Papers

In the 1980s, Stanley and J. Watanabe proved that over a field of characteristic zero every monomial complete intersection has the strong Lefschetz property. Determining which monomial complete intersections in positive characteristic have the weak Lefschetz property has been the subject of ongoing research. In this paper, we give a complete description of the locus of exponent vectors corresponding to monomial complete intersections over a field of positive characteristic that fail the weak Lefschetz property. We describe this locus explicitly in terms of translates of positive-orthant lattice balls in the taxicab metric.

We construct quantum locally testable codes (LTCs) with constant rate, distance, soundness and locality under a variant of a product expansion conjecture of Bafna and Vyas about Reed-Solomon codes. In particular, we use the high-dimensional expansion framework of Dinur, Lin and Vidick for constructing quantum LTCs, instantiated with the non-Abelian cubical complexes of Rungtanapirom, Stix and Vdovina. Our code is obtained by equipping the complex with carefully chosen Reed-Solomon local codes whose symmetries are compatible with those of the complex.

We prove the Mumford–Shah conjecture, formulated in 1989, by completing the classification of blow-up limits of minimizers in the plane. The remaining case concerns limits whose discontinuity set is noncompact and has connected complement. In this setting, we rule out bounded connected components using second-variation inequalities obtained from combined inner and outer variations associated with two perpendicular translations of a compact subset separated from the rest of the discontinuity set. Established results then imply that the discontinuity set is connected, and the known classification for connected discontinuity sets completes the proof.

In this paper, using the equivalence of two extensions, we obtain an improved upper bound for the perturbation index, improving the bounds of Quy and N. V. Trung.

For every $n\ge2$, we construct $4n-2$ nondegenerate Gaussian measures on $\mathbb{R}^{6n-3}$ that cannot be simultaneously equipartitioned by two affine hyperplanes. This disproves the Ramos conjecture for two hyperplanes. Combined with known upper bounds, the construction shows that $3\cdot2^{s-1}-2$ is the least dimension guaranteeing a common two-hyperplane equipartition of $2^s-2$ absolutely continuous probability measures, for every $s\ge3$. We characterize the Gaussian equipartition threshold in terms of the least number of positive definite quadratic measurements needed for phase retrieval. Modified complex polynomial multiplication gives $2r-2$ positive definite measurements in every even dimension $r\ge4$. This number is optimal when $r=2^k+2$, $k\ge1$.

In 2002, Montiel and Urbano conjectured that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore-type functional $ \mathcal{W}^-=\int_{T^2}(2+|H|^2)\,dA,$ either among all tori or among all Lagrangian tori. In this paper, we confirm this conjecture in the Lagrangian setting and disprove it in the general setting. We establish that every oriented closed Lagrangian surface of genus $g\geq 1$ in $\mathbb{C}P^2$ has $\mathcal{W}^-$-energy no less than that of the Clifford torus. Moreover, we construct non-Lagrangian deformations of the Clifford torus along which $\mathcal{W}^-$ strictly decreases.

We present an analytical organizing principle underlying the dynamics of the periodically forced Duffing oscillator. In the massless limit the system reduces to a time-dependent cubic equation whose discriminant and Jacobian classify the canonical Duffing regimes and determine the number and stability of instantaneous equilibria. For the double-well oscillator, the same construction gives an exact forcing threshold $F_{\mathrm{bif}}$ separating intra-well and inter-well dynamics. We show that this algebraic quantity governs the finite-mass dynamics far beyond the singular limit from which it originates: the boundary between intra-well and inter-well motion converges to $F_{\mathrm{bif}}$ as $m\to0$, and the families of pitchfork bifurcations responsible for asymmetric inter-well states accumulate at a common point. Numerical simulations further show how the discontinuous jumps of the degenerate model are regularized at finite mass into short oscillatory transients that shrink as $m\to0$. These results reveal the massless Duffing equation as the organizing center of the full finite-mass dynamics and establish a direct link between the algebraic structure of the degenerate problem and the global bifurcation structure of the oscillator.

