Papers

Local differential privacy (LDP) protects individuals in a dataset by perturbing each measurement before release. For real-valued data, a widely used mechanism is additive Laplace noise. We study the problem of estimating the distribution of the latent confidential data from the privatized observations via the nonparametric maximum likelihood estimator (NPMLE) under an i.i.d. sampling model. We first show that under the Laplace convolution model, the NPMLE admits a finite-dimensional reformulation in which the support is restricted to the observation set. This reduces the original infinite-dimensional optimization over all mixing distributions to an $n$-dimensional convex optimization over mixture weights, where $n$ is the sample size. We then study the statistical convergence of the NPMLE under the 1-Wasserstein distance, and explicitly connect its convergence rate with the privacy noise scale. Allowing the privacy noise level to change with the sample size, our analysis shows that the NPMLE remains consistent when the Laplace noise grows at a rate slower than $n^{3/16}$. Conversely, when the Laplace noise is of the order $\sqrt n$ or larger, no estimator can achieve uniformly consistent recovery of the latent distribution.

We show that the nilpotent cone associated to a simply connected semisimple algebraic group, together with all its crepant projective partial resolutions, can be constructed as Hamiltonian reductions of the affine closure of the cotangent bundle of base affine space for suitable choices of stability parameter of a maximal torus. We also realize the base change of the universal Poisson deformation of each of these crepant projective partial resolutions as the GIT quotient of the affine closure of the cotangent bundle of base affine space at the same stability parameter.

Traditional plasticity theory is commonly organized through an elastic law, a yield condition, a flow rule, hardening relations, and loading–unloading conditions. This paper formulates these relations through directional stationarity of one scalar functional evaluated over one-sided admissible plastic paths. The first variation determines stress, internal-variable forces, and activity resistance. Restriction to an admissible plastic tangent defines a reduced directional force for each mechanism; its sign and one-sided stationarity give the elastic inequality, complementarity, loading–unloading conditions, and active consistency. Associated response corresponds to alignment between the admissible tangent and the normal to the resulting yield boundary, whereas a nonparallel tangent represents non-associated flow. A self-similarity analysis identifies positively homogeneous stress gauges as a broad associated family and separates the roles of yield-surface shape, isotropic expansion, and kinematic translation. The storage and resistance terms cover isotropic, kinematic, coupled, and gradient hardening, while multiple activity variables describe independently activated mechanisms. A viscous potential extends the construction to rate-dependent evolution. Four closed-form solutions illustrate multi-activity regions, kinematic-hardening fields, pressure-sensitive limit states, and the distinct displacement fields associated with normal and non-normal flow. Time-discrete constitutive integration and dimensional checks are collected in the appendices.

We compare the sharp constants in Lee-type Schatten norm inequalities for finite families of complex matrices with the corresponding constants obtained after applying an arbitrary nonnegative concave function to the relevant absolute values. We prove that the nonlinear problem has exactly the same best constant as the underlying linear problem for every finite matrix dimension, every number of summands, and every Schatten exponent, including the operator-norm endpoint. For finite exponents, the proof proceeds through a lossless single-cap reduction, an exact representation as a positive mixture for finite cap combinations, and a noncommutative reassembly argument based on a weighted Schatten contraction and the Araki-Lieb-Thirring inequality. Finite-spectrum cap interpolation and a perturbation argument at zero then yield the general concave case. Thus every sharp linear result transfers without loss to the full class of nonnegative concave functions.

In $2011$, Osin and Thom conjectured that the first $\ell^2$-Betti number of a torsion-free discrete group is bounded above by the normal rank of the group minus one. The conjecture has surprising consequences for some fundamental problems in group theory and topology. These include the Wiegold problem on perfect groups, the Levin conjecture, the torsion-free case of the Kervaire conjecture, and an important special case of the Whitehead asphericity conjecture. In this article, we construct for each $n\in \mathbb{N}$ a countable torsion-free group $\Gamma_n$ such that $\beta^{(2)}_1(\Gamma_n)=n$ and so that the normal rank, $n(\Gamma_n)$, equals one. This disproves the conjecture. Our counterexamples are locally free and hence locally indicable. However, they are not finitely generated.

