Papers

Every Lipschitz map from a closed subset of a real Hilbert space into a real Hilbert space has an absolutely minimising Lipschitz extension: an extension whose Lipschitz constant on every open subset of the complement, bounded or not, equals the Lipschitz constant of its restriction to the boundary of that subset. Neither space need be separable. This answers a question of Naor and Sheffield. For targets of dimension at least two we prove a density theorem on connected domains: every optimal extension of a boundary datum can be approximated, uniformly on compact sets, by extensions whose Lipschitz constant on every nonempty open subset, and on its boundary, equals the constant of the datum. Consequently, in contrast with the scalar case, such a datum has a unique absolutely minimising extension exactly when its optimal extension is unique, and absolutely minimising extensions need not be closed under uniform convergence, even for fixed smooth data on a planar disk. The extensions are built by finite piecewise isometric interpolation: on nested polyhedra with dense union they are piecewise affine and preserve distances in at least two directions on each simplex, a structure described through leaves of 1-Lipschitz maps. The boundary constraints are handled by a midpoint argument for admissible values, and a Kuratowski-Zorn argument removes the separability assumption while preserving previously prescribed values. We also characterise the bounded linear maps that are absolutely minimising extensions of all data determining their operator norm.

Fix integers $k,b,q$ with $k\ge 2,b\ge 0,q\ge 2$. Define the function $p$ to be: $p(x)=kx+b$. We call a set $S$ of integers $(k,b,q)$-linear-free if $x\in S$ implies $p^i (x)\notin S$ for all $i=1,2,\dots,q-1$, where $p^i (x)=p\left(p^{i-1}(x)\right)$ and $p^0 (x)=x$. Such a set $S$ is maximal in $[n]:=\{1,2,\dots,n\}$ if $S\cup \{t\},\forall t\in [n]\backslash S$ is not $(k,b,q)$-linear-free. Let $M_{k,b,q} (n)$ be the set of all maximal $(k,b,q)$-linear-free subsets of $[n]$, and define $g_{k,b,q} (n)=\min_{S\in M_{k,b,q} (n)}|S|$ and $f_{k,b,q} (n)=\max_{S\in M_{k,b,q} (n)}|S|$. In this paper, formulae for $g_{k,b,q} (n)$ and $f_{k,b,q} (n)$ are proposed. Also, it is proven that there is at least one maximal $(k,b,q)$-linear-free subset of $[n]$ with exactly $x$ elements for any integer $x$ between $g_{k,b,q} (n)$ and $f_{k,b,q} (n)$, inclusively.

We prove local well-posedness for the modified Benjamin-Ono equation in $H^s(\mathbb{R})$ for $s>3/2$. This equation arises as a modified counterpart of the Benjamin-Ono equation through a Miura-type transformation, in close analogy with the classical relation between mKdV and KdV equations. One of the main obstacles is the apparent derivative loss in the quadratic part of the equation. To overcome this difficulty, we introduce a Cole-Hopf transform adapted to the structure of the equation. The map removes the quadratic derivative term and converts the equation into a Schrödinger equation with a transport and a commutator terms that can be controlled without derivative loss. Solutions are then constructed by parabolic regularization, while frequency envelopes are used to obtain the continuity of the data-to-solution map. In the degenerate case $\gamma=0$, the transform completely linearizes the equation and yields local well-posedness for every $s>1/2$, down to the scaling-critical threshold.

We study algebraic relations among functions in star-invariant subspaces. Let $\theta$ be an inner function and suppose that $f,g,h\in K_{\theta}^p$ satisfy $P(f,g,h)=0$, in which $P$ is a homogeneous polynomial and the associated projective plane curve is smooth. Then $\dim\operatorname{span}\{f,g,h\}\leq1$ if either $\operatorname{deg} P\geq 4$ and the boundary spectrum of $\theta$ has logarithmic capacity zero, or $\operatorname{deg} P = 3$ and the boundary spectrum contains at most one point. This settles new cases of a problem of Dyakonov, subsumes the previously known results, and recovers the classical fact that polynomial parametrizations of a smooth projective plane curve of degree at least three are projectively constant.

Let $X$ be a smooth projective variety over the complex numbers, and let $\mathbb{V}$ be an irreducible privileged $\mathrm{SL}_3$-local system on $X$ in the sense of Esnault and Groechenig. We show that $\mathbb{V}$ is of geometric origin. We also give an example of a privileged but non-rigid $\mathrm{SL}_3$-local system.