Let $(M^3,g)$ be a complete, connected, noncompact Riemannian three-manifold without boundary and with nonnegative sectional curvature. We prove that its scalar-curvature integral over geodesic balls, divided by the radius, has a limit. In the one-ended case, \[ \lim_{r\to\infty}\frac1r\int_{B_p(r)}\operatorname{Scal}\,d V =8\pi\bigl(\chi(M)-V_M\bigr)\le8\pi(1-V_M), \] where $V_M$ is the asymptotic volume ratio. In the one-ended case, we show that $\chi(M)\in\{0,1\}$. Thus positive asymptotic volume ratio gives the value $8\pi(1-V_M)$, while in the collapsed case the value is determined by the topology of the end. No pole or scalar-curvature bound is assumed. The proof combines separate smooth approximations of the Busemann and distance functions, integrable negative curvature errors, and a determinant estimate on the level surfaces. An averaged boundary estimate then gives convergence of the integrated extrinsic curvature. In the two-ended case, the splitting theorem gives the exact limit $8\pi\chi(N)$ for the compact surface factor $N$. The one-ended upper bound is attained for every prescribed asymptotic volume ratio in $[0,1]$, and the two-ended bound is also sharp.

We classify compact Kähler surfaces with semipositive anticanonical bundle. In particular, we resolve a question of Demailly–Peternell–Schneider: any blowup of $\mathbb{P}^2$ at 9 points on a smooth cubic has semipositive anticanonical bundle.

We completely characterize when a general curve of genus $g$ admits a non-degenerate degree $d$ map to projective space $\mathbb{P}^r$, that is $k$-secant along a linear space $\mathbb{P}^s \subset \mathbb{P}^r$ and deforms in a smooth family of expected dimension. This gives an optimal improvement of a theorem of Farkas, who gave necessary conditions, and proved sufficiency under additional numerical assumptions. Our main theorem shows that the natural necessary conditions are in fact sufficient, yielding a generalization of the classical Brill-Noether existence theorem. The proof relies on deformation theory of stable maps, and is valid in arbitrary characteristic.

Alpöge's construction produces a period family of compact complex threefolds fibred over $\mathbb{P}^1$, with complex two-tori as general fibres. We give a direct proof that every admissible member is Oka, using only the complex-analytic construction. Du–Guo–Kusakabe–Xie independently established the Oka property and determined the exact biholomorphism classes in this family. Assuming Alpöge's identification of the underlying smooth manifolds with $S^6$, these results yield uncountably many pairwise nonbiholomorphic Oka complex structures on the six-sphere.

A fundamental question in physics is: When does classical behavior emerge from quantum systems? Bosonic Gaussian states provide a natural setting to explore this quantum-classical boundary, as they capture both the classical field behavior and the intrinsic quantum nature of light. Here, we address this problem from a learning-theoretic perspective by asking: When are bosonic Gaussian states classical to learn? That is, under what conditions (if any) can an n-mode bosonic Gaussian state be learned with as few samples, and with operations as simple, as are needed to learn a classical 2n-variate Gaussian distribution? We establish a smooth crossover in learnability governed by the state's thermal fluctuations: - Cold Gaussian states are non-classical to learn: When the covariance matrix satisfies $\Sigma\le(\frac12+O(\frac1n))I$, i.e. close to the vacuum covariance, tomography under single-copy (i.e., non-entangled) measurements fundamentally requires $\Omega(n^3)$ copies, strictly exceeding the sample complexity $\Theta(n^2)$ of learning classical Gaussian distributions. We show that this hardness persists even when few-copy entangled measurements are allowed. - Warm Gaussian states are classical to learn: When thermal fluctuations exceed the vacuum noise, parameterized by $\Sigma\ge(\frac12+\nu)I$ for any parameter $\nu>0$, we prove that single-copy tomography requires $N=\Theta\left(n^2\min(n,1+\nu^{-1})\right)$ copies. This bound is tight and is achieved by simple, non-adaptive, unentangled heterodyne measurements. Crucially, for $\nu=\Omega(1)$, the sample complexity drops to $\Theta(n^2)$, matching the classical case. Our results tightly characterize a quantum-to-classical crossover in the learnability of bosonic Gaussian states, reveal a novel connection between fundamental physics and statistical learning theory, and have implications for real-world sensing experiments.

