Papers

We prove that there exists an absolute constant $c > 0$ such that the following holds for every $k \ge 3$ and $p \ge n^{-c / \log \log k}$. Letting $\delta > 0$ and $D \sim [n]_p$, with high probability every ${S \subset [n]}$ with $|S| \ge \delta n$ contains a $k$-term arithmetic progression whose common difference lies in $D$. This improves the best known bounds for the threshold of Szemerédi's theorem with random differences for all sufficiently large $k$, previously of the form ${p \ge n^{-2/k+o(1)}}$, due to Briët and Castro-Silva. We obtain our improvement via a small cover lemma, in the sense of Alon and Pham's work on random Cayley graphs, and its proof crucially relies on a construction of Green and Ruzsa in the context of the arithmetic Kakeya conjecture. The usage of the latter was suggested to us by GPT 6 Astra.

Persistent homology summarizes the topological evolution of a graph as persistence diagrams (PDs). Generating graphs whose topology matches a target descriptor is increasingly important, yet existing topology-aware generative models rely on soft regularization, producing graphs whose PDs may only be approximately similar. We initiate a principled investigation of the generation problem: produce graphs that realize a target PD exactly. We propose two approaches guided by the implicit local and global constraints a target PD imposes. The local approach builds degree-based PD-equivalent graphs iteratively, and we prove that every output realizes the target PD and that every degree-based PD-equivalent graph is reachable. However, as shown thereafter, local generation necessarily trades off between sample diversity and reaching infeasible states. We therefore recast generation as a global constraint satisfaction problem with a complete encoding solvable by modern constraint programming solvers. This is applicable to any vertex-based permutation equivariant filtration scheme and is minimal under the degree filtration. Experiments on four standard graph generation benchmarks confirm that both methods realize the target PD exactly. Together, our methods provide a principled treatment of strict topology-compliant graph generation. Code is available at https://github.com/Aalto-QuML/ExactCompliance.

We consider the Asymmetric Simple Exclusion Process (ASEP) at its critical dimension $d=2$, started from the Bernoulli product measure with density $\rho \in (0,1)$. We first derive the sharp $\log(t)^{2/3}$ divergence of the bulk diffusivity, predicted via mode-coupling theory in [H. van Beijeren, R. Kutner, H. Spohn, Phys. Rev. Lett., 1985], including the exact prefactor, thereby removing the subleading diverging multiplicative corrections in the Tauberian bounds of [H.-T. Yau, Ann. of Math., 2004]. More importantly, we establish an invariance principle for the density fluctuation field on the corresponding superdiffusive scales. On the time scale $\tau$, we rescale space by $\sqrt{\tau} \log(\tau)^{1/3}$ in the direction of the drift and by $\sqrt{\tau}$ in the transverse direction, and prove functional convergence of the rescaled fluctuation process to the solution of a conservative stochastic heat equation with additive noise. This is the same explicit Gaussian Fixed Point obtained in [G. Cannizzaro, Q. Moulard, F. Toninelli, arXiv:2501.00344, 2025] for the two-dimensional Stochastic Burgers Equation (SBE). We also deduce a superdiffusive central limit theorem for the second-class particle under the same anisotropic rescaling. The result holds for any finite-range jump kernel with nonzero drift and uniaxial asymmetry. This is the first space-time scaling limit result for a nonequilibrium interacting particle system at the critical dimension, where Gaussian fluctuations emerge despite logarithmic corrections to the scaling. The proof uses an approximate fluctuation-dissipation relation, which is obtained by combining a sharp quantitative comparison between the ASEP and SBE generators with refined estimates on the SBE Ansatz. An additional key novelty is a tightness argument that overcomes the limitations of the Itô trick on superdiffusive scales.

We prove that closed arithmetic hyperbolic $n$-manifolds of bounded rank have bounded adjoint trace field degree, and consequently have lower bounded injectivity radius. From this, we obtain a number of corollaries. In particular, we obtain that closed arithmetic hyperbolic $3$-manifolds of bounded rank are contained in finitely many commensurability classes, and describe these commensurability classes. Additionally, it follows that for closed arithmetic hyperbolic $3$-manifolds, Heegaard genus is equal to rank for all but finitely many manifolds of each rank.