Let $K/\mathbb{Q}_p$ be a finite extension, and let $X/K$ be an algebraic variety. We present a new proof that the unipotent completion of the étale fundamental group of $X_{\bar{K}}$ is a pro-de Rham representation, and show that it is pro-semi-stable whenever $X$ has semi-stable étale cohomology. We also reprove, using syntomic methods, that it is pro-crystalline at integral base points whenever $X$ admits a smooth proper formal model. All three proofs use the bar construction and are inspired by Wojtkowiak's approach to unipotent fundamental groups.

We show that finite-order exponential mixing does not imply the central limit theorem for smooth diffeomorphisms. For every integer $R\geq2$, we construct a $\mathcal{C}^{\infty}$ diffeomorphism $T:M\to M$ of a compact smooth manifold $M$ and a $T$-invariant probability measure $\mu$ such that $(T,\mu)$ is exponentially mixing of order $R$ for $\mathcal{C}^1(M)$ observables, but not of order $2R+2$. For some centred observable $f_0\in\mathcal{C}^{\infty}(M)$, no subsequence of the normalised sums $\frac{1}{\sqrt n}\sum_{i=0}^{n-1}f_0\circ T^i$ converges in distribution to a Gaussian law. In particular, exponential decay of two-point correlations implies neither exponential mixing of all orders nor the central limit theorem. We realise $(T,\mu)$ in two ways. In the first, $T$ is a fixed linear Anosov diffeomorphism of $\mathbb T^2$ and $\mu$ is singular with respect to Lebesgue measure. In the second, $M$ can be any compact smooth surface without boundary and $\mu$ is a physical measure whose statistical basin contains a nonempty open set. The main theorems have been formalised in Lean 4.

Let lambda(n) denote the infimum of -min_x sum_{a in A} cos(ax) over all sets A of n positive integers. While the classical Sidon-difference construction yields lambda(n) <= (1 + o(1))sqrt(n)/2, we construct sets achieving a leading constant of 1/2: lambda(n) <= (1/2)sqrt(n) + O(n^(1/4)sqrt(log n)), establishing limsup_{n->infinity} lambda(n)/sqrt(n) <= 1/2. This construction combines a two-sided Sidon square identity, a Singer difference set, a Fejer-kernel majorant, a parity-biased deletion, and an exact-cardinality probabilistic selection. We also provide an L^4 criterion and a multiplicity bound constraining any construction with a smaller leading constant. In the opposite direction, we incorporate the L^1 norm into Bedert's boundary estimate with an explicit constant, yielding two lower bounds for f_A(x) = sum_{a in A} cos(ax): (1) -min f_A >= cn^(1/4) for every n-element set A subset of [1, Cn]; and (2) (-min f_A)^3 ||f_A||_1^2 >= cn for every n-element set A subset of N. The second bound recovers the 1/5 exponent of Bedert and Shankar and improves it whenever ||f_A||_1 is small relative to -min f_A. Numerical values of the construction are reported up to |A| = 25,440.

A classical theorem of Kaiser and Rabinovich implies that if every subfamily of size at most $\lfloor\log_2(d+1)\rfloor+1$ of a finite family of compact connected sets in $\mathbb{R}^d$ has a common axis-parallel hyperplane transversal, then the entire family has an orthogonal star transversal, i.e., a union of $d$ hyperplanes, one perpendicular to each coordinate axis. We study colorful variants of this result. First, we use standard KKM machinery to establish a colorful analogue with $d+1$ color classes and a $\lfloor\log_2(d+1)\rfloor+1$ transversal hypothesis for rainbow subfamiles. Then we focus on reducing the required number of colors. To this end, we introduce the rainbow pushability $\pi_r(P)$ of a polytope and we use it to state and prove few-color variants of the colorful Carathéodory theorem and the colorful polytopal KKMS theorem. Applying the latter to the $d$-dimensional cube, we show that $\lfloor\log_2 d\rfloor+2$ color classes suffice for our transversal problem, provided that every rainbow choice has a common axis-parallel hyperplane transversal. We also extend both colorful results to the setting of convex $(n,D)$-bodies and we show that three colors are needed for $d\geq 2$.

We investigate the regularity of Lyapunov exponents for products of random invertible matrices of arbitrary dimension $d\ge 2$. We prove that the top Lyapunov exponent is pointwise Hölder continuous with respect to the Wasserstein-Hausdorff distance in the space of probability measures with compact support, in the case where this exponent is simple and the equator has dimension equal to or less than one.