We study the positivity of the tropical multidegrees of a tropical variety contained in a product of real vector spaces. These multidegrees are obtained by stably intersecting the tropical variety with pullbacks of positive tropical divisors. We introduce projection-purity and facet-selectability, two conditions under which positivity is determined by the dimensions of the natural projections, and the support of the tropical multidegrees is precisely the set of lattice points of a polymatroid base polytope. This extends He's theorem for translation-admissible tropical varieties. We also show that these conditions alone do not force the corresponding tropical volume polynomial to be Lorentzian. By contrast, for the augmented Bergman fan of any polymatroid, the positive multidegrees are supported precisely on the lattice points of the polymatroid base polytope, and the tropical volume polynomial is Lorentzian for every sequence of positive tropical divisors.

We give a brief and partially informal introduction to the Birch and Swinnerton-Dyer conjecture. After a short review on basic Diophantine equations, we define the natural abelian group structure on an elliptic curve, leading to the definition of the corresponding algebraic rank. Afterwards, we sketch how to define the analytic rank of an elliptic curve and we state the conjecture, that is, the equality of these two ranks.

Let $n\geq3$ and $0\leq\tau\leq2$. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive $\tau$-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindrical necks, the key ingredients are a preserved quantitative spectral pinching condition, which converts the intrinsic hypothesis into uniform two-convexity, and cylindrical and derivative estimates that remain valid across the hyperbolic standard neck replacement. At the endpoint $n=3$, $\tau=2$, positive $\tau$-bi-Ricci curvature is positive Ricci curvature and forces strict convexity, so the ordinary mean curvature flow converges to a round point. Consequently, the underlying manifold is diffeomorphic to a sphere or to a finite connected sum of copies of $\mathbb{S}^{n-1}\times\mathbb{S}^1$. If the initial hypersurface is embedded and bounds a compact domain, that domain is a one-handlebody, namely a ball with finitely many one-handles attached.

We summarize our advances in the algorithmic development and hardware utilization for risk-averse optimization problems in random materials. This includes risk-averse optimization using the entropic risk measure, as well as recently developed sampling techniques for random materials, that are interoperable with the optimization framework. Furthermore, we discuss recent progress in the efficient utilization of modern hybrid hardware architectures for these methods to solve three-dimensional partial differential equations.

Since Hastings' proof of superadditivity of classical communication over quantum channels, considerable effort has been devoted to finding a structural explanation of this phenomenon that was originally established by concentration of measure for Haar random unitaries. The main observation of this work is that Haar randomness can be replaced by random permutations without changing the limiting geometry responsible for nonadditivity. This replacement turns a continuous problem over unitary matrices into a discrete combinatorial problem over zero–one permutation matrices, and thereby opens a path toward derandomization. The theorem of Bordenave and Collins shows that random permutations have the required limiting behavior and the algorithm of O'Donnell and Wu then provides a deterministic asymptotic construction, running in polynomial time in the size when the channel parameters and accuracy are fixed. Thus the random construction can be derandomized in an asymptotic algorithmic sense. Finally, a quantitative random permutation estimate by Chen, Garza-Vargas, Tropp and van Handel gives a fully numerical estimate: there exists a tuple of 57,836,025 permutations acting on a set of size \[ N \le 5.422\times 10^{116216}\] such that the associated finite dimensional channel exhibits nonadditivity. This enormous value remains an obstacle to a practical construction.

We present recent developments in multilevel sampling methods under resource constraints. Over the past 15 years, multilevel methods have become widely used for uncertainty quantification. However, scaling them to high-dimensional problems and high-performance computing (HPC) environments remains challenging. In this work, we discuss two algorithms designed to address these issues: the budgeted Multilevel Monte Carlo (MLMC) method and the Multilevel Stochastic Gradient Descent (MLSGD) method. We demonstrate their effectiveness on HPC systems under consideration of the available computational resources for applications in forward uncertainty quantification (UQ) and optimal control (OC) under uncertainty.

