Papers

Physics-Informed Neural Networks (PINNs) build neural representations of time-dependent PDE solutions, naturally incorporating physics knowledge and observational data, which makes them well suited to both forward and inverse PDE problems. PINNs, however, are known to suffer from spectral bias and lack of causality. Neuro-Spectral Architectures (NeuSA), a recently proposed alternative to PINNs, mitigate both issues, but their numerical integration becomes unstable for stiff differential equations arising in many relevant physical problems. This study proposes Neuro-Spectral Exponential Time Differencing Architectures (NEXT), which combines the spectral representation of the PDE solution in NeuSA with high-order exponential integrators. Within this approach, the linear stiff part of the vector field induced by the PDE is integrated exactly through matrix exponentials, while the possibly nonlinear remainder is modeled by a neural network. The effectiveness of NEXT is verified through benchmark experiments on a set of stiff PDEs, in which NEXT is stable and accurate while NeuSA diverges numerically. It is also shown that NEXT can be applied to inverse problems, where the model has to learn unknown parameters or boundary conditions from sparse data. All code used in this work is publicly available at: https://github.com/marcioh2m/next.git .

For an $n$-Lie–Rinehart algebra, $n>2$, the anchor is defined on $\Aext{n-1}L$, so its kernel does not determine a distinguished submodule of $L$. We introduce the anchor radical $\Rad(\rho)\subset L$ and prove that it is an adjoint-stable ideal on which the bracket is $A$-multilinear. We then derive an explicit obstruction to the $A$-linearity of metric compatibility in fundamental directions. This obstruction vanishes on $\Rad(\rho)$, whereas a strongly nondegenerate compatible metric on all of $L$ forces the anchor to vanish. These results lead to a natural notion of orthogonal structure on the anchor radical. For anchors induced by Nambu tensors, the radical is identified with the cotangent annihilator of the tensor.

We pin down the largest entries possible in positive integral friezes of types $D_n$ for $n\geq 4$ and $B_n$ for $n\geq 2$, addressing a conjecture of Robin Zhang. For type $D_n$, the sharp upper bound is $F_nF_{n+1}-1$, and for type $B_n$, it is $F_{n+1}F_{n+2}-1$, where $F_k$ are the Virahanka–Fibonacci numbers.

On a network with a fixed set of verified sources, discounted averaging induces an equilibrium support $h_i^S=\mathbb{E}_i[\lambda^{T_S}]$, the discounted probability that a random walk reaches $S$ before attenuation. We define the discounted hitting domination number $\delta_{\lambda,\tau}(G)$ as the minimum number of sources required to guarantee $h_i^S\ge\tau$ at every vertex. Although this potential is known through penalized and group hitting probabilities, the associated minimum-cardinality uniform-coverage problem appears to be new. Aggregate support is monotone submodular, while the uniform-floor problem is an exact submodular-cover problem. Moreover, if $\lambda^{r+1}<\tau\le\left(\frac{\lambda}{\Delta}\right)^r$, then $\delta_{\lambda,\tau}(G)$ equals the distance-$r$ domination number. This yields NP-completeness and APX-completeness at $(\lambda,\tau)=(1/4,1/14)$ on graphs of maximum degree three. For spiders, we obtain an exact finite-state characterization and a polynomial-time algorithm for every fixed rational pair $(\lambda,\tau)$, and show that the branching vertex need not belong to a minimum source set. Finally, an exact mixed-integer linear formulation certifies optimal placements on a real network and a synthetic graph and demonstrates substantial differences from degree, closeness, and classical domination.

Let $E$ be an elliptic curve defined over rational numbers. Further assume $E$ has good ordinary reduction at $p=2$ and $E[4]$ is reducible as a $G_\mathbb{Q}$-representation. In this paper, we offer sufficient and necessary computational criteria for the algebraic Iwasawa $\mu$-invariant over the cyclotomic $\mathbb{Z}_2$-extension of rational number to be $0$, $1$ or $\geq 2$. We also show that elliptic curves with isomorphic mod-$4$ representations have algebraic $\mu$-invariant equal to $0$ if one of the curves does.

