Papers

We give an exact algorithm for minimizing an arbitrary rational quadratic polynomial $x^T Q x + c^T x + \gamma$ over the integer points of a bounded rational polyhedron $\{x : Ax \le b\}$. For $n$ variables and $m$ inequalities, the running time is $$2^{O(n \log(n+1))} (m+1)^{O(n)} \beta^{O(n)} (1+L)^{O(1)},$$ where $\beta$ is one plus the maximum binary encoding length of an entry of $A$ or $Q$, and $L$ is the full input encoding length. The polynomial degree in $L$ is absolute; in particular, the encoding of $b$, $c$, and $\gamma$ enters only this fixed-degree factor. Our algorithm refines the integral parallelepiped cover of Goemans and Rothvoss into cells with controlled displacement sets. On each cell, either an integer displacement of negative quadratic value rules out every point as a global minimizer, or the quadratic identity supplies a separation oracle for the convex hull of each integer objective sublevel set. The integer-query feasibility theorem of Hildebrand and Goess then finds the cell minimum. A refinement through scaled lattices solves the displacement search, and a localization argument separates the encoding dependence of the constraint matrix and quadratic part from that of the remaining data. Combining the bounded algorithm with the unboundedness test of Ari and Hildebrand extends the method to arbitrary rational polyhedra.

We prove sharp upper and lower bounds for the Morse index and nullity of linearly full branched minimal 2-spheres in the round 2n-sphere. In particular, we show that the Morse index of a linearly full minimal 2-sphere in the round 2n-sphere is at least n(n-1)(2n+1), with equality if and only if the 2-sphere is twistor-equivalent to the Boruvka sphere. Our techniques also apply more generally to give bounds for the Morse index and nullity for totally isotropic surfaces in the 2n-sphere.

We propose pointwise representations of the Stein solution to a weighted first-order Stein equation associated with a univariate absolutely continuous distribution. The representation extends to all the derivatives of the solution and with respect to any derivative of the test function. The mechanism behind these representations is an array of kernels that satisfy two key identities, yielding a specific calculus for the Stein framework considered here. The representations record every correction term appearing by differentiation, explaining how choosing the Stein kernel as weight has a special role in Taylor and pairing arguments. The calculus of the kernel extends these representations to any order and exhibits natural and explicit conditions to obtain a one term representation of the $n$-th derivative of the solution with respect to exactly one of the three neighbouring test-function orders of derivatives, \(n-1,n,n+1\). These conditions are simple to ensure by choosing the weight accordingly. The representations yield pointwise envelopes, uniform and weighted Stein factors, and ensure sharpness of the constants. We recover the known sharp results for the Gaussian and the Gamma distribution, improve known factors for the rest of the integrated Pearson family, which is the family under which every correction term cancels. We also study the Subbotin and symmetrized Maxwell families, for which the formulas retain additional pointwise terms. Applications sharpen constants in known distributional approximations and use the higher-order calculus to obtain Edgeworth corrections for the Beta approximation of the Pólya–Eggenberger urn and the quartic Subbotin approximation in the critical Curie–Weiss model.

We derive explicit Gamma–Stein bounds for the Satterthwaite approximation of weighted sums of independent chi-square random variables and, more generally, of independent Gamma random variables. The approximating Gamma distribution is chosen by matching the first two moments. Our approach relies on new kernel representations of the solution to the Gamma–Stein equation and of its first derivatives on the positive real line. These representations yield weighted Stein-factor bounds adapted to the moment-matching structure and show that the quality of the approximation is controlled directly by the discrepancies between the individual scale parameters and the matched scale parameter. Kolmogorov bounds are obtained by smoothing. The results provide quantitative guarantees for the widely used Satterthwaite approximation and are applied to pooled variances, sample variances of stationary Gaussian time series, Wishart traces and sums of Wishart traces, and high-dimensional repeated-measures statistics.