We show that a Kähler surface with opposite Hermitian conformally Kähler structure defined by a foliation of complex curves is a QCH Kähler surface of Calabi type.

Suppose we are given an arbitrary line bundle $\mathcal{L}$ on a complex algebraic variety $X$, not necessarily smooth. For a positive integer $N$, suppose there exists a global section $s \in \Gamma(X, \mathcal{L}^{N})$ that defines an effective Cartier divisor $D$. If we denote $\pi: Y \rightarrow X$ to be the $N$-fold cyclic covering of the divisor $D$ resulting from the global section $s$, we describe the cyclic covering in terms of Morihiko Saito's theory of mixed Hodge modules.

$X$ is a Nakajima quiver variety. In this note we report the observation that the space parameterizing quasimaps from $\mathbb{P}^1$ to $X$, sending infinity to a given point, may be presented globally as the critical locus of an explicit function.

The Turing jump has no fixed point on the Turing degrees: $\mathbf a <_T \mathbf a'$ for every degree $\mathbf a$. After passing to the ideal completion, however, a natural fixed-point phenomenon appears. We study the Scott-continuous lifting $\Gamma:\operatorname{Idl}(\mathbf D_T)\to\operatorname{Idl}(\mathbf D_T)$, given by $\Gamma(I)=\downarrow\{\mathbf a':\mathbf a\in I\}$. Starting from the computable degree, Kleene iteration reaches its first fixed point at stage $\omega$, namely the Turing ideal of arithmetical degrees; more generally, above $\mathbf a$ the least fixed point is the ideal of degrees arithmetical in $\mathbf a$. To pass beyond this fixed point, we introduce a limit-uniformization operator. Although the ideal of finite jumps contains every $\mathbf 0^{(n)}$, it does not contain the uniform limit oracle $\mathbf 0^{(\omega)}=\deg_T\!\left(\bigoplus_{n < \omega}0^{(n)}\right)$. The uniformization operator adjoins this oracle only when all finite jump degrees are present. It is monotone but not Scott-continuous. Composing jump closure with one such gate yields closure ordinal $\omega\cdot 2$; gates at $\omega,2\omega,3\omega,\ldots$ yield closure ordinal $\omega^2$. Thus non-uniform closure under relativized halting problems is Scott-continuous and reaches fixed ideals, while uniform coding of an entire prior hierarchy is infinitary, discontinuous, and reopens diagonalization. This gives a domain-theoretic semantics for the successor/limit distinction in transfinite Turing-jump hierarchies and links failures of Scott continuity with closure ordinals.

In this study, we construct the first orthogonal basis for additively consistent subspace in pairwise comparisons theory. This construction is based on our representation of additively consistent best approximations of skew-symmetric matrices with respect to a tensor basis having minimal support. The orthogonal basis establishes the logarithmic consistent projection for the orthogonal windowing of pairwise comparisons matrices. It is compared with the windowing of the Saaty and SVD types. These comparisons resulted in new composite formulae for logarithmic, Saaty, and SVD projections. The theoretical considerations presented in the paper are accompanied by numerous examples.

Let $X$ be an $n$-dimensional generalized Grassmannian not isomorphic to $\mathbb{P}^n$. We prove that $k(X)\le n-1$, where $k(X)$ denotes the maximal integer such that every uniform bundle on $X$ of rank at most $k(X)$ is homogeneous. In particular, for smooth quadrics $\mathbb{Q}^n$, we have $k(\mathbb{Q}^n)=n-1$ for odd $n$, and $n-2\le k(\mathbb{Q}^n)\le n-1$ for even $n$. We classify uniform rank $n$ bundles on $\mathbb{Q}^{n}$ for $n=3$, $5$. Furthermore, we characterize projective spaces among generalized Grassmannians in terms of uniform bundles.