We study first-order differential quantities built from solutions of second-order black-hole wave equations. These quantities can satisfy new differential equations that contain extra singular points, even when the original physical solutions remain regular. We show that an isolated zero of the transformation determinant can produce such an apparent singularity, but it does not by itself create a new resonance or quasinormal-mode frequency. We then study the corresponding Green functions and show that the physical transformed response has the same spectral poles as the original problem, provided that the transformation does not become globally degenerate. For the Regge–Wheeler and Zerilli equations, this means that ordinary turning points can appear as singular points of the derivative equation without changing the nonzero-frequency quasinormal-mode spectrum. The zero-frequency case is different and must be studied separately.

Let $E$ be an elliptic curve define over $\mathbb{Q}$. Further assume $E$ has good ordinary reduction at $p=2$. In this article, we prove Greenberg's conjecture on the value of the algebraic Iwasawa $\mu$-invariant when $E$ has a rational isogeny of degree 2, and obtain a sharp absolute upper bound of the $2$-adic $\mu$-invariants for all such curves. We also prove the vanishing of the algebraic $\mu$-invariant of the fine Selmer group introduced by Coates and Sujatha at $p=2$.

Bi-invariant geodesic regression generalizes linear regression to Lie groups. Its main feature is that it respects the symmetries of the group so that the resulting estimator is independent of arbitrary choices such as a reference frame. However, the local existence and uniqueness of the underlying estimator have not been shown until now. Furthermore, the convergence properties of the proposed algorithm for computing the estimator are not known. In this work, we investigate these questions. We prove that, locally, a unique estimator exists and give explicit bounds on the size of this neighborhood. We also show that the proposed iterative algorithm converges linearly to this estimator.

We study the profinite non-rigidity of arithmetic lattices and its implications for the Kähler property. In the first part, we characterize the absolute Dynkin types that admit non-isomorphic real forms of higher-rank Lie groups containing torsion-free arithmetic lattices with isomorphic profinite completions. In the second part, as a geometric application, we answer a question asked independently by Arapura and Libgober, by showing that the Kählerness of finitely presented, residually finite groups is not determined by its profinite completion.

We develop several tools to study the local and global structure of the Teichmüller and moduli spaces for complex nilmanifolds. As a starting point, we prove a structure theorem for holomorphic maps between complex nilmanifolds, which allows for an explicit description of the Teichmüller and moduli spaces. We then introduce two period maps, and give general criteria for the existence of a complex structure on these spaces. To illustrate the theory, we prove a global Torelli theorem for various classes of nilmanifolds, including principal torus bundles over tori and almost abelian nilmanifolds. In particular, we show that there are nilmanifolds of arbitrary dimension and nilpotency index for which the Teichmüller space admits the structure of a complex manifold.

We investigate the long-time behavior of small solutions to the one-dimensional Hartree equation with the soft Coulomb interaction. The Hartree equation arises as an effective mean-field model for the evolution of many-body quantum systems, while the soft Coulomb potential provides a regularized one-dimensional approximation of the Coulomb interaction that removes the singularity at the origin while preserving its long-range character. Owing to the slow spatial decay of the interaction kernel, the equation exhibits long-range nonlinear effects. We prove global existence and modified scattering for sufficiently small, exponentially localized initial data. A notable feature is that, in contrast to the logarithmic phase correction arising in the well-known three-dimensional Coulomb Hartree equation, the one-dimensional soft Coulomb interaction produces a leading phase correction of order $(\log t)^2$, followed by a lower-order logarithmic correction. The proof is carried out in analytic function spaces. The exponential localization of the initial data is converted into analyticity of the transformed profile, and a method based on a generator function with a time-dependent radius of analyticity is used to compensate for the derivative loss arising from the nonlinear phase equation.