Let $\Gamma$ be the mapping class group of the plane minus a Cantor set, acting on the ray graph $\mathcal{R}$, and for $\gamma \in \Gamma$ let $\tau(\gamma)$ denote the translation length of $\gamma$ on $\mathcal{R}$. We prove that $\tau(\gamma) \le h(f)/\log(2)$ for every $C^\infty$ diffeomorphism $f$ of $S^2$ representing $\gamma$, where $h$ denotes topological entropy. The proof falls into two parts: a combinatorial argument bounding distance between two rays in $\mathcal{R}$ by the logarithm of the geometric intersection number; and a geometric argument that promotes geometric control (coarse length of iterates of a fixed ray) to combinatorial control (geometric intersection number). We conjecture that the $C^\infty$ hypothesis on $f$ can be removed.

We present an exposition of a proof, discovered by GPT-6 Astra, of the Erdős–Sós Conjecture, which states that every graph with average degree greater than $t-2$ contains every tree on $t\geq 2$ vertices.

We prove that broad families of finite Dirichlet sums exhibit deterministic concentration of normalized vertical zero density on a single line. The main result gives a general criterion under which the normalized Jessen potentials converge locally uniformly to a piecewise-linear convex function with a single corner. The associated normalized measures describing the distribution of real parts of zeros in vertical mean density then converge weakly to a Dirac mass. Thus, for each fixed truncation, zeros may occupy a nontrivial range of real parts, while asymptotically their normalized vertical density concentrates on one line. The proof combines the finite Bohr lift with a translation-uniform anti-concentration estimate for isolated prime coordinates. We apply the criterion to partial sums of the Riemann zeta function, fixed Dirichlet $L$-functions, and primitive holomorphic Hecke eigenforms of fixed level, trivial Dirichlet character, and without complex multiplication. The concentration line is $\operatorname{Re} s=1/2$ in the normalized setting and becomes $\operatorname{Re} s=k/2$ for the classical Fourier coefficients of a form of weight $k$.

We study linearizations of dynamical systems and some of its topological properties. Special attention is paid to the case of Lipschitz-free operators, and they are shown to model the dynamics of very general linearizations. We provide a new criterion, called the targeting property, for the (weakly) mixing property in a linear dynamical system through a (possibly) non-linear restriction of it. We then apply this criterion to the linearization $T_f$ of a Lipschitz map $f\colon M\to M$, where $M$ is a metric space and the operator $T_f$ is defined on the corresponding Lipschitz-free space $\mathcal F (M)$—the so-called Lipschitz-free operators. We show that, under some natural assumptions on the distance considered in $M$, the operator $T_f$ is hypercyclic if and only if the map $f$ has the targeting property.

We study negative spheres obtained from exceptional sections of Gurtas Lefschetz fibrations. Starting with the known system of $4n$ disjoint $(-1)$-sections, we show that a marked double sum contains $4n$ symplectic $(-2)$-spheres and $2n$ smooth $(-4)$-spheres obtained by pairwise tubing. By carrying the same sections through a fourfold sum, we instead obtain $4n$ disjoint symplectic $(-4)$-spheres and hence symplectic rational blowdowns. We then consider the Lefschetz fibrations on knot-surgered elliptic surfaces. Their branched-cover description gives $2n$ branch sections together with simultaneous geometric duals arising from node resolution. In the knot-surgery double these sections glue to symplectic $(-4)$-spheres, while the duals make the complement of every subcollection simply connected. The resulting rational blowdowns provide simply connected exotic symplectic four-manifolds under the parity condition stated in the main theorem. We also record the boundary-multitwist relation determined by the Gurtas sections.