Global synchronization has been largely investigated in static higher-order networks and little attention has been brought to directed, temporal higher-order networks. We hereby make a step in this direction by studying the emergence of global synchronization in periodically fast switching systems with directed higher-order interactions. By working in the framework of the Master Stability Function, we show that the linearized dynamics includes two time-dependent Jacobian matrices, both evolving on the timescale of the synchronous solution, together with a generalized Laplacian evolving on a fast switching timescale, resulting from the hypergraph time evolution. To address this two-timescale structure, we establish a fast-switching stability criterion for directed balanced higher-order structures. The latter condition allows to decouple the parallel and transverse perturbations to the synchronization manifold. By relying on the average theory for dynamical systems, we are able to prove exponential stability of global synchronization in the fast switching higher-order network. The proposed theory is then illustrated by using a higher-order Stuart–Landau model, which provides a natural benchmark for the theory. Directionality can move Laplacian eigenvalues into instability regions off the real axis, hence destabilizing synchronization; our results nevertheless reveal that temporal switching can counteract this destabilizing effect and restore global synchronization when the directed structures alternate sufficiently rapidly, even though the instantaneous directed higher-order structures are individually unstable.

We prove the non-characteristicness of the completed self-sum of the universal nilpotent cone in $T^*\mathrm{Bun}_G$ relative to the moduli of curves, for reductive groups $G$ of semi-simple rank $1$. In particular, this implies the non-characteristicness of the $\mathcal{H}om$-sheaf between Betti automorphic sheaves for groups of semi-simple rank $1$. We also construct an example of a collision of universal nilpotent Higgs fields, proving that the completed self-sum is not non-characteristic for $\mathrm{GL}_n$, $n\ge 3$, relative to the moduli of curves of genus $g\ge 4$.

We prove a one-sided central limit theorem for $\log |L(\tfrac 12,\chi)|$ in a thin family of $n$-th order Hecke characters, for $n \geq 3$. The proof combines the method of Radziwiłł and Soundararajan with a recent asymptotic formula of Diaconu, Ion, Paşol, and Popa for the twisted second moment with a power-saving error term. We also obtain upper bounds of the conjectured order of magnitude for the $2k$-th moments for $0< k< 1$.

We study periodic homogenization of fully-nonlinear second-order Hamilton–Jacobi–Bellman and Isaacs equations subject to a homogeneous Dirichlet boundary condition and the numerical approximation of the effective Hamiltonian, where the diffusion matrix is assumed to satisfy the Cordes condition. In the first part, assuming the Cordes parameter is sufficiently large, we show existence and uniqueness of $W^{2,p}$ solutions to the true cell problem, prove homogenization under minimal regularity assumptions, and establish a quantitative homogenization result. In the second part, we propose and rigorously analyze $W^{2,p}$-conforming and mixed finite element methods for the approximation of the corrector, which are then used in the approximation of the effective Hamiltonian. The key tool used throughout this work is Campanato's concept of near operators.

We prove sharp arithmetic localization for one-dimensional quasiperiodic potentials with jump discontinuities, beyond uniform transversality. Our results include every non-constant $C^1$ monotone sampling function, even with arbitrarily large flat regions, and also certain non-monotone potentials. A key new mechanism is a symbolic unique-continuation principle: first-return words to the region where $f'>0$ have no common Dirichlet resonance, forcing eigenfunction mass to transfer away from flat or negatively sloped regions and recovering quantitative monotonicity of finite-volume eigenvalue branches.

Periodically driven quantum systems exhibit a wide range of complex dynamics, including chaos. Yet, at special parameter values, their evolution can recur exactly. Several quantum protocols have recently been proposed that exploit exact Floquet recurrences to demonstrate genuine quantum advantage. Such recurrences can be studied through the arithmetic structure of the Floquet unitary. In this work, we establish a necessary and sufficient arithmetic criterion for exact recurrence and determine the minimal recurrence period when one exists. Because recurrence is a global property of the unitary, it holds for every initial state. Applying our framework requires only knowledge of the number field associated with the Hamiltonian parameters. We demonstrate the method using the kicked Ising chain, establishing exact recurrences in both Clifford and non-Clifford regimes. We further show that, in the recurrent non-Clifford regime, magic–a resource for quantum computation–is generated at intermediate times and undergoes exact periodic revivals enforced by the recurrence of the full Floquet unitary. By connecting number theory, quantum many-body dynamics, and quantum information, our work shows that exact recurrence extends beyond Clifford circuits and opens new avenues for magic generation and quantum hardware benchmarking.