Finite-temperature many-body theories are typically formulated in imaginary time, where they are amenable to efficient numerical treatment. However, the extraction of spectral properties from the imaginary axis is difficult and requires numerical analytic continuation, while a direct formulation on the real axis requires prohibitively expensive numerical quadrature. In this paper, we represent the objects of many-body perturbation theory using systematically improvable pole and moment expansions and develop the corresponding framework for evaluating diagrammatic contributions directly on the real axis. We introduce a controlled recompression procedure that prevents the proliferation of poles under algebraic and diagrammatic operations. We benchmark the approach on paradigmatic quantum impurity problems and apply it to self-consistent calculations of the uniform electron gas. The resulting spectral quantities converge systematically towards numerical accuracy. This work provides a practical framework for controlled and systematically improvable calculations of spectral quantities within diagrammatic many-body theories, including self-consistent conserving approaches.

We show that a great circle in $\mathbb{S}^3$ bounds embedded non-orientable minimal surfaces of unbounded genera and all possible non-zero Euler numbers. This answers a question of R. Hardt and H. Rosenberg (1990). It also gives the first example of a real analytic Jordan curve bounding infinitely many embedded minimal surfaces of distinct topological types. Such surfaces may arise as the links of boundary singularities of non-orientable minimal hypersurfaces in $\mathbb{R}^4$.

Aldous and Fill (2002) conjectured the asymptotic maximum relaxation time of a random walk on a connected regular graph. Since the relaxation time of a $d$-regular graph $G$ is $d/\mu(G)$, where $\mu(G)$ denotes its algebraic connectivity, this conjecture is closely related to the problem of minimizing algebraic connectivity among regular graphs. Guiduli and Mohar (1996) conjectured that, for every fixed minimum degree $\delta=d\ge3$ and all sufficiently large orders, graphs with minimum algebraic connectivity are path-like and, apart from bounded portions near their two ends, have a prescribed block structure. Abdi and Ghorbani (2024) proposed an analogous structural conjecture for $d$-regular graphs with minimum algebraic connectivity and fixed degree $d\ge3$. In Part~I, we proved the Aldous–Fill conjecture, the Guiduli–Mohar conjecture, and for odd degrees, the Abdi Ghorbani conjecture. In this paper, we settle the remaining even-degree case, thereby completing the structural characterization of regular graphs with minimum algebraic connectivity. We also prove that, for every fixed even $d\ge4$, the minimum algebraic connectivity at order $n$ is $2(d-2)\pi^2/n^2+O_d(n^{-3})$, and that every minimizing graph has diameter $3n/(d+1)+O_d(1)$. For every fixed even $d\ge6$, $d$-regular graphs whose algebraic connectivity is asymptotically minimum have asymptotically maximum diameter. Finally, we obtain a sharp normalized-gap bound for all even regular degrees, including degrees that grow with $n$.

Aldous and Fill (2002) conjectured that the maximum relaxation time of a random walk on a connected regular graph with $n$ vertices is bounded above by $(1+o(1))\frac{3n^2}{2\pi^2}$, with asymptotic equality for even $n$. Since the relaxation time of a $d$-regular graph $G$ is $d/\mu(G)$, where $\mu(G)$ denotes its algebraic connectivity, this conjecture is closely related to the problem of minimizing algebraic connectivity among regular graphs. Guiduli and Mohar (1996) conjectured that, for every fixed minimum degree $\delta=d\ge 3$ and all sufficiently large orders, graphs with minimum algebraic connectivity are path-like and, apart from bounded portions near their two ends, have a prescribed block structure. For fixed odd degree $d\ge 3$, Abdi and Ghorbani (2004) conjectured that $d$-regular graphs with minimum algebraic connectivity have the same structure. We prove the Aldous–Fill conjecture and the Guiduli–Mohar conjecture, as well as the corresponding conjecture for $d$-regular graphs of fixed odd degree. Finally, we prove that, for every fixed odd degree $d\ge 3$, $d$-regular graphs, as well as graphs of fixed minimum degree $d\ge 3$, whose algebraic connectivity is asymptotically minimum have asymptotically maximum diameter. This establishes the corresponding cases of another conjecture of Abdi and Ghorbani.