We develop fully discrete multi-entropy-stable high-order compact Runge–Kutta discontinuous Galerkin (cRKDG) methods for hyperbolic conservation laws. Our main result is a weak cell entropy inequality for multi-dimensional systems after one full high-order Runge–Kutta (RK) time step. The proof exploits the cell-local intermediate stages of cRKDG: their perturbations are controlled by the initial in-cell variation. The analysis does not require a strong stability preserving decomposition or stagewise entropy correction. For a sufficiently dissipative Lax–Friedrichs flux and explicit RK methods with nonnegative output weights, the inequality holds under a suitable time-step restriction for strongly convex entropy pairs. For a prescribed, uniformly conditioned family of such entropies, common dissipation and time-step conditions ensure weak cell entropy stability for all entropy pairs. A single cell-average-preserving scaling at the end of each step then enforces all corresponding strong cell entropy inequalities simultaneously. We also prove a Lax–Wendroff-type theorem: under the stated assumptions, any strongly convergent sequence of numerical solutions converges to a weak solution satisfying all prescribed entropy inequalities. Numerical experiments in one and two dimensions demonstrate high-order accuracy, the benefit of multiple entropy constraints for nonconvex scalar problems, and robust performance on shock problems.

An exterior free-boundary minimal surface is a properly immersed minimal surface in the complement of the open unit ball of $\mathbb{R}^3$ whose boundary lies on the unit sphere and meets it orthogonally. We prove that for all integers $\ell\geq2$, $b\geq1$ and $\gamma\geq0$ there is a connected, orientable, properly embedded exterior free-boundary minimal surface with $\ell$ ends, $b$ boundary components and genus $\gamma$. Its total curvature is finite, and its ends are parallel catenoidal ends of multiplicity one. By the theorem of Mazet and Mendes, a surface of this kind with one end is a rotational annulus. Hence the topological types of such embedded surfaces are exactly the annulus with one end and all types with at least two ends. There are two gluing constructions. For $\ell\geq3$ we stack $\ell$ sheets over a stable catenoidal annulus that meets the sphere along a circle of radius $4/5$, and we join adjacent sheets by half-catenoidal bridges sitting on the sphere. The sheets are small normal graphs over this annulus whose displacements contain different multiples of the Jacobi field of the family of such annuli, so their boundary circles and the logarithmic growth of their ends differ slightly. For $\ell=2$ we double a catenoidal annulus near the critical angle $\pi/4$, where it acquires a bounded rotationally invariant Jacobi field; the distance to the critical annulus is coupled to the size of the bridges. In gluing constructions inside the unit ball that double or stack a compact surface, the bridges themselves must balance the sheets, which forces many bridges. Here the logarithmic growth of the ends absorbs this balancing. A fixed number of bridges therefore suffices, and every topological type is reached, including the small ones.

Loss and noise limit the use of light to distribute entanglement, transmit quantum information, and establish secret keys. Gaussian states have a compact description in terms of means and covariance matrices, but their entanglement still requires global optimization. We develop a framework for computing Gaussian-restricted versions of the relative entropy of entanglement and Rains entanglement. An integral representation reveals a convex geometry under matrix geometric means and reduces Gaussian Rains entanglement to relative entropy minimized over Gaussian states with positive partial transpose. We obtain unique optimal reference states and additivity for independent Gaussian input states. Our convex-majorization algorithm has an explicit geometric convergence rate, uniform in the faithful regularization parameter and valid for pure input states. Controlled numerical integration and approximate convex subproblem solutions give a bit-complexity bound that is polynomial in accuracy bits for each fixed rational input covariance matrix, with explicit dependence on covariance size. These calculations bound extractable entanglement and, through the relative entropy of entanglement, extractable secret key. Teleportation simulation, energy allocation, and a convex optimization of passive-basis loss cores give bounds on quantum and private communication under an input-energy constraint, allowing arbitrary two-way classically assisted protocols. Two- and three-mode examples compare the bounds with achievable rates, including matching pure-loss benchmarks. Finally, we separate the Gaussian-restricted quantities from their unrestricted counterparts and give a photon-number truncation method with controlled error for computing the latter.

