Papers

Conditionally on the finiteness of the relevant Tate-Shafarevich groups, we prove a local-to-global result for zero-cycles of degree $1$ on the surfaces given by $y^2 = f_1(x_1)f_2(x_2)$, where the polynomials $f_1$ and $f_2$ are algebraically general. The proof combines the fibration method, parity results for $2$-Selmer groups in quadratic twist families, and a variant of a theorem of Morgan on the variation of the Cassels-Tate pairing.

In this paper, we study a bootstrap percolation process on the finite triangular grid $\mathfrak{T}_n$ of side length $n$. We say that a subset $\eta$ of points in $\mathfrak{T}_n$ percolates if the final configuration, starting from $\eta$, is the whole grid $\mathfrak{T}_n$. A basis of size $n$ is then a subset of points of $\mathfrak{T}_n$ of minimum cardinality which percolates. In this paper, we first prove that the generating function counting bases satisfies an algebraic differential equation. Then, by analysing a modified version of this equation, we prove that the number $t_n$ of bases of size $n$ exhibits a stretched exponential asymptotic behaviour. More precisely, we show that $t_n \sim c n!e^{\sqrt{12n}}n^{5/12}$, for some constant $c>0$. These bases were recently shown by the second author to be in bijection with $3$-permutations avoiding the patterns $(12, 12)$ and $(231, 312)$, so this represents to our knowledge the first proven example of an asymptotic stretched exponential appearing in the study of pattern avoiding permutations.

A tensor of degenerate geometric rank corresponds to a linear space of matrices in which some locus of matrices of rank at most $r$ has unexpectedly large dimension. We prove a sufficient criterion for when such a tensor cannot be lifted to a larger tensor and characterize those tensors for which the degeneracy is caused by the rank $r=1$ locus. We also provide new examples of tensors with degenerate geometric rank stemming from nonlinear loci.

Inspired by recent works on Diophantine tuples over finite fields, in this paper we consider Diophantine tuples related to primitive elements of finite fields. Let $\mathbb{F}_q$ be the finite field with $q$ elements and $\mathbb{F}_q^*=\mathbb{F}_q\setminus\{0\}$ be the multiplicative cyclic group of all non-zero elements over $\mathbb{F}_q$. An element $g\in\mathbb{F}_q$ is called primitive if $g$ generates the group $\mathbb{F}_q^*$. A set $\{x_1,x_2,\cdots,x_m\}\subseteq\mathbb{F}_q^*$ of $m$ elements is said to be a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ if $x_ix_j+1$ is primitive for any $1\le i\le j\le m$. Let $N_m$ denote the number of $\mathcal{P}$-Diophantine tuples over $\mathbb{F}_q$. Then we obtain the asymptotic formula $$m!\cdot N_m=\left(\frac{\varphi(q-1)}{q-1}\right)^{m(m+1)/2}q^m+O_{m,r}\left(q^{m-\frac{1}{2}+r}\right),$$ where $\varphi(\cdot)$ is the Euler totient function and $r\in(0, 1/2)$ is an arbitrary real number. Moreover, we prove that there exists a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ whenever $q\ge \exp(\exp(m(m+1)))$.

This paper investigates max-min user fairness optimization for flexible intelligent metasurface (FIM) and non-orthogonal multiple access (NOMA)-assisted integrated sensing and communication (ISAC) systems with active self-localization. To tackle the multi-user interference, performance imbalance, and neglected sensing accuracy problems encountered in conventional ISAC designs, we derive the closed-form Cramer-Rao lower bound (CRLB) for angle-of-departure (AoD) estimation in target sensing and embed it into a max-min fairness optimization framework. The optimization problem, which jointly designs the base station transmit beamforming, FIM reflection coefficients, and surface deformation, is non-convex and solved by an alternating optimization (AO) algorithm. Notably, the devised optimization framework facilitates superior performance trade-off in terms of spectrum resource utilization between communication and sensing tasks. Simulation results validate that the proposed scheme significantly improves user fairness, balances communication performance and sensing precision effectively, and reveals the coupling characteristic between signal-to-interference-plus-noise ratio (SINR) and sensing CRLB. This work provides a feasible solution for FIM-aided ISAC system optimization in 6G networks.

Let $X$ be the union of $k$ generic linear spaces in $\PP^n$ of codimension at least $2$. We prove that the Castelnuovo-Mumford regularity of the vanishing ideal of $X$ grows in order not more than $\sqrt{k}$, and that this is an order-tight bound when the codimension equals $2$.