The goal of personalized immunogenomics is to recover an individual's germline immunoglobulin gene segments from 'repertoire sequences' altered by trimming, extension, and mutation. In this work, we study the fundamental trace complexity (or the number of samples/traces required for accurate reconstruction) of D-gene reconstruction under three biologically motivated trace-generation models introduced by Bhardwaj et al. (2021) and develop practical algorithms for reconstruction problems left open in the original work. First, for the TrimSuffixAndExtend model, we establish the optimal trace complexity to be {\Theta(n)}, and develop a low-complexity Prefix-Filtered Mode (PFM) decoder that achieves this scaling. Second, for the closely related two-sided TrimAndExtend model, we show that the optimal trace complexity is instead {\Theta(n^2)}, and is achieved by the simple Bit-Wise Mode (BWM) decoder. Third, for the SuffixExtend-t(TrimSuffix) model, we establish polynomially separated lower and upper bounds on trace complexity. Our results follow from information-theoretic lower bounds coupled with tight analyses of the proposed reconstruction algorithms.

We establish sharp weak-type bounds for commutators of Calderón–Zygmund operators and $\text{BMO}(\mathbb{R}^n)$ symbols with respect to $A_p$ weights. We prove that the bound \looseness=-1 $$ \|[b,T]\|_{L^p(w)\rightarrow L^{p,\infty}(w)}\lesssim [w]_{A_p}^{\max(2,p')} $$ for $p \in (1,\infty)$ and $w \in A_p$ is sharp for general $T \in \text{CZO}(\mathbb{R}^n)$ and $b \in \text{BMO}(\mathbb{R}^n)$. In the case of the standard logarithmic symbol, we show that the estimate improves to $$ \|[\log|\cdot|,T]\|_{L^p(w)\rightarrow L^{p,\infty}(w)}\lesssim [w]_{A_p}^{\max\big(2,\frac{1}{p}+\frac{p'}{p}\big)} $$ and that this dependence is optimal for this symbol. Our upper bound for the logarithmic symbol follows from the pointwise control of the commutator by a sum of the Hardy–Littlewood maximal operator and the squares of the radial Hardy operator and its adjoint, while our lower bounds are consequences of a general result for singular integral operators.

The notion of sum-freedom of binary functions was introduced recently by C. Carlet as a generalization of the APN functions used in cryptography; the $q$-ary version of the notion is a natural extension. For each integer $k$ with $0\le k\le n$, there is a $k$th order sum-free function on $\Bbb F_{2^n}$. It is also known that when $k/n$ is not close to 0 or 1, the multiplicative inverse function on $\Bbb F_{2^n}$ is not $k$th order sum-free. We generalize these two results to $q$-ary functions. APN functions have a coding theoretic characterization. We generalize the characterization to sum-free functions of arbitrary order over any finite field. It is well known that the Welch functions is 2nd order sum-free. We give an alternative proof for this result which leads to a more general algebraic question. We also investigate that the 3rd order sum-freedom of the Welch function and power functions of algebraic degree 3. We formulate a conjecture about the 3rd order sum-freedom of the Welch function which is supported by strong numerical evidence.

We investigate a class of backward stochastic differential equations (BSDEs) with at most quadratic growth which explode at a possibly unbounded random horizon, defined through the first hitting of zero of an adapted Ito process. In contrast with the classical theory of singular BSDEs, the explosion is generated by the nonlinear dependence of the generator on the martingale integrand, rather than by a superlinear coercivity condition in the solution component. We construct a minimal singular solution and derive two-sided estimates on its explosion, together with weighted BMO estimates for the martingale integrand. We also obtain the uniqueness and exact explosion rates under additional structural assumptions. For Hamiltonian generators, we establish a verification theorem for an infinite-horizon stochastic optimal control problem with possible non-Markovian state constraints, showing that the BSDE feedback induces the unique optimal constrained law. We finally specialize the theory to exit times of uniformly elliptic Markov diffusions and recover the connection with large solutions of viscous Hamilton-Jacobi equations.