It is proved that every Borel ensemble of pure states on a separable bipartite Hilbert space with finite average entanglement admits a countable ensemble with the same barycenter and arbitrarily small additional average entanglement. Consequently, the discrete and continuous versions of the Entanglement of Formation coincide on the whole space of states of infinite-dimensional bipartite quantum system (with infinite values allowed). The main idea of the proof (realized with the AI-assistance) is the lifting a Borel ensemble of pure bipartite states to a Borel probability measure on the unit sphere in the space of Hilbert–Schmidt operators followed by dividing the support of this measure into appropriate cells and replacing each cell by its conditional barycenter. The resulting countable ensemble is obtained by a recursive construction.

We prove a case of a conjecture of Gundlach on the number of Galois extensions over $\QQ$ with bounded multi-invariants for the group $G=Q_8$. We also obtain results when imposing local conditions at the wildly ramified place. As an application, we prove asymptotics for the number of $Q_8$-extensions with bounded product of ramified primes.

We give a general formula for minimal projective presentations and for minimal discrete-injective presentations of finite-dimensional modules over basic pseudocompact algebras. We also show that the representation-theoretic g-vector of a module coincides with its DWZ-g-vector. These results generalize the description that Derksen, Weyman and Zelevinsky gave of minimal projective presentations for modules over certain finite-dimensional Jacobian algebras. In particular, we can drop their finiteness assumption, and we may work over fields of arbitrary characteristic.

Building on work of Kauffman–Saleur and earlier work of the second author, we derive a determinant formula for the Links–Gould polynomial of links. As an application, we prove a 20-year-old conjecture of Ishii: for any alternating link, the Links–Gould invariant is an alternating polynomial.

We resolve the long-standing non-extinction problem for Brownian Fleming-Viot systems in bounded domains. For every fixed N >= 3, we construct a bounded connected domain in R3 in which branching explosion occurs with positive probability before every prescribed positive time, from every interior initial configuration. On this event all particles converge to one boundary point. Together with two-particle non-extinction, this gives the sharp population threshold for bounded-domain Brownian explosion. The mechanism is directed transport generated by uniform copying in a periodic absorbing environment. We compute a positive cubic current for the same two-harmonic killing profile at every N >= 3, with an explicit population coefficient and a controlled fourth-order remainder. To realize this transport by spherical absorption, we develop a marked genealogical transfer theorem that retains the Euclidean displacement inherited at each copy. Common ancestry makes the integer winding reconstructible from phase records and yields convergence of the stationary signed current. A boundary restart law with P'olya weights supplies the continuity needed to turn that current into stable escape. Diffusion coefficients adapted to a common Kelvin clock preserve the joint Brownian law and uniform donor marks while compressing escape into finite lifetime. Disjoint spherical holes in a ball accumulate only at the explosion point. The boundary is smooth elsewhere, with finite area and perimeter. Boundary area within distance r of the singularity is of order r. For three particles, the transfer theorem covers every smooth positive periodic killing rate.

We introduce a diagram algebra $\mathcal{K}_L$ based on Kuperberg's $A_2$-webs and prove a Schur–Weyl duality $\mathcal{K}_L \cong \operatorname{End}_{U_q(\mathfrak{sl}_3)}((V^+ \oplus V^-)^{\otimes L})$, where $V^\pm$ are the fundamental representations of $U_q(\mathfrak{sl}_3)$. Thus, $\mathcal{K}_L$ is a natural $A_2$ analogue of the Temperley–Lieb algebra. We construct an explicit cellular structure for $\mathcal{K}_L$, with the web basis as its cellular basis and cell modules indexed by dominant $U_q(\mathfrak{sl}_3)$-weights. The bases of these modules are parametrized by dominant lattice paths via a recursive growth algorithm. Using this structure, we determine local generators of $\mathcal{K}_L$ and establish web formulations of the first and second fundamental theorems of invariant theory. In particular, we show that the braid-group image, augmented by local idempotents, does not generate $\mathcal{K}_L$, and identify an additional local generator. Finally, we derive a bimodule decomposition of $(V^+ \oplus V^-)^{\otimes L}$, yielding a complete set of irreducible $\mathcal{K}_L$-modules for formal or generic $q$.

We prove the global existence of solutions to the Navier-Stokes equations for any divergence-free distributional initial data. The solutions are smooth in both time and space at positive times. When the initial data belong to the Sobolev spaces $W^{-1,p}$ for $1<p<\infty$, our constructed solutions are weak in the classical sense. We also characterize the distributional initial data admitting weak solutions and obtain the sharp ranges in Besov and Triebel-Lizorkin spaces.