We achieve linearly many parallelizable and addressable transversal logical multi-controlled-Z gates with asymptotically optimal parameters simultaneously on good quantum low-density parity check codes and quantum locally testable codes, by applying the gate framework of [arXiv:2604.01874] to good qLTC constructions. To establish the nontriviality of the logical operation, we express the cup product pairing as a coefficient in a product of Moore determinants and prove that this coefficient is nonzero. Moreover, we use the symmetries of the covering spaces to construct large spaces of cocycles. By multiplying the initial cocycles by suitable elements of these spaces, we obtain linearly many independent pairings, each selected by a corresponding cycle constructed using Poincaré duality. This yields linearly many parallelizable and addressable multi-controlled-Z gates and enables asymptotically constant-overhead magic state distillation with good qLDPC codes.

We prove that, over a commutative Noetherian ring, every finitely generated module of finite complete intersection dimension has finite quasi-projective dimension. Our proof adapts Bergh's technique of reducing complexity, originally used to establish virtual smallness for complexes of finite complete intersection dimension over local rings, by successively enlarging the perfect locus of the cones and controlling their homology. This gives an affirmative answer to part of a question of Jorgensen, Takahashi, and the third author. We also establish a dual injective version of our result for rings admitting a dualizing complex. Finally, we show that, over such rings, the three injective analogues of complete intersection dimension appearing in the literature coincide for bounded complexes with finitely generated homology, while two of them coincide even without a dualizing complex. We also give an example showing that the finite-generation hypothesis is necessary, thereby providing a full answer to a question of Sather-Wagstaff. We conclude by studying the localization behavior of these dimensions, obtaining a partial answer to a question of Sather-Wagstaff and Totushek.

In this note we investigate the construction of near-group fusion categories via building explicit representations of them in certain graph planar algebras. To achieve this we first give a presentation for a cyclic near-group fusion category. Our presentation is “subfactor-centric” in the sense that we use a Q-system as one of our generating morphisms. Using this presentation we then obtain a rational system of equations, a solution of which corresponds to a faithful embedding of a near-group fusion category into a certain 2-coloured graph planar algebra. We end the note by providing solutions to these systems of equations for the odd cyclic groups up to order 13. This gives an alternate construction to the Evans-Gannon construction of the corresponding near-group fusion categories.

Row-sparse projection provides a useful tool in machine learning (ML) when it comes to, for example, feature selection, aiming to choose most relevant features for various ML objectives. One way to seek a high quality row-sparse projection is to combine an ML objective, such as the ones for PCA, LDA, and OCCA, with the matrix $(2,1)$-norm regularization which is nonsmooth. Such combinations result in challenging optimization problems on the Stiefel manifold that need to be solved efficiently. In this paper, a unifying NEPv framework is established to efficiently deal with optimization on the Stiefel manifold with the $(2,1)$-norm regularization. The effect of the $(2,1)$-norm regularization is also investigated. The wide applicability of the framework is demonstrated through the combinations of common learning objectives in today's data science applications with the $(2,1)$-norm regularization. Numerical experiments are presented to illustrate the use of the NEPv approach and to gain insights as to what a proper regularizing parameter should have in real-world applications.

For every positive sequence that tends to zero faster than every fixed exponential, we construct a Hausdorff topological Vector Group topology on the additive group of real numbers. It lies strictly between the minimal Hausdorff topological Vector Group topology and the usual topology. The construction is illustrated by factorial powers, quadratic exponential decay, and prime radicals divided by a quadratic exponential. We also record a finite scalar covering criterion for comparing two such topologies.