In this paper, we investigate integral solutions satisfying the system of two forms in many variables of different degrees. Let $d_1$ and $d_2$ be natural numbers with $d_2>d_1\geq 2$. Let $F_{i}(\boldsymbol{x}) \ (i=1,2)$ be forms in $n$ variables of degrees $d_i\ (i=1,2)$, respectively. Define $$N(\boldsymbol{F};P):=\#\{\boldsymbol{x}\in [-P,P]^n\cap \mathbb{Z}^n:\ F_{i}(\boldsymbol{x})=0\ (i=1,2)\}.$$ When each dimension of singular loci of $F_1=0$ and $F_2=0$ is small, we obtain a number $n_0:=n_0(\boldsymbol{F})$ such that whenever $n> n_0$ one has the expected asymptotic formula \begin{equation*} N(\boldsymbol{F};P)=c_{\boldsymbol{F}}\cdot P^{n-d_1-d_2}+O(P^{n-d_1-d_2-\delta}),\ \text{for some }\delta>0, \end{equation*} where the constant $c_{\boldsymbol{F}}$ is the product of local densities. We note that this asymptotic formula agrees with the Manin-Peyre conjecture. Compared to the previous work, we lower the admissible threshold $n_0(\boldsymbol{F})$ in most cases, with the exception of case $d_2-d_1=1$. In particular, if $F_1$ and $F_2$ are non-singular forms, then we obtain $$n_0(\boldsymbol{F})=3(d_2-1)2^{d_2-1}+(d_1-1)2^{d_1},$$ provided that $d_2\geq 5d_1$ with $d_1\geq2$. This yields a substantial improvement over the previous bound $n_0(\boldsymbol{F})=(d_1+2)(d_2-1)2^{d_2-1}+d_12^{d_1-1}$. To achieve this, we develop a new differencing argument together with the van der Corput differencing argument, delivering an efficient upper-bound estimate for mean values of exponential sums associated with two forms in many variables of different degrees, when the difference between degrees is sufficiently large. Furthermore, the method described in this paper is flexible enough to apply to forms in many variables of differing degrees in general.

The systematic generalisation of the well-established $S$-matrix positivity and bootstrap bounds from $2 \to 2$ scattering processes to higher-point amplitudes has, to date, remained elusive. To address this, we consider a large class of tree-level UV-completions of a single-scalar EFT. Specifically, we consider the most general renormalisable $N$-scalar theory in $D=4$. By explicitly computing tree-level amplitudes for four-, five- and six-point scattering and expanding at low energies, we can search for positivity relations among the expansion coefficients. We find that quite generally positivity bounds are unavailable due to contamination from a specific class of Feynman diagrams that are linear in sign-indefinite cubic couplings. However, by isolating a sector of the full crossing-symmetric low-energy amplitude that is unaffected by these sign-indefinite cubic couplings, we find that selected six-, five- and four-point Wilson coefficients organise into positive semi-definite matrices. While a violation of these positivity statements only implies that a given EFT is inconsistent with our class of UV completions, the result helps to guide towards what more general positivity bounds for higher-point interactions could, and could not, look like.

In this paper we consider affine selfmaps of compact flat manifold pairs. We derive an explicit formula for the relative Nielsen number in terms of the ambient holonomy and the holonomy carried by the stabilizer of the lifted submanifold. The formula extends the flat specialization of Wong's nilmanifold-pair formula to finite holonomy, agrees with the author's earlier model-solvmanifold-pair formula on the flat overlap, and extends beyond it. We also show that the stabilizer holonomy cannot in general be replaced by the intrinsic holonomy of the submanifold. Examples exhibit interesting Schirmer theory and include flat pairs outside the model-solvmanifold class.

This paper considers feedback optimization for optimal steady-state operation of nonlinear discrete-time systems when the steady-state input-output map and its sensitivity are expensive to compute. We propose a hybrid extension of sequential feedback optimization (SFO) that augments the model-based SFO gradient with correction terms through a convex combination with summable diminishing weights. Two variants are studied: one based on recursive least-squares (RLS) sensitivity estimation, and another on extremum seeking control (ESC) gradient estimation. Under contractivity and smoothness assumptions, we show that both hybrid schemes preserve the convergence of SFO to a neighborhood of the optimal steady state. The proposed methods are validated through a wind farm power maximization problem using a medium-fidelity model, demonstrating improved early transient performance compared to pure SFO.