Let $\Gamma<SO(4)$ be a nontrivial finite group acting freely on $S^3$, and let $M=S^3/\Gamma$. We prove that every smooth conformal class $C$ on $M$ satisfies \[ Y(M,C)\le |\Gamma|^{-2/3}Y(S^3), \] with equality for the quotient-round conformal class. Consequently, \[ \sigma(M)=|\Gamma|^{-2/3}Y(S^3). \] Our approach uses four-dimensional Poincaré–Einstein fillings of the ball. We first establish compactness for fillings whose conformal infinities range over a compact family of positive-scalar-curvature metrics. Since compactness alone does not guarantee continuation through degenerate Einstein metrics, we use equivariant Fredholm parity to obtain a filling with the prescribed boundary conformal class and group action. A common interior fixed point then makes the Li–Qing–Shi boundary test function invariant, and its descent to $M$ gives the sharp covering factor.

We study the invariant rank-four deformed Hermitian–Yang–Mills equation associated with the chain $\mathcal{O}(1)\to\mathcal{O}(2)^{\oplus2}\to\mathcal{O}(3)$ over $\mathbb P^1$, with vertical weights $(4,2,0)$ and a fixed product Kähler class and phase. For arbitrary linear arrows $\psi=Bz$ and $\phi=(Az)^T$, smooth strictly calibrated reduced solutions exist exactly when $A$ is invertible and $z^TA^TBz$ is not identically zero. The same condition characterizes stability, semistability and polystability in the specified equivariant coherent category. The classification includes all eleven nonempty coefficient strata and both coincident and distinct divisor points on the rank-one lower boundary. The analytic proof retains the ordered, noncommuting middle equation. After constructing positive-parameter solutions, it rules out unbounded metric values by combining the source budgets with moving spectral tests; on the rank-one lower boundary, the divisor mass must also be retained. Global value, gradient and symbol estimates then give a smooth strictly calibrated endpoint on the original chain. No equivalence with full coherent or derived-category stability, uniqueness, or estimate uniform across changing chains is asserted.

We develop and analyze an entropic descent method for a nonconvex optimization problem on probability simplices arising in classical and quantum secure communication. The optimization variables describe a randomized encoder, allowing its input distribution and stochastic preprocessing to be optimized jointly. An exact relative-entropy identity yields a channel-independent majorization bound and an explicit mirror-descent iteration with a fixed unit step size. We establish quantitative descent, convergence of objective values, and stationarity of accumulation points on their support faces for arbitrary finite classical channels, without assuming degradedness or less noisiness. For strictly positive transition matrices, a square-root parametrization and a Kurdyka–Łojasiewicz argument establish finite length of the transformed trajectory and convergence of the entire sequence, including limits on the simplex boundary. For less-noisy channels, a restricted convex formulation admits global convergence with an explicit \(O(1/t)\) objective-error bound. We extend the relative-entropy construction to noncommuting quantum outputs and derive monotone alternating updates for covariant encoders, including correlated preprocessing over a fixed block. Numerical experiments compare the method with analytic benchmarks and constrained optimization, reveal multiple attracting values, and demonstrate strict improvements from stochastic preprocessing.

In this short note we show the existence of stably exotic knotted surfaces in 4-manifolds in the sense that the embedded surfaces are topologically equivalent but are not smoothly equivalent even after any number of internal and external stabilisations.

We consider solutions to the two-dimensional Euler equations with a logarithmically regularized velocity given by $u=\nabla^{\perp}(-\Delta)^{-1}\ln^{-\gamma}(|\nabla|+e)\omega$ and initial vorticity in $\dot{H}^1 \cap \dot{H}^{-1}(\mathbb{R}^2)$. We show that if $\gamma>1/2$, then global well-posedness holds.