In this paper, we give conditions on a unital non-associative ring $R$ and additive map $\sigma: R\longrightarrow R$ so that the non-associative skew power series ring, denoted by $R[[X;\sigma]]$ and the non-associative skew Laurent series ring denoted by $R((X;\sigma))$ are left (resp. right) Noetherian, respectively. Consequently, we establish versions of Hilbert Basis Theorem of $R[[X;\sigma]]$ and $R((X;\sigma))$, respectively, for left (resp. right) Noetherianity . We also give a counterexample showing that the Hilbert Basis Theorem does not hold in general in the non-associative case. In establishing the Hilbert Basis Theorems for $R[[X;\sigma]]$ and $R((X;\sigma))$, we introduce and utilize the right (resp. left) quasi-associative property of $R$, which is equivalent to $rR$ (resp. $Rc$) being one-sided ideals for all $r\in R$ (resp. $c\in R$).

In 2025, Jordan et al. introduced Decoded Quantum Interferometry (DQI), a quantum algorithm for combinatorial optimization based on decoding. They considered Reed-Solomon decoding and its associated optimization problem, called Optimal Polynomial Intersection (OPI), which may be viewed as a polynomial regression problem over a finite field. DQI exhibits provable speedups on certain problem instances and establishes a connection between decoding problems and optimization tasks. Leveraging the well-understood dual structure and decoding theory of algebraic geometry codes from other curve families, we introduce COFI: Curve-based Optimal Function Intersection. By considering two-point Hermitian codes, Suzuki codes, and extended norm-trace codes, we broaden the range of algebraic geometry codes used in DQI and identify families that offer further improvements over one-point Hermitian codes in the Hermitian Optimal Polynomial Intersection framework considered by Jordan and Gu. Depending on the family and parameter regime, these curves can reduce quantum resource requirements or increase the number of constraints that can be considered.

We introduce a new method for constructing Zoll metrics that admit first integrals polynomial in the momenta. The method uses Funk's form of the metric together with an algebraic relation between the first integrals. This reduces the problem to an algebraic system from which all unknown functions can be determined explicitly. In the case of a non-trivial cubic integral, we present an explicit two-parameter family of metrics that covers all such rotationally symmetric Zoll metrics on the $2$-sphere: every such metric is isometric to a member of this family. We compare these metrics with the earlier results of Valent, Duval, and Shevchishin.

In this work, we enumerate the ternary perfect forms over all imaginary quadratic fields of absolute discriminant up to $91$ and study the number and combinatorial types of their associated polytopes. We prove a bound on the number of combinatorial types that can arise regardless of the discriminant. We analyze the data and make several observations concerning the number and complexity of the perfect forms that arise in the scope of our calculation.

The classical ADE classification implies that $E_8$ is the unique irreducible unimodular Euclidean lattice generated by vectors of norm $2$. In sharp contrast, for every positive integer $m$, we construct an irreducible odd unimodular lattice of rank $24m$ and minimum norm $3$ that is generated by its minimal vectors.

We generalise a mixing result due to Pommerenke for non-autonomous dynamical systems of forward compositions of the boundary values of centred inner functions; the proof depends on a precise bound on the orbits of such forward compositions in the unit disc. This enables us to give new criteria for such systems to have positive metric entropy, lower entropy and topological entropy. In the opposite direction, we construct a forward composition of centred degree two Blaschke products with topological entropy zero.

Outliers are essential for evaluating and improving the robustness of machine learning systems, especially when future distributions may differ significantly from historical training data. In high-stakes applications, robustness often depends on rare cases that finite datasets fail to capture, making simple resampling or perturbation insufficient for stress scenario generation. Existing outlier synthesis methods typically rely on sparse neighborhoods, low support latent regions, or classifier boundary crossings, which can be heuristic, unstable, and tied to specific modalities or architectures. We therefore propose Sinkhorn Boundary Outlier Generation (SBOG), a structured framework for latent-space outlier generation that couples Sinkhorn optimal transport geometry with distributionally robust boundary modeling. The resulting Sinkhorn-induced support cost guides the sampler toward weakly supported boundary regions, while semantic constraints prevent uncontrolled drift from the intended context, yielding controlled deviations from the in-distribution reference measure rather than arbitrary sparse-region samples. Experiments on time series anomaly generation and image outlier synthesis show that our framework produces informative, semantically controlled outliers and improves downstream robustness evaluation across modalities, providing a foundation for stress scenario generation beyond empirical support.