Understanding the relation between $\dim R$ and $\dim R[[x]]$ is a classical problem in commutative algebra. For a Noetherian ring $R$, one has $\dim R[[x]]=\dim R+1$, but the general case is considerably more delicate. In 1973, Arnold proved that finite power-series dimension requires the strong finite type (SFT) condition, whereas, in 2002, Coykendall constructed a one-dimensional SFT domain whose power series ring has infinite dimension. The question of Coykendall and Gilmer whether $\dim R[[x]]<\infty$ forces $\dim R[[x]]\le2\dim R+1$ was answered negatively by Kang and Park in 2009. In this paper, we prove that, as $R$ ranges over the nonzero commutative rings with identity, the finite pairs $(\dim R,\dim R[[x]])$ are exactly $(0,1)$ and the pairs $(n,m)$ with $1\le n<m$.

We provide a universal characterisation of the Banach spaces of equivalence classes of $L^p$ Bochner integrable functions, under equality almost everywhere, as initial objects in appropriate categories of bifunctors, for all $p \in [1,\infty)$. As a by-product of our construction, we also obtain a unique characterisation of the Bochner integral.

We propose a variational mollification approach to circular deconvolution based on the reconstruction of a mollified target object rather than the exact solution itself. The method is formulated as a convex variational problem whose solution admits an explicit Fourier representation. We establish the consistency of the proposed reconstruction as the target resolution increases and derive convergence rates for deterministic data perturbations. For ordinary smooth kernels, the method achieves the classical order-optimal algebraic convergence rates under Besov-Nikolskii smoothness assumptions, while for supersmooth kernels it attains the corresponding order-optimal logarithmic rates. Numerical experiments on synthetic and wind-direction data illustrate the effectiveness of the proposed approach and confirm the theoretical predictions.

Let the exact homogeneous hard-margin support vector machine be trained on \(m\) independent observations from a Borel probability law on a real Hilbert space. We prove that, with score zero counted as an error, there is a universal numerical constant \(C\) such that \[ \Pp\left( \gamma_m>0,\quad \Risk(u_m)> \frac{C}{m} \left( K_m+\log\frac1\delta \right) \right) \le \delta . \] Here \(\gamma_m\) is the empirical homogeneous margin, \(u_m\) is the exact minimum-norm unit-margin separator, \(r_m\) is the largest training radius, and \(K_m:=r_m^2\norm{u_m}^2=r_m^2/\gamma_m^2\) on \(\{\gamma_m>0\}\). The proof is driven by a deterministic deletion problem. Given vectors \(x_1,\ldots,x_n\) in the unit ball, delete a set \(B\) of constraints and let \(u_B\) be the closest point to the origin that satisfies every retained unit-margin constraint. Suppose that \(\norm{u_B}^2\le k\) and that every deleted vector has nonpositive score under \(u_B\). We prove that a family of such deletion sets of cardinality \(q\) has size at most \(\exp(8k+2q)\). The conceptual step is an exact identity obtained from the KKT representation of \(u_B\). For a random deletion set, the identity converts the mean squared spread of the separators into a weighted sum of score deficits. It therefore forces a coordinate whose deletion status separates the two conditional means by a quantitatively large amount. Revealing that coordinate decreases the conditional separator variance enough to control the binary entropy of the split. An entropy induction gives the deletion count, and an exact factorial ghost-sample identity converts that count into the stated high-probability SVM bound.