By Montel's theorem, the continuous functions on a compact subset of a complex manifold that admit uniformly bounded holomorphic extensions to the manifold form a compact set in the uniform topology. We render this compactness quantitative by determining the asymptotics of the associated metric entropy, thereby giving a new solution to a problem of Kolmogorov for domains in $\mathbb{C}^n$ and extending the solution to domains in arbitrary Stein manifolds.

In this paper, we study the prime graph, also known as Gruenberg-Kegel graph, realizable by finite solvable uniformly semi-rational groups. This is primarily achieved by classifying the prime graphs realizable by metanilpotent groups. We also introduce a new class of groups, called MP* groups, which extends the class of groups with Magnus property and forms a natural subclass of uniformly semi-rational groups. For MP* groups, we give a complete classification of the prime graphs realized by them. In the process, we obtain the realizability of prime graphs by uniformly semi-rational groups, assuming them to be in varied substantial subclasses of solvable groups, namely, 2-Frobenius, metacyclic, metabelian, nilpotent-by-abelian, abelian-by-cyclic and cyclic-by-abelian. We also classify the so called N-prime graphs for all these classes of groups. The prime graph question for uniformly semi-rational groups has also been addressed.

Systems of hard rods of equal length in a one-dimensional harmonic trap have been observed to exhibit peculiar non-ergodic behavior that might suggest the existence of a novel hidden conserved quantity beyond the two well known ones, i.e., the total energy and the center-of-mass energy. In this work, we investigate this possibility by systematically constraining the forms of the conserved quantities, and we rigorously rule out the existence of any extra hidden conserved quantity that is analytic in the positions and momenta of the rods involved. We do so by showing two key results: conservation during free motion demands the $U(1)$ invariance of these quantities under rotations of the position and momenta of each rod, and conservation during collisions demand an $S_N$ invariance under the permutation of the momenta of the rods as long as one of the rods have non-zero length. We then show that these conditions imply that any conserved quantity is functionally dependent on the two known conserved quantities. In addition, we show that in the special case where all rods have zero length (i.e., when they are point particles), conservation under collisions only requires invariance under a smaller $S_N$ group of permutations of the labels of the rods, which leads to a much larger set of analytic conserved quantities that we explicitly write down. In all, this rigorously clarifies the structure of conserved quantities in the hard rod problem, and motivates the application of such systematic methods to other classical systems.

We establish the absolute convergence of the spectral side of the twisted trace formula for reductive algebraic groups. More precisely, we extend the absolute convergence theorem of the spectral side due to Finis–Lapid–Müller to the twisted setting. This paper provides a detailed proof of the absolute convergence theorem, whose proof is known to experts but does not seem to have appeared in the literature. The resulting formulas are motivated by applications to limit multiplicity problems and to Weyl laws for self-dual automorphic representations of $\mathrm{GL}_n$.

We prove strong convergence rates for tamed Euler schemes of multidimensional stochastic differential equations with singular drift, multiplicative Brownian noise, and multiplicative Levy noise. The drift is assumed to satisfy a Ladyzhenskaya-Prodi-Serrin type condition, while the diffusion and jump coefficients have Sobolev-type spatial regularity. The proof is based on a nonlocal Zvonkin transform, Krylov-type estimates, and a stochastic Gronwall argument. The main novelty is an error decomposition that isolates two jump-induced contributions: a compensated jump martingale error and a nonlocal compensator error caused by Euler freezing. This decomposition yields explicit rates and recovers the Brownian-type estimate when the jump coefficient vanishes.

Sufficient dimension reduction seeks a low-dimensional covariate representation that preserves the conditional law of a response. We introduce SDR-COT, which represents that law by conditional optimal transport from an independent reference response. On separable Hilbert spaces, sufficiency forces the response component of the optimal triangular map to factor through the reduction. For quadratic cost, the induced interpolation has a Borel current-state velocity on every truncated time interval, without global injectivity of the terminal map, and this velocity has the same factorisation. These results motivate a conditional-flow-matching criterion. For linear reductions, we prove consistency using a suitably tuned relaxed empirical coupling. Euclidean responses are treated through slicewise Caffarelli bounds; Hilbert-valued responses are treated through Gaussian Sobolev regularity, interpolation compression and uniqueness of a Gaussian continuity equation. Numerical studies with Euclidean and functional data show competitive performance, especially when sufficient information is not solely contained in the conditional mean.