The Hilbert–Poincaré series of the Chow rings of the toric varieties of matroids, often called the Chow polynomials, and the $f$-polynomial of their boundary complexes, called the chain polynomials, have been conjectured to only have real zeros. We construct several explicit infinite families of matroids from finite geometries, whose multichain counts fail to be log-concave. They share a common geometric construction called hyperplane coalescence, together with several classical operations in projective geometry and combinatorial designs. This induces failures of log-concavity for Chow, augmented Chow, chain, and $h$-polynomials of several explicit infinite families of matroids, disproving the real-rootedness conjectures of Ferroni–Schröter, Huh–Stevens, and Athanasiadis–Kalampogia-Evangelinou from the lowest possible ranks. As an application of the same mechanism, we also give counterexamples to the log-concavity of $h$-polynomials of graph associahedra, answering in the negative a question of Sack.

In this paper we deal with the theory of Regular Lagrangian Flows, as initiated in [29] and axiomatized in [2]. In the first part of the paper we show in Theorem 4.3 (see also Corollary 4.5) that the decomposition of $\mathcal{L}^{1+d}$-almost all the space time in a family of (essentially) disjoint characteristics, the technique at the basis of the well posedness in [1] (for planar, divergence-free and autonomous vector fields), and in [13] (for vector fields having a suitable regularity in space, including BV vector fields) is not only sufficient for well posedness, but also necessary. This shows that in many cases the typical uniqueness statement of the DiPerna-Lions theory, namely that the path $\mathbb{X}(\cdot,x)$ is (essentially) unique given the initial position $x$, can, as in the classical theory, be strengthened by saying that the paths $\mathbb{X}(\cdot,x)$ (essentially) do not branch. As the simple Example 1.1 shows, when we consider vector fields with singular divergence well posedness results may hold only for solutions that do not change sign. In the second part of the paper we revisit [8,9] obtaining a well posedness result for special vector fields of bounded deformation and nonnegative solutions, under a mild regularity assumption (5.3) with respect to the time variable, weaker than the space-time $SBD$ regularity assumption made in [9] for nearly incompressible vector fields. Finally, we revisit carefully the constructions of Section 9 of [13], to see how they provide existence and uniqueness for (1.2) assuming $BV$ regularity in space, in the class of nonnegative bounded solutions.

We consider the Frank–Wolfe algorithm for solving variational inequalities over compact polytopes under a monotone $C^1$ operator $\Phi$. The linear minimization oracle returns vertices, and we assume a coercivity condition on the Jacobian of $\Phi$ at a solution. We prove last-iterate convergence to the unique solution at rate $O(k^{-\min\{\alpha,1\}})$ for step sizes $\gamma_k=\alpha/(k+\beta)$ with $\beta\geq\alpha>0$. The bound holds uniformly over starting points and vertex tie-breaking. In particular, both $1/(k+1)$ and $2/(k+2)$ give $O(k^{-1})$ distance. A matching lower bound shows that this distance guarantee is worst-case order-optimal among prescribed admissible step-size schedules. We also obtain rates $O(k^{-\min\{\alpha,1\}})$ for the Frank–Wolfe gap and $O(k^{-\min\{\alpha,2\}})$ for the dual gap. These yield $O(\varepsilon^{-1})$ and $O(\varepsilon^{-1/2})$ calls to each oracle for sustained gap accuracy $\varepsilon$ when $\alpha\geq1$ and $\alpha\geq2$, respectively, with instance-dependent constants. We also treat more general prescribed step-size schedules.

We consider the existence and uniqueness of positive, bounded stationary solutions of the Fisher-KPP equation with Dirichlet boundary conditions. We establish that stationary solutions are unique, and exist if and only if the generalised principal eigenvalue is strictly positive (supercriticality), in dimension at most $6$. This is sharp: we exhibit examples both of non-uniqueness and of existence at criticality in dimension $7$ or higher, including for the classical reaction $f(u)=u-u^2$. This proves a conjecture of Berestycki and Graham in the affirmative for dimension at most $6$, and in the negative for dimension at least $7$. We thereby show that branching Brownian motion cannot survive globally without surviving locally in dimension $6$ or lower, but can do so in dimension $7$ or higher, mirroring the classical transition of Brownian motion from recurrence to transience from dimension $2$ to $3$. We further show that relaxing the strong-KPP condition results in a more general sharp dimension threshold for the uniqueness/non-existence result. These results are valid without any regularity assumptions on the domain boundary. As a corollary, we show that solutions of the evolution reaction-diffusion equation are always convergent in the long run in dimension at most $6$, whilst we exhibit non-convergent solutions in dimension larger than $6$.

