characterize absolutely continuous curves in the $\infty$-Wasserstein space and prove a Benamou–Brenier formula for $W_\infty$.