We present full-dimensional quantum simulations of $H_2^+$ dissociative ionization driven by strong orthogonal laser fields. We consider equal-frequency orthogonal components, which generate elliptical or circular polarization depending on their relative phase and amplitude, as well as orthogonal $800$- and $400$-nm two-color fields. These two-dimensional fields strongly modify the fragmentation dynamics. Most notably, we identify a high-energy peak in the proton kinetic-energy-release (KER) spectrum at approximately $4-5$ eV that is absent from the corresponding single-color, linearly polarized calculations. The yield of this peak can be coherently controlled by varying the relative carrier-envelope phase of the perpendicular field component. The perpendicular field also disrupts the clear electron-proton energy-sharing pattern observed in the main $3-3.5$ eV dissociation channel, indicating more complex multichannel dynamics. Time-dependent state projections and calculations initiated from individual excited states attribute the additional peak to laser-induced vibrational excitation of $H_2^+$. Furthermore, the perpendicular field rotates the fragment angular distributions, causing the most probable proton and electron emission directions to deviate substantially from the principal $z$ axis. These findings demonstrate that the spatial and temporal geometry of orthogonal laser fields provides an additional degree of freedom for controlling ultrafast electron-nuclear dynamics.

The isomorphism problem for generalized Baumslag-Solitar (GBS) groups is a long-standing open problem. We prove that, in the random model considered here, a GBS graph with many edges is flexible with high probability. Since the isomorphism problem is decidable for flexible GBS graphs, this gives a high-probability decidability result for random GBS groups in this regime.

We give a generic construction of the n-generated free Hilbert algebras and Brouwerian semilattices by a uniform method applicable to any variety of Hilbert algebras or Brouwerian semilattices, as long as a manageable description of subdirectly irreducible algebras in that variety is available. A slight modification yields analogous constructions for varieties of Hilbert algebras and Brouwerian semilattices with zero. As examples we construct free algebras in varieties of bounded height and of bounded width; we also find a closed formula for the free spectrum of linear Hilbert algebras. Finally, we obtain a few results on structural completeness of varieties of Hilbert algebras with zero, in particular, we give a sufficient condition for a quasi-equation to be equivalent to an equation.

Determinantal point processes (DPPs) are widely used as probabilistic models for diverse random subsets, but their approximation error under model misspecification has not been fully characterized. We study the population-level approximation of a strictly positive target distribution p* by DPPs under the forward Kullback-Leibler divergence. Using information-geometric analysis and the standard quality-diversity decomposition of an L-ensemble kernel, in which the diagonal quality component Q encodes item-specific weights and the diversity component D controls repulsive interactions among items, we show that the quality component can be chosen uniquely to match all first-order inclusion probabilities of p*. The DPP approximation problem therefore reduces to the optimization of the diversity component. This reduction yields a global optimality result for attractively dependent targets: under conditions including weak positive association, the independent product distribution with the same first-order marginals, corresponding to D=I, is an optimal DPP approximation. For more general target distributions, positively correlated pairs yield lower bounds on the approximation error. On the repulsive side, a matching of disjoint negatively correlated pairs yields an upper bound on the approximation error, or equivalently a guaranteed improvement over the independent approximation. We further study local optimality around D=I and, more generally, around block-diagonal diversity matrices by analyzing perturbations between their blocks.

In this paper, we investigate automorphism groups preserving Cartan subalgebras of C*-algebras. First, we describe these groups in terms of automorphisms and 1-cocycles of twisted étale groupoids. As a consequence, we obtain a C*-algebraic analogue of a theorem of Feldman and Moore on Cartan-preserving automorphisms of von Neumann algebras. We then study Cartan-fixing automorphism groups. We show that every UCT Kirchberg algebra admits a Cartan subalgebra whose Cartan-fixing automorphism group contains every second countable compact abelian group. In contrast, for C*-algebras arising from expansive effective groupoids, we prove that compact Cartan-fixing automorphism groups must have finitely generated Pontryagin duals. As an application, we establish the existence of inequivalent Cartan subalgebras for Kirchberg algebras arising from expansive effective groupoids.

We investigate the transformation of Killing-Yano (KY) $p$-forms under Abelian T-duality in the presence of a non-trivial three-form torsion. Using a torsionful, metric-compatible connection and the Buscher rules of T-duality, we derive compact, coordinate-free transformation laws applicable to KY forms of arbitrary degree under explicit matching assumptions. In particular, we show that a Killing 1-form is preserved by the direct duality map whenever