We provide a purely geometric necessary and sufficient condition, in terms of a certain Cesàro mean of the volume function, for the existence of the Maz'ya-Shaposhnikova functional limit with respect to Gagliardo seminorms on metric measure spaces (under a mild measure condition). As an application, we establish the Maz'ya-Shaposhnikova convergence on the unbounded Sierpiński gasket. We also show that Ahlfors regularity does not imply the existence of this functional limit by constructing a concrete example.

We prove that every closed cooriented hypersurface immersion in $\R^{n+1}$, $n\ge 2$, with positive definite first Newton transformation bounds a compact immersed manifold with the prescribed boundary map and outward coorientation. Combined with the rigidity results of Ros and Pinkall, this filling theorem implies that every closed connected hypersurface immersed in $\R^{n+1}$ with constant scalar curvature is a round sphere.

We present a new algorithm to compute a stream for the length spectrum of an oriented, finite-volume hyperbolic 3-manifold $M$. The advantage over the existing algorithm by Hodgson-Weeks is that our algorithm does not require finding a Dirichlet domain and thus can use interval arithmetic to verify the result. Our algorithm is also significantly faster in some cases. We also present a new algorithm to compute the optimal Margulis number of $M$. The authors have made these algorithms available in SnapPy.

We prove that every spanning subgraph $H$ of an $n$-vertex $(p,\beta)$-bijumbled graph, satisfying $\delta(H)\geq(1/2+\varepsilon)pn$, where $\varepsilon > 0$ is arbitrarily small yet fixed, admits a rainbow Hamilton cycle under any globally $\mu pn$-bounded colouring, provided that $\beta\leq cpn$ and $pn\geq M$, where $\mu$, $c$, and $M$ are positive constants depending solely on $\varepsilon$; in particular, $pn$ may be taken to be a sufficiently large constant. This answers a question of Coulson, Keevash, Perarnau and Yepremyan for random graphs and extends it to deterministic pseudorandom hosts. In fact, we prove that $H$ admits at least $(apn)^n$ rainbow Hamilton cycles, where $a > 0$ is a constant depending solely on $\varepsilon$. If $pn\geq D\log n$, then, upon fixing a coloured $H$, retaining each edge independently with probability $D\log n/(pn)$ preserves rainbow Hamiltonicity asymptotically almost surely. The logarithmic degree requirement is needed only for this percolation conclusion. All three conclusions also hold with rainbowness replaced by avoidance of prescribed pairs of edges, provided that each edge has at most $\mu pn$ conflicting partners. Our proofs use spread measures and the lopsided local lemma.

We present a distributed framework for large-scale Bayesian inverse problems governed by the eikonal equation, with a specific focus on seismic traveltime tomography. Traditional deterministic approaches often fail to provide the uncertainty quantification (UQ) necessary for ill-posed problems, while conventional Bayesian sampling methods such as Markov chain Monte Carlo (MCMC) suffer from the curse of dimensionality and slow convergence in high-dimensional model spaces. The proposed framework addresses these challenges through a three-tier computational strategy. First, we utilize the Fast Marching Method (FMM) to solve the eikonal equation, ensuring high numerical accuracy. Second, we reformulate the global tomographic objective into a decentralized consensus form, allowing the inversion to be decomposed into independent subproblems solved in parallel via the Alternating Direction Method of Multipliers (ADMM). This architecture eliminates the need for the explicit construction of large-scale sensitivity matrices, significantly reducing the memory footprint for 3D surveys. Finally, we integrate Stein Variational Gradient Descent (SVGD) within the ADMM workers to perform approximate posterior sampling. By evolving a set of model particles along a functional gradient direction that balances data-fitting forces with a repulsive kernel-based diversity force, we obtain an ensemble from which posterior summaries are computed. We derive a data-space Gauss-Newton update using the Woodbury matrix identity to further accelerate the particle evolution in large-scale 3D problems. Numerical experiments on complex 2D and 3D models demonstrate that the algorithm achieves stable convergence, produces high-fidelity velocity reconstructions, and provides posterior uncertainty maps.