Randomization testing with permutations is a very common nonparametric approach to hypothesis testing. However, randomization testing can be done with other group transformations including random rotations. In this work, we consider the problem of invariance in quadratic form statistics under a unified framework with closed form p-values. In particular, we propose a nonparametric variant of the classic Durbin-Watson test for testing for autocorrelation in time series data at arbitrary lags. Our test is performed by integrating over a group of invariances of the test statistic, and easy-to-compute analytic formulae for the p-value are derived from concentration inequalities on compact groups on a per-lag basis. Thus, the usual necessity of large-scale Monte Carlo simulations is rendered unnecessary. Our tests outperform the classic Breusch-Godfrey and Ljung-Box tests on simulated data with respect to statistical power to identify significant autocorrelation. They also can be used to identify the presence of autocorrelation at large lags such as in monthly solar intensity data, which follows an approximate 11 year (132 month) cycle. The general formulation of this approach can be easily adapted to other quadratic form statistics.

We investigate the statistics of the time-integrated current in an infinite one-dimensional gas of independent active Ornstein–Uhlenbeck particles, as a model system for studying an active generalisation of the corresponding passive (diffusive) phenomenology. Unlike the latter, where current fluctuations exhibit universal sub-diffusive scaling, active systems display diffusive, super-diffusive, and sub-diffusive regimes over different time scales. We fully characterise the distribution of current in terms of large-deviation asymptotics, showing that all of these scaling regimes are described by a single scaled cumulant generating function. Moreover, the statistics retain a dependence on the initial condition even at large times, revealing a persistent memory of the initial state. We further obtain the joint large-deviation statistics of currents measured at two distinct times, characterising temporal correlations. Our analytical predictions are verified using rare-event importance sampling, which resolves probabilities as small as $10^{-1000}$.

The defining feature of one-sided device-independent quantum key distribution is its asymmetric trust model in which only one party is characterized. This scenario sets an interesting middle ground between high key rates achievable by characterizing devices and the security of full device-independence. Here, we provide new tools, methods, and benchmarks for calculating key rates in this setting. To achieve this, we develop and compare two extensions of the NPA hierarchy and derive finite-size security bounds against general attacks with arbitrary device memory. The latter is based on the Generalized Entropy Accumulation Theorem. We then investigate the performance of various protocols: the BB84 protocol both with and without losses, a qutrit mutually unbiased bases protocol, and protocols based on Bell inequalties such as CHSH and $I_{3322}$. We find that the choice of which party is characterized can strongly affects the key rate, surprisingly without a universal ordering. Our work thus offers a general toolbox for calculating key rates for one-sided device-independent QKD protocols.

In this paper, we investigate the neighbourhood convexity ($n$-convexity) on graphs, a new finite convexity space grounded in the common closed neighbourhood closure operator. Unlike standard path-based graph convexities, $n$-convexity shows a non-canonical behaviour, giving rise to compelling structural properties and being almost never hereditary. Focusing on the properties of graphs that form $n$-convex geometries, a parity distinction emerges: an $n$-convex geometry contains a star vertex if and only if the number of its vertices is odd. Every odd-order $n$-convex geometry can be uniquely constructed by attaching a star vertex to an even-order one. We introduce the concept of quasi-stars (vertices of degree $\vert{}V\vert{}-2$) and prove a reduction property that allows systematically reducing an $n$-convex geometry by removing a pair of vertices, one of which is a quasi-star. Finally, we explore the connections between $n$-convexity and $P(G)$, the neighbourhood preorder, demonstrating that $n$-convex sets are upsets of $P(G)$ and that, in star-free $n$-convex geometries, quasi-stars correspond precisely to the maximal elements of $P(G)$. We complete our study by classifying quasi-threshold and threshold $n$-convex geometries.