We study plank coverings of cube annuli $Q\setminus\delta Q\subset\mathbb{R}^d$, where $Q$ is the $d$-dimensional unit cube and $\delta Q$ is a homothetic copy lying in its interior. A. Bezdek proved that, in dimension $2$, every plank covering of such an annulus has total width at least~1 whenever $\delta\le1-1/\sqrt2$. Smurov, Bogataya, and Bogatyi extended this result to all $d\ge2$ under the condition $\delta\le1-1/\sqrt d$. It is conjectured that the same lower bound holds for $\delta\le1-1/d$. We provide evidence for this conjecture in dimensions $2$ and $3$ by proving that there exist positive constants $\varepsilon_2$ and $\varepsilon_3$ such that the bound remains valid whenever $\delta\le1-1/\sqrt d+\varepsilon_d$.

We associate each h-convex body in hyperbolic space $\mathbb{H}^{n+1}(n\geq 2)$ with a Radon measure valued $(0,2)$-tensor on the sphere $\mathbb{S}^n$ through its horospherical function. Using spherical convolution, we prove that any h-convex body can be approximated by a sequence of smooth strictly h-convex bodies in the Hausdorff sense. This approximation result extends the inequality parts of the Alexandrov-Fenchel inequality and the modified Alexandrov-Fenchel inequality to general h-convex bodies, but equality rigidity cannot be obtained directly. We prove that the equality holds in the Alexandrov-Fenchel inequality and the modified Alexandrov-Fenchel inequality if and only if the h-convex bodies are geodesic balls. For the Alexandrov-Fenchel inequality, we establish the Hausdorff stability independent of curvature bounds described by the Alexandrov-Fenchel deficits for h-convex bodies, which implies the equality rigidity of the Alexandrov-Fenchel inequality. For the modified Alexandrov-Fenchel inequality, we introduce a class of globally constrained $k$-th modified quermassintegrals preserving flows with general h-convex initial hypersurfaces to obtain the equality rigidity. Moreover, we prove the flows above preserve h-convexity, make the $(k+1)$-th modified quermassintegrals non-increasing and converge to geodesic spheres. We prove a rigidity theorem for modified area measures for h-convex bodies and use this rigidity theorem to characterize the geodesic balls via modified support measures for h-convex bodies, from which we prove that the $(k+1)$-th modified quermassintegrals strictly decrease in a short time along the flows with non-spherical initial hypersurfaces. Then we obtain the equality rigidity of the modified Alexandrov-Fenchel inequality.

Sensitivity analysis of coupled systems typically propagates derivatives of single-valued maps, whereas multiobjective optimization generally treats sensitivity at the system level rather than through interconnected subsystems. We develop a mathematical framework for compositional sensitivity analysis in multiobjective system design, building on monotone co-design. Subsystems are represented as design problems (DPs), interconnected through series, parallel, union, intersection, and feedback composition. We introduce tangential DPs (TDPs) to describe first-order variations around an operating point, show that they preserve the structure of DPs, and characterize their queries through graphical derivatives. To account for multiple internal realizations of a system-level operating point, tangential co-design problems compose families of local TDPs by taking a union over these realizations while retaining the original interconnection structure. We show that these compositions yield families of TDPs and establish conditions under which sensitivity analysis commutes with individual composition operations. Beyond infinitesimal sensitivity, we show that bounded tangential models of convex DPs provide tighter outer approximations of feasible finite changes, which can be propagated through co-design problems without union compositions. We conclude with several convex and nonconvex numerical examples that showcase the framework's accuracy and computational benefits.

We study optimal transport with a non-convex polyhedral target in $\mathbb{R}^n$. The closure of the singular set is locally a smooth hypersurface outside a relatively closed set of Hausdorff dimension at most $n-2$. For partial transport between domains separated by a hyperplane, the source free boundary has the same regularity.

We propose a of definition conjugate stack for a class of suitable $\mathbb{E}_3$-$\mathrm{MU}$-algebras that includes the truncated Brown-Peterson spectra $\mathrm{BP}\langle n \rangle$ for all heights $n$. Then we calculate part of $\mathrm{BP}\langle n \rangle^{\mathrm{conj}}$ as a quotient stack in terms of Witt vectors. To perform this computation we study generalized Thom spectra and their relative Topological Hochschild Homology. We exhibit the Eilenberg-Mac Lane spectrum $\mathrm{H}\mathbb{F}_2$ as a $\mathbb{E}_3$-$\mathrm{MU}$-Thom spectrum, the $p$-local Eilenberg-Mac Lane spectra $\mathrm{H}\mathbb{Z}/p^n$ (except for $\mathrm{H}\mathbb{Z}/4$) and $\mathrm{H}\mathbb{Z}_{(p)}$ as $\mathbb{E}_3$-$\mathrm{MU}_{(p)}$-Thom spectra, and the connective complex K-theory spectrum $ku$ as $\mathbb{E}_1$-$\mathrm{MU}$-Thom spectra. We accompany these “go" results with “no-go" results that show these “go" results do not admit $\mathbb{E}_{\infty}$-refinements.