Commonly, classifiers and monitoring procedures are trained from labeled data by optimizing an objective such as the misclassification rate. This may lead to unrepresentative conditional distributions of the outcome (the labels) given important external variables, different from the conditional laws in the population. We show how to modify any given threshold-type classifier resp. monitoring rule to achieve representative conditional label prediction by using adapting the threshold to a covariate $Z$ (the context) to distribute sensitivity while maintaining the false alarm rate. In case that the alarm event is unknown, this approach also allows to (approximately) infer the event in terms of a thresholding rule. The approach is implemented by a computationally cheap nonparametric estimation procedure, and its properties are studied in terms of nonasymptotic error bounds and asymptotic distribution theory including empirical process theory. These results allow to construct uniform confidence bands, functional hypothesis tests and change-detection procedures. For the well known FICOS credit scoring example, often used in interpretable machine learning, threshold adaptation leads to an easily interpretable decision rule which can compete with state of the art methods including transformers, in terms of common classification metrics.

A length-$n$ binary $k$-deletion code is a set of binary strings such that if we delete any $k$ bits of a string, leaving a length-$(n-k)$ binary string, we can uniquely recover the codeword. In this paper, we consider $t$-list decodable deletion codes, where after $k$ bits of a codeword are deleted, we can identify a list of size at most $t$ such that the original codeword lies in the list. We prove a lower bound of $\Omega_k(2^n t\log^{1/t}n/n^{k+k/t})$ on the optimal size of a $t$-list decodable $k$-deletion code, giving a $\sqrt{\log n}$ improvement over the previously best known bounds for $2$-list decodable $2$-deletion codes [GH21] and providing the first nontrivial lower bound when $t>2$ or $k>2$. Our bound holds for all $t\leq n^k$, showing that $t=\Omega(\log n)-$list decodable deletion codes have optimal size $\Theta_k(2^n t/n^k)$, asymptotically matching the known upper bound. We also prove upper bounds on the number of common subsequences and common supersequences of a given length for any two binary strings.

The current density in a rectangle in the $D=2$ dimensional Euclidean space, where opposite point charges are placed in a source $s$ and a destination $d$, is computed as the gradient of a potential that obeys the Poisson equation (first law of Maxwell). Although the computation of the current density in a 2D rectangle is a classical problem, several barriers (e.g. very slow convergence of double Fourier series) were eventually alleviated by a $q$-analysis, related to Gaussian polynomials and Jacobi's theta-functions, that form the basic building blocks for any elliptic function. We present an exact and computationally very efficient analytic formula of the potential and the magnitude of the current density in a rectangle.

We study statistical rates in entropic optimal transport in the semi-discrete regime where one measure has finite support and the other is subGaussian. Our main result establishes parametric convergence rates for the empirical dual potentials to their population counterparts, with no dimension dependence in the leading term. Our result relies on tailored strong concavity analysis of the semi-dual objective, coupled with specialized bounds for the semi-discrete potentials. As a consequence, we obtain fast rates for downstream quantities derived from the optimal coupling. Chiefly, the empirical barycentric projection achieves a squared-error rate $n^{-1}$, matching the fully compact case and improving over the less favorable $n^{-1/2}$ rate known for fully subGaussian settings. Altogether, these results may indicate a lower complexity adaptation phenomenon whereby the statistical complexity of the barycentric projection is governed by the discrete measure. As an application, we analyze Sinkhorn-EM, an EM-type algorithm in which the E-step is replaced by an entropic optimal transport problem. In a well-specified and balanced two-component Gaussian mixture model, we prove $\sqrt{n}$-consistency of the empirical iterates to their population counterparts for any fixed number of iterations, matching classical EM rates up to a $\sqrt{\log n}$ factor. Simulations support the theory.