Shapley values, a solution concept from cooperative game theory, have recently become a standard tool for feature credit allocation in machine learning. They provide an axiomatically justified method to fairly distribute a model's prediction among the data features. Shapley values have not yet been applied to explain machine learning models trained on features from topological data analysis. We develop what we believe is the first such approach, focusing on the persistence landscape featurization of persistence diagrams. Because each landscape coordinate is a rank statistic, crediting a model's prediction back to individual persistent homology classes (persistence diagram points) is nontrivial. We introduce LandscapeSHAP, a method for fair credit allocation to persistence diagram points based on a model's prediction. For linear models on persistence landscapes, LandscapeSHAP has a closed form expression that gives the exact Shapley value of every persistence diagram point. In particular, there is no coalition sampling required. We further prove that the four Shapley "fairness" axioms uniquely characterize this credit allocation for any model, not only linear ones. For a general nonlinear model, this unique value can only be calculated exactly from its defining coalition averaging formula, which requires considering all $2^N$ many coalitions, where $N$ is the number of points in the persistence diagram. This is computationally intractable for persistence diagrams of realistic size. We complement the exact linear model result with an efficient Monte Carlo sampling of persistence diagram coalitions. We give convergence rates in terms of number of samples needed to approximate to a desired degree of accuracy. We also prove stability results for the LandscapeSHAP credit allocation, for any model.

Transformers have become a central architecture for in-context learning (ICL), particularly through their state-of-the-art performance in large language models. This success motivates understanding how transformers exploit task-relevant structure in geometrically heterogeneous data. However, existing nonparametric ICL theory has largely focused on Euclidean domains or single-manifold models. To address this gap, we study the prediction problem under unknown local geometry, modeled by sample size-dependent mixtures of manifolds with heterogeneous dimensions, smoothness, and sampling masses. Under local separation and small-perturbation conditions, we establish a minimax lower bound capturing the aggregate difficulty of the components and construct an oracle tangent local-polynomial estimator with a matching upper bound. This estimator is connected to a structure-informed, two-stage softmax transformer with a geometric preconditioner and chartwise reduced local-polynomial solvers. The transformer achieves negligible approximation error relative to the minimax rate with logarithmic depth and polynomial size. Finally, we derive an in-context generalization bound for near empirical risk minimizers over this class. Together, these results identify conditions under which the resulting predictor exploits local geometry and attains the aggregate minimax rate.

We show that there exists a constant $\epsilon > 0$ such that if $C$ is a conjugacy class in $A_n$ of size at least $|A_n|^{1-\epsilon}$, then $A_n \setminus \{1\} \subseteq C^2$. This proves the robust version of Thompson's conjecture for the alternating group, conjectured by Shalev (Annals of Math., 2009). Our proof combines character bounds and combinatorial cancellation techniques with the recent theory of hypercontractivity for functions over symmetric groups.

The goal of this paper is to clarify the relation between the Kerler-Lyubashenko TQFT $J_\mathcal{E}$ associated with the adjoint end $\mathcal{E}$ of a (not necessarily semisimple) modular category $\mathcal{C}$ and its renormalized version $V_\mathcal{E}$ based on the modified trace supported by the ideal $\mathrm{Proj}(\mathcal{C})$ of projective objects of $\mathcal{C}$. More precisely, we construct a $3$-dimensional ETQFT $\boldsymbol{\hat{A}}_\mathcal{E}$ that contains both $J_\mathcal{E}$ and $V_\mathcal{E}$. The ETQFT $\boldsymbol{\hat{A}}_\mathcal{E}$ is given by a $2$-functor whose source is the $2$-category of admissible $3$-dimensional cobordisms, and whose target is the $2$-category of finitely complete linear categories. We improve on earlier versions of the construction by dropping the admissibility condition for surfaces. By doing so, we obtain an ETQFT whose circle category $\boldsymbol{\hat{A}}_\mathcal{E}(\boldsymbol{S}^1)$ is equivalent to the category $\mathcal{C}$, as opposed to the ideal $\mathrm{Proj}(\mathcal{C})$.