We continue the study of beautiful pairs of henselian valued fields, initiated in joint work of the second author with Cubides Kovacsics and Ye. As our main result, we show that separably closed non-trivially valued fields of arbitrary imperfection degree admit beautiful pairs which are elementary pairs satisfying certain natural axioms. They satisfy beauty transfer, i.e., saturated models of their theory are again beautiful. The same holds for beautiful pairs associated to natural subclasses of all definable types, most notably for all definable types orthogonal to the value group and for all bounded definable types. This yields strict pro-definability of the space of the definable types from any of these classes, in the theory SCVF of separably closed non-trivially valued fields. We also show a separable analog of a classical result of Keisler's, namely that the first order theory of an infinite separable extension $L/K$ of fields - considered in the pairs language and where the large field $L$ is separably closed, is determined by the theory of the small field $K$ together with the imperfection degree of $L$ over $K$.

In [1], we surveyed classical and recent results on the construction of exponential bases on planar domains with explicit control over the associated frame bounds. We also constructed well-conditioned exponential bases on certain planar sets that multi-tile the plane. In this revision, we identify an error in the choice of basis used in the proof of Theorem 3.11 and we provide a new proof using a simpler exponential basis.

We study the long-time, large-distance asymptotics of finite-temperature dynamical Fredholm determinants using effective form factors. The method expresses these asymptotics as a functional of an effective phase shift. We determine its two branches on either side of a quadratic stationary point and the transition profile joining them. The profile is expressed through parabolic-cylinder functions. Its connection to the high-temperature Gaussian $\beta$ ensemble fixes the finite normalization, in agreement with an independent Riemann–Hilbert analysis. We also show that the leading exponential asymptotics suffice to recover the Euler-scale prediction of generalized hydrodynamics for the dynamical structure factor of hard rods.

We consider vector mean-field spin glass models with independent and identically distributed spins without assuming the covariance function is convex. We identify the limit free energy as the minimum of a functional that can be written as the sum of a relative entropy and the solution of a Hamilton–Jacobi equation. The proof of this result relies on the observation that when the spin configurations of the system are constrained to have a given empirical measure, the corresponding free energy approximately satisfies a Hamilton–Jacobi equation with a uniformly bounded error.

Let $V_i(n)$ denote the minimum worst-case number of comparisons needed to find the $i$-th smallest of $n$ distinct elements. We construct algorithms showing $V_7(16)\le28$ and $V_8(16)\le29$, improving the previously reported upper bounds of 30 and 31. Together with known lower bounds, these results establish $V_7(16)=28$ and leave $V_8(16)\in\{28,29\}$. For 17 elements, we establish $V_5(17)=27$ and improve the upper bounds for ranks 6 through 9 to 29, 30, 31, and 31, respectively. Every new upper bound has an explicit decision-graph certificate verified by a static checker developed independently of the search. We also derive a lower bound for selection from a partially ordered state by combining the Fussenegger-Gabow weight with its order-reversed counterpart. All new upper bounds and this lower bound are formally verified in Lean 4. For $V_8(16)$, we show that a lower bound based solely on the total number of decision-tree leaves cannot establish $V_8(16)\ge29$. The algorithms were found using fixed-budget search, information-guided comparison ordering, and proof-number search. Proof-number search was decisive in finding the 28-comparison algorithm; in our experiments, refuting a comparison budget remained substantially more expensive than finding an algorithm within it.

We develop a rate-transfer method that converts finite-time Lyapunov estimates for an associated stochastic differential equation (SDE) into finite-time convergence bounds for stochastic gradient algorithms. Under generic moment and smoothness conditions, the convergence analysis of the stochastic gradient method reduces to a calculus argument verifying a Lyapunov generator inequality for the SDE. The method recovers established rates for classical stochastic gradient methods and provides a way to establish convergence rates for new optimization algorithms through calculus arguments applied to their associated SDEs.