Recent frameworks in differential privacy (DP), such as privacy loss distributions and $f$-DP, provide lossless representations of the privacy leakage of mechanisms. These approaches are often considered to yield tighter $(\varepsilon, \delta)$-DP guarantees than those obtained from relaxed notions such as Rényi DP. But is that always the case? We study this question for differentially private stochastic gradient descent (DP-SGD) with fixed-size batches under replace adjacency. Our key observation is that in this setting, privacy accounting based on Rényi DP can use a pair of worst-case distributions (i.e., a dominating pair) that differs from the one used in the lossless frameworks. In particular, we characterize the worst-case distributions as two-dimensional Gaussian mixtures and derive an improved Rényi DP bound for DP-SGD. Numerical and asymptotic comparisons show that across various hyperparameter settings, a Rényi DP accountant based on the new bound can yield better privacy guarantees than those obtained from the lossless frameworks.

We prove that, on any compact Riemann surface, there are only finitely many meromorphic projective connections with prescribed singular points and real exponent differences whose projective monodromy is unitarizable. We also establish properness and growth estimates for holonomy, including a relative version for smooth marked families, and derive finiteness results for spherical conical metrics and ramification divisors of stable extensions.

We characterise all (undirected) cacti graphs, and forests of cacti, with an edge-preserving near-unanimity operation, by means of forbidden induced subgraphs. Indeed all such graphs also have an edge-preserving majority operation. We show that said majority can be chosen to be symmetric and explicitly construct it for all forests of cacti without the forbidden induced subgraphs.

We study the geometry and stability of solution mappings in regularized composite optimization, allowing multiple minimizers. We establish a composite representation that separates optimal linear images from their corresponding primal solutions through an image mapping and a generalized basis pursuit mapping. Under a fiber-attainment condition, the image mapping admits a reduced optimization formulation; neither representation requires convexity. For convex regularizers, suitable qualification conditions yield a Fenchel-conjugate representation of the reduced value function and a geometric characterization of the generalized basis pursuit mapping through affine fibers and a fixed lifting set. Using these representations, we establish sufficient conditions for local Hausdorff Lipschitz continuity of possibly set-valued solution mappings. We further show that the primal-space sufficient conditions imply the corresponding image-space conditions, with an additional injectivity requirement arising in the former. The results cover $\ell_1$-type and structured piecewise linear-quadratic regularizers under suitable assumptions. For classical $\ell_1$ regularization, our approach recovers an existing stability result without a finite sign-pattern decomposition. Numerical examples illustrate the recovery of complete solution sets, including a case with a nonpolyhedral graph of the generalized basis pursuit mapping.

This paper establishes a local turnpike property for a two-population, second-order mean-field game system with a quadratic Hamiltonian, in which the running cost for each population may depend on both population densities. In the case of two populations, imposing a scalar stability condition separately on each species is not sufficient because it neglects cross-interaction effects. The appropriate requirement is instead a matrix-valued coercivity condition on vector perturbations $(\mu_1,\mu_2)$, which involves the symmetric part of the linearized interaction matrix together with weighted Poincaré inequalities associated with the two stationary measures. Consequently, in the multi-population framework, diffusion still acts independently on each population, but the interactions are controlled by the full coupling matrix. A naive scalar energy functional fails to capture mixed second-variation terms arising from costs that depend on both densities. We prove that matrix coercivity is precisely equivalent to the strict positivity of this second variation, and we provide a spectral characterization in terms of the principal spectral value of the coupled stationary linearized operator. For linear models with cross-interactions, this yields explicit algebraic conditions that separate stable parameter ranges from unstable ones. In the non-potential framework, the skew-symmetric part of the interaction matrix leaves the coercivity threshold unchanged, but it can induce transient oscillatory dynamics. Moreover, the scalar turnpike theory extends in a natural, though nontrivial, manner to multi-population systems, where stability is governed by the positivity of a coupled matrix-valued second variation that encodes both self-interactions and cross-interactions.