This paper studies Veech groups of dilation surfaces whose fundamental group is not finitely generated. We prove that if $ S$ is a surface with self-similar end space and every end is accumulated by genus, then every countable subgroup of $SL(2,\mathbb{R}) $ can be realized as the Veech group of a dilation surface homeomorphic to $ S$. Additionally, we can construct this dilation surface such that it realizes horizon saddle connections in a prescribed set of directions. This contrasts with the case of closed dilation surfaces, where horizon saddle connections impose restrictions on the algebraic structure of the Veech group.

We build on the recent development that toposes with enough points can be represented as categories of points equipped with ultraconvergence data (a.k.a. virtual ultracategories), characterizing usual classes of geometric morphisms — surjections, embeddings, hyperconnected maps, and localic maps — in terms of their points. In particular, the characterisation of geometric surjections relies on a notion of "virtual ultraretractions", which generalises usual retractions and filtered colimits, categorifies Johnstone's notion of subclosure for topological spaces, and provides a new characterisation of separating classes of points of a topos.

Low-order refined (LOR) preconditioning replaces a high-order operator with a low-order discretisation on a refined nodal mesh. For tensor-product elements, the two operators are spectrally equivalent with bounds independent of the polynomial order $P$, but the construction does not extend directly to simplex and mixed-element discretisations. This work makes two contributions: it extends LOR preconditioning to simplex and mixed-element discretisations, including triangular, tetrahedral, and prismatic elements, and establishes a generalised Vandermonde transformation linking the modal and nodal LOR formulations, showing that the resulting preconditioned spectra and Krylov convergence are independent of the high-order basis. Numerical experiments show controlled iteration growth on triangular meshes despite increasing condition number, and controlled iteration counts up to $P=5$ on tetrahedral, prismatic, and mixed-element meshes. A single algebraic multigrid V-cycle per outer iteration gives the best balance of iteration count and cost. The method is applied to a production incompressible Navier-Stokes simulation of a race-car front-wing and wheel configuration, discretised on a mesh of $2.87\times10^6$ mixed prismatic and tetrahedral elements giving $32.2\times10^6$ pressure degrees of freedom at $P=3$. LOR reduces the mean pressure conjugate gradient (CG) iteration count from $235.3$ to $5.5$, and the pressure-solve time over 1000 timesteps by 16.1%, relative to the default production static-condensation diagonal preconditioner in Nektar++. The one-time cost of constructing the LOR preconditioner is amortised over the production simulation.

We study the order dimension of Cartesian products of finite posets. Kelly and Trotter conjectured in 1982 that $\dim(P\times Q)\ge\dim P+\dim Q-2$ for all finite posets $P$ and $Q$. For $m\ge3$, let $R_m$ denote the incidence poset of the complete graph on $m$ vertices. We prove that there is a constant $C$ such that, for all sufficiently large $m$, $\dim(R_m\times R_m)\le\left(1+\frac{2}{\log_2 6}\right)\dim R_m+C$. Since $1+2/\log_2 6<2$, this disproves the Kelly–Trotter conjecture and shows that $2\dim R_m-2-\dim(R_m\times R_m)$ can grow linearly with $\dim R_m$. For every integer $d\ge8$, we construct an incidence poset $Q_d$ such that $\dim Q_d=d$ and $Q_d$ has the $(3,d)$-covering property. Consequently, $\dim(P\times Q_d)\le\dim P+d-3$ for every poset $P$ with $\dim P\ge3$. Thus, for every integer $d\ge8$, the poset $Q_d$ violates the Kelly–Trotter conjecture with every poset of dimension at least $3$. Finally, we prove that a poset $Q$ has an $(r,s)$-covering if and only if $\dim(Q\times C^r)\le s$ for every finite chain $C$ with at least two elements.