This note supplies genuinely non-fibred examples for the manuscript: Rigidity on the Two-Torus and Sarnak's Conjecture. For every $0<\delta<\tfrac12$, we construct $C^\infty$ Lebesgue-area-preserving pseudo-rotations of $\mathbb{T}^2$ which satisfy the $(C,\delta)$-deviation condition but do not have bounded mean motion. We give both semi-irrational and totally irrational rotation vectors and two realizations: a controlled weakly mixing Anosov–Katok construction and an explicit weakly mixing special flow construction. Weak mixing is used as a conjugacy-invariant obstruction to every continuous circle-rotation factor. Consequently, none of the resulting maps is topologically conjugate, by a linear or nonlinear change of coordinates, to a skew product over a circle rotation. In the special-flow realization, the same lacunary Fourier series simultaneously gives weak mixing, the sharp upper bound $O(n^\delta)$, and unbounded deviations; in fact no smaller deviation exponent is possible. The semi-irrational examples meet the assumptions of Theorems~1 and~2 of the cited manuscript, whereas the totally irrational examples meet those of Theorem~1. Each construction produces continuum many maps and continuum many topological conjugacy classes of each rotation type.

Thousands of sangaku tablets, inscribed with colorful geometry problems, were posted in shrines and temples across the Japanese archipelago starting in the Edo period (1603–1868), representing an intersection of artistic, religious, and mathematical traditions. The Sangaku Archive is an attempt to carefully document these objects using the latest digital tools, and to tell an unusual and compelling story about mathematics—one that enriches computations and proofs with historical context and cultural appreciation.

Let $s_2(n)$ be the binary sum-of-digits function and let $c_t$ be the natural density of the integers $n\ge0$ for which $s_2(n+t)\ge s_2(n)$. Earlier work of the author proved the universal exponential bound $$c_t-\frac12\ge 2^{-2s_2(t)-1},$$ thereby resolving Cusick's conjecture for every $t$. This estimate, however, does not reflect the true size of the smallest possible bias at a given large Hamming weight. In this paper, we determine this extremal scale sharply: $$\inf_{s_2(t)=k}\left(c_t-\frac12\right) \sim \frac{1}{2\sqrt\pi} \left(\frac{\log_2 k}{k}\right)^{3/2} \qquad(k\to\infty).$$ Thus the optimal fixed-weight gap is polynomial-logarithmic rather than exponential, with the explicit sharp leading constant $1/(2\sqrt\pi)$. The proof combines the five-cumulant Edgeworth expansion of Spiegelhofer and Wallner with a new extremal rigidity mechanism for near-extremal binary block patterns. We also prove a stability theorem for asymptotic extremizers and give a separate shadow-energy interpretation of the same constant.

Lacey used sparse domination to study the sharp weighted norm estimate of the maximal function of predictable multipliers in discrete time filtration spaces. Domelevo, Petermichl, and Škreb developed the self similarity argument known as sparse domination in an abstract martingale setting with a continuous time parameter. In our investigation, we establish sparse domination for discrete-time martingale transforms, introducing the novel concept of conditional sparsity as a core property of our approach. The conditional sparsity framework enables derivation of sharp weighted estimates and a mixed-norm estimate \( A_p^\alpha A_r^\beta \) that improves upon known sharp \( L^p \) bounds. Moreover, we develop dedicated sparse domination specifically for Doob's maximal operator, recovering the sharp bound as a direct application. Finally, we focus on the application of sparse theory to quantitative two-weight estimates.

A celebrated result of Bayer and Stillman from 1987 states that for a homogeneous ideal $I$ of a polynomial ring $S$, the regularities of $S/I$ and $S/\textrm{GIN}(I)$ are the same under the reverse lexicographic monomial ordering, where $\textrm{GIN}(I)$ is the generic initial ideal. If $R$ is a polynomial ring whose variables are subdivided into disjoint blocks of variables $X_1,\dots,X_c$, there is a natural multi-grading on $R$, and one can analogously define a multi-graded version of the generic initial ideal for any multi-homogeneous ideal $I$ of $R$. However, the full strength of the Bayer–Stillman Theorem fails in the multi-graded setting; there are multi-homogeneous ideals $I$ such that the regularities are not preserved after passing to the multi-graded generic initial ideal no matter the choice of monomial ordering. We prove lower bounds on the regularity of $R/I$ in terms of almost regular sequences of the multi-graded generic initial ideal of $I$ restricted to each block of variables. Again, we use the reverse lexicographic monomial ordering, but interestingly, the lower bound result requires a particular choice of ordering on the variables. As an application, we prove the optimal fractional Helly theorem for $d$-Leray simplicial complexes, a problem stemming from the work of Kim in 2017.