We develop a hybrid modeling framework for coupling pre-trained numerics-informed neural networks (NINNs) with classical full order models (FOMs) using the overlapping Schwarz alternating method. We consider the two-dimensional advection-diffusion equation in the advection-dominated, Peclet-number 10^6 regime. We first demonstrate that, unlike the corresponding physics-informed neural network (PINN), a monolithic NINN can be accurately trained on our model problem without domain decomposition. We then employ overlapping multiplicative Schwarz as a deployment mechanism for coupling a pre-trained, subdomain-local NINN with a neighboring FOM, with the NINN weights held fixed throughout the Schwarz iteration. We consider two training approaches for the subdomain-local NINNs: a top-down approach, in which boundary data are obtained from a coupled Schwarz solve on the full domain with a FOM on each subdomain (FOM-FOM Schwarz), and a bottom-up approach, in which boundary traces are generated synthetically on the NINN subdomain without requiring any full-domain solves. The resulting hybrid NINN-FOM solutions agree closely with the corresponding FOM-FOM Schwarz solutions, with the top-down and bottom-up training approaches yielding comparable accuracy.

Hybrid domain decomposition methods provide a flexible framework for coupling full order models (FOMs) and reduced order models (ROMs), but typically assume the model assigned to each subdomain is fixed throughout a simulation. This is limiting for transient problems in which localized features propagate through the domain and the regions requiring high-fidelity resolution change over time. We introduce a reinforcement learning (RL)-based approach for online adaptation of FOM-ROM models coupled via the overlapping Schwarz alternating method (O-SAM), an iterative domain decomposition method that solves subdomain-local problems while exchanging solution information through transmission boundary conditions on overlapping interfaces. Deep Q-networks (DQNs) are trained offline to select among subdomain-local FOMs and pre-trained Operator Inference (OpInf) ROMs using a reward balancing accuracy, cost, and model-switching frequency. Once trained, the policies are deployed predictively on problem instances not seen during training, without requiring a reference FOM solution. We demonstrate the approach on two examples: a 1D advection-diffusion problem with a moving front, and a 3D linear elastic wave propagation problem implemented in the Norma.jl solid mechanics code. For the advection-diffusion benchmark, the learned policy dynamically allocates high-fidelity resolution as the front propagates and outperforms static FOM/ROM assignments; letting the agent also adapt the domain decomposition provides no further benefit. For the elastic wave benchmark, learned policies for two and three subdomain decompositions track the propagating wave by assigning FOMs to subdomains containing the wave and ROMs elsewhere, as expected. Our results demonstrate the potential of RL to enable predictive online adaptation of model fidelity within Schwarz-based hybrid simulations.

We study the Quantum Approximate Optimization Algorithm with a Grover mixer and independently sampled cost and mixing angles. Under a lattice condition on the cost values carried by the initial state, we prove a depth-independent lower bound for the variance of the loss at every depth, together with the same bound for the derivative with respect to the final mixing angle. The estimate is instance-dependent and, for fixed locality, is inverse polynomial in the number of qubits for integer-valued local objective functions with uniformly bounded local terms, in particular for MaxCut. We also establish Grover-type reachability bounds showing that the depth required to approximate a prescribed carried eigenspace is bounded below by a constant multiple of the inverse square root of its initial probability.

By using the Dirac and symplectic formalisms, the analysis of the Holst action in the $G \rightarrow 0$ limit is performed. We study the theory's full canonical structure and symmetries. In particular, we focus on the cases $\gamma = i$ and $\gamma \neq i$, aiming to obtain Smolin's auto-self-dual model in terms of the Ashtekar variables and a Barbero-type formulation. We then perform the symplectic analysis and discuss the principal insight of these formalisms.

This paper gives results which relate the definable Galois theory of [28] to the definable Galois cohomology of [18] in the setting of a totally transcendental first order theory. Firstly, we show that under some common assumptions the definable Galois cohomology of the extrinsic definable Galois group associated to fixed internality data classifies the number of definable Galois extensions inside a copy of the prime model. This generalizes a well-known result for Picard-Vessiot differential Galois theory shown originally via tannakian methods [3]. We then collate some triviality results for differential Galois cohomology [21] and recent results on iterated Picard-Vessiot extensions [10] [15] to give colimit formulas for differential Galois cohomology. Precisely, for an ordinary differential field $K$ of characteristic $0$ and a linear differential algebraic group $G$ over $K$, the differential Galois cohomology $\text{H}^1_{\delta}(K,G)$, is given by a colimit over the family of normal closures of iterated Picard-Vessiot extensions of $K$ of finite type. We then propose a new notion of boundedness for a differential field. We finally show that the minimal closure of a set of parameters $A$, the intersection of all elementary embeddings of a copy of the prime model over $A$ into itself, contains all of the definable Galois cohomological information for definable groups over $A$ satisfying also some strong but common conditions. Lastly, we give some colimit formulas for definable Galois cohomology and propose a model-theoretic definition of a bounded set of parameters.