We propose a model-based trust region method for derivative-free optimization with simulation-based inequality constraints. We are motivated by settings of simulation optimization in which both function evaluations at infeasible points may be unreliable and constraint evaluations are expensive. Our method maintains feasible incumbents and computes steps from a subproblem suitable for a feasible-direction method in order to encourage generation of feasible trial points. To reduce the computational cost of modeling the constraints, we describe an active-set estimation scheme that permits modeling with only a subset of constraints in each subproblem, as opposed to maintaining models of all constraints. We show that every limit point of the incumbents generated by our method is a Fritz-John point, and is additionally a KKT point under the Mangasarian-Fromovitz constraint qualification. Under standard nondegeneracy conditions, the active-set estimate identifies, within finite time, all constraints active at the limit point of the incumbents. Experiments on CUTEst problems show that our method generates feasible trial points far more often than a comparable SQP-based approach and solves more problems within a given budget with fewer function evaluations.

We study block sensitivity and certificate complexity of rotation symmetric Boolean functions, with particular emphasis on monomial rotation symmetric Boolean functions. We first show that every odd-degree monomial rotation symmetric Boolean function on $n$ variables satisfies $s=bs=C=n$. We then determine sensitivity, block sensitivity, and certificate complexity for every quadratic monomial rotation symmetric Boolean function. For the quadratic function $Q_{n,k}$ generated by $x_0x_k$, we prove, outside the short-orbit case, that \[ C(Q_{n,k})= \begin{cases} n,&4\mid n/\gcd(n,k),\\ n-\gcd(n,k),&4\nmid n/\gcd(n,k). \end{cases} \] Moreover, \(bs(Q_{n,k})=C(Q_{n,k})\) for every quadratic MRS function, and outside the short-orbit case, strict separation \(s<bs=C\) occurs exactly when \[ \frac{n}{\gcd(n,k)}\equiv2\pmod4. \] Finally, for the consecutive quartic family \[ R_n(x)=\bigoplus_{i\in\mathbb Z_n}x_ix_{i+1}x_{i+2}x_{i+3}, \] we determine all three measures in every residue class modulo $6$, using the sensitivity formulas and, in one residue class, finite-state results established in our companion preprint~\cite{LiZhangRSBFSensitivity}. In particular, for $r=2,3$ and $6q+r\ge5$ we obtain strict separation, \[ \boxed{s(R_{6q+r})=6q<bs(R_{6q+r})=C(R_{6q+r})=6q+1,\qquad r=2,3.} \] Thus this family gives an infinite collection of rotation symmetric Boolean functions for which sensitivity is strictly smaller than block sensitivity, while block sensitivity and certificate complexity coincide.

On 6 October 2026 OpenAI published an initial 722 AI-written mathematical manuscripts, organized into 372 problem families, in the public repository github.com/openai/math. The release includes claimed solutions to many prominent open problems. We approach this collection as a treasure mine of mathematical ideas. Our aim is to extract new lemmas, constructions, inequalities and proof mechanisms introduced in the source arguments, and to distinguish them from established tools and new combinations of those tools. We call these mathematical instruments. Extraction comes first; their validity, novelty and possible reuse then become subjects for further study. The resulting descriptions also help readers understand the selected papers without having to reconstruct every argument at the outset. We examine 38 families in two groups of 17 and 21, spanning number theory, geometry, analysis, combinatorics, complexity, algebra, operator algebras, topology and mathematical physics. Each entry records the claim, extracted instruments, antecedents and caveats. Eleven cross-cutting patterns connect the entries, and a scoreboard records formalization status and dependencies. Novelty assessments are provisional and sometimes mixed. Selected source comparisons correct mathematical statements, hypotheses and counts; they do not independently establish priority or certify the underlying proofs. No Lean development was compiled or kernel-checked for this survey. We also briefly describe a subsequent application of the signed restriction lemma from family 112 to circuit lower bounds at every fixed depth.

We study privacy-preserving algorithms that redact information from a correlated dataset. The goal is to reduce the information leakage about sensitive parts. We analytically characterize the optimal tradeoff between privacy (leakage through correlation) and utility (number of redactions) for local mechanisms and binary Markovian correlations. Our result shows how to decompose the mechanism design problem into a sequence of local mechanism design problems and reveals their sequential relation. The optimal mechanism is the solution to a Finite-Horizon Deterministic Markov Decision Process.

We compute the index and nullity of any harmonic morphism from $S^3$ to $S^2$.