Detailed multicellular growth simulations based on subcellular element models (SEMs) can capture complex tissue development, but their element-level interactions impose substantial computational cost. This work presents a scalable multi-GPU framework for 3D multicellular growth simulation that combines GPU acceleration, spatial binning, domain decomposition, and workload-aware partitioning. Cell movement, growth, and division continuously reshape the spatial workload distribution, causing initially balanced partitions to become inefficient over time. To address this, we introduce an RNN-based load-balancing controller that observes recent per-rank execution times and partition states and learns residual corrections to a reactive boundary-adjustment rule. The controller is trained offline in a differentiable surrogate of the load-balancing loop with randomized workload dynamics, requiring no measured execution traces for training. We evaluate the framework in terms of single-GPU acceleration, multi-GPU computation scaling, controller-level load-balancing behavior, and end-to-end simulation performance, with comparisons against static partitioning, reactive load balancing, and conventional time-series prediction baselines. A representative embryonic epidermal development use case further demonstrates the type of spatially and temporally evolving workload targeted by the framework. In our evaluation, GPU acceleration with spatial binning accelerates the interaction computation by roughly three orders of magnitude over a serial CPU baseline. RNN-guided load balancing reduces the mean global imbalance from 11.3% under static partitioning to 3.5%, lowers end-to-end runtime by 9.0% relative to static partitioning, and reduces slice migration by 7.7x compared with the reactive baseline, showing that history-aware control can improve workload balance while avoiding unnecessary repartitioning.

We construct infinitely many new pairwise nondiffeomorphic smooth structures on infinitely many closed simply connected spin 4-manifolds with positive signature. Our construction builds on an infinite family of simply connected complex surfaces of general type due to Roulleau and Urzúa that populate points arbitrarily near the Bogomolov-Miyaoka-Yau line in the complex geography plane. We also discuss how the conjectural symplectic Bogomolov-Miyaoka-Yau inequality implies that our results are close to being optimal.

We determine, up to absolute constants, the worst-case number of point evaluations required for a weighted $L_2$ Marcinkiewicz-Zygmund inequality for an $m$-dimensional complex function space. If $0<\varepsilon<1$ is the relative distortion, this number is $$\Theta\Big(\min\Big\{m^2,\frac{m}{\varepsilon^2}\Big\}\Big),$$ and exact discretization has the sharp worst-case value $m^2$. While the upper bounds follow from recent constructions, our contribution is the construction of function spaces that are hard to discretize and yield matching lower bounds. We use a trace-variance inequality for weighted subframes of unit-norm tight frames. One such instance is the complete-graph edge frame, which yields a construction in every dimension. Singer equiangular tight frames improve the constant when $m-1$ is a prime power, while maximal equiangular tight frames give the strongest bound possible using our method whenever they exist. We also derive consequences for the conditioning of weighted least-squares systems and for standard condition-number-based iteration estimates when these systems are solved by LSQR.

The Serre problem asked if a locally trivial holomorphic fibration with Stein base and Stein fiber is itself Stein. \v Skoda constructed the first counterexample for this problem. Later, Cœuré and Loeb constructed another remarkable counterexample, with bounded domain of holomorphy as fiber. The total space of their fibration has global holomorphic functions which separate points and provide local coordinates. However, it does not admit a Stein envelope of holomorphy. In this paper, we prove that the Cœuré–Loeb fibration can be realized as an open subset of a Stein space, more precisely it is the complement of an analytic set in a Stein space.