Matrix models express partition functions of random surfaces with prescribed boundary lengths through string equations. We give a probabilistic realization of this structure in Liouville quantum gravity (LQG). For $\gamma^2\in(8/3,4)$, we construct the quantum $n$-hole sphere from a quantum disk decorated by an independent conformal loop ensemble (CLE) and compute its area Laplace transform at fixed boundary lengths. The transform depends only on the total boundary length apart from a simple prefactor and is determined by a hypergeometric string equation. We derive this equation from stable Lévy excursions using LQG/CLE coupling. The conformal welding equation yields symmetric polynomials that represent integrated LQG/CLE coupling observables. These polynomials converge to Weil–Petersson volumes of spheres with $n$ geodesic boundaries as $\gamma\downarrow0$.

In this paper, we study supersingularity of modular Jacobians $J_0(N)$ and abelian varieties $A_f$ of $\mathrm{GL}_{2}$-type from both the vertical perspective (where the prime $p$ is fixed, and the level $N$ varies) and the horizontal perspective (where the newform $f$ is fixed, and the prime $p$ varies). Vertically, we prove that $J_{0}(N)$ is supersingular modulo a given prime $p$ for only finitely many levels $N$. Horizontally, we show that for sufficiently large $p$, the reduction of $A_f$ modulo $p$ is supersingular if and only if it isogenous to a power of a supersingular elliptic curve.

We study wetting on a doubly periodic rough surface in three dimensions. Although the liquid-vapor interface meets the solid at the local Young's angle, microscale roughness can cause the macroscopic apparent angle to differ substantially from this value. We formulate the directional contact angle hysteresis (CAH) interval in terms of apparent angles associated with pinned microscopic configurations. To approximate its receding and advancing endpoints, we evolve capillary mean curvature flow (CMCF) toward extremal stationary states. Computing these states is difficult because the pinning that produces hysteresis is generated at the scale of the roughness, whereas the apparent angle is only meaningful at the macroscopic scale, so a single uniform grid must resolve both. We therefore introduce a two-scale alternating (TSA) method based on a Schwarz decomposition: a Merriman-Bence-Osher (MBO) diffusion-generated scheme resolves the contact-line near region, while a linearized minimal-surface problem updates the far region. For an idealized reference iteration, we prove decay of an approximate interfacial energy. Numerical experiments on a representative doubly periodic surface show a strongly anisotropic CAH interval whose width varies by more than a factor of three with contact-line orientation and changes sharply near the diagonal directions. Because a stationary droplet must meet the solid at an apparent angle inside this interval, the computed anisotropy constrains which macroscopic wetted regions the surface can support; it is consistent with a square-like stationary droplet whose sides align with the diagonal directions.

For a fixed prime $p$, let $\mu_p$ denote the $p$-adic Plancherel measure. Then the first main goal of this paper is to prove effective (and moreover explicit) $\mu_p$-equidistribution of the $p$-th Hecke eigenvalues over the Atkin–Lehner subspaces $S_k^\sigma(N) \subseteq S_k(N)$ and $S_k^{\operatorname{new}, \sigma}(N) \subseteq S_k^{\operatorname{new}}(N)$. We then highlight five applications of this explicit equidistribution result. For the first application, we generalize Kim's vertical analog of the Atkin–Serre conjecture to the Atkin–Lehner setting. For the second application, we obtain explicit bounds on the number of newforms $f \in S_k^{\operatorname{new}, \sigma}(N)$ for which $p$ is extremal over $S_k^{\operatorname{new}, \sigma}(N)$. For the third application, we prove explicit asymptotics for the number of $\mathbb{F}_{p^r}$-points on the modular Jacobian $J_0(N),$ as well as on its factors $J_0^{\operatorname{new}}(N),$ $J_0^\sigma(N),$ and $J_0^{\operatorname{new}, \sigma}(N)$. We also make explicit an asymptotic result of Serre concerning point counts of the modular curves $X_0(N)$. For the fourth application, we generalize lower bounds due to Murty and Sinha on the sizes of large $\mathbb{Q}$-simple factors of $J_0(N)$ to analogous bounds for $J_0^\sigma(N)$. Finally, for the fifth application (the details of which are given in a separate paper), we use our explicit equidistribution result to prove that only finitely many modular Jacobians are supersingular modulo any fixed prime.

The problem of inverting the Radon integral transform in finite-dimensional Euclidean space is considered. The relevance of this topic for probing issues is indicated. It is noted that for the latter direction, it is natural to consider the integrand as discontinuous function. However, the available inversion formulas are only proven for differentiable functions. Therefore, the question of obtaining formulas for discontinuous functions arises. It is set that the required results can be obtained by some modification of available proofs for smooth functions.

Measures on function spaces arise throughout Bayesian inverse problems and generative modeling, often with low-dimensional structure relative to a tractable reference measure. We introduce the class $\mathcal{P}_\psi(\mu)$ of measures that differ from a reference measure $\mu$ only through a finite-dimensional map $\psi$ while preserving the reference conditionals on its fibers. Class members are determined by their $d$-dimensional pushforwards under $\psi$ and admit convenient block-triangular transport map representations. Draws are taken from this $d$-dimensional distribution and then completed to function space through sampling of the reference conditionals. For Gaussian references, these transport maps are finite rank perturbations of the identity. In contrast, optimal transport maps do not preserve this low-dimensional structure. For covariance perturbations that are trace class in the Cameron-Martin geometry of the reference, we show the optimal map is a trace class perturbation of the identity. Even finite rank perturbations yield corrections whose rank, governed by an invariant subspace, is typically infinite. We develop approximation theory for $\mathcal{P}_\psi(\mu)$ and error analysis for fitting within it, splitting the total error into an irreducible class error and a finite-dimensional marginal term set by the dimension of $\psi$ rather than the ambient discretization. We present numerical results including inference from low-dimensional nonlinear observation maps, deconvolution under a jump process prior, and state estimation for Navier-Stokes flows.

We study the geometry of parameter spaces of homogeneous ideals in k[x,y,z] minimally generated by three forms of the same degree, stratified by the degree of their base scheme and by the initial degree of their syzygies. We realize these strata through Quot and Hilbert schemes, construct universal families of Bourbaki schemes, and prove smoothness and irreducibility results for the free loci. We then compare these spaces with the loci of gradient ideals and plane curves. Using slope semistability of the associated syzygy bundles, we give a vector-bundle interpretation of the refined du Plessis–Wall bounds and obtain the same numerical restriction for arbitrary three-generated ideals. Finally, in the case of triples of cubic forms, we determine all nonempty degree strata, prove that they are smooth and irreducible, and identify their gradient loci in terms of the Bourbaki stratification of reduced quartic plane curves.

We consider the problem of finding stationary points of stochastic convex functions and related variational inequalities. For each, we show that regularized empirical risk minimization, coupled with a random tilting perturbation, obtains stationarity residual order $\sqrt{d/n}$ for $d$-dimensional problems given $n$ observations. We present a few complementary results that show that some dimension dependence is necessary, in distinction from standard stochastic optimization and empirical risk minimization, by providing minimax lower bounds scaling as $\sqrt{\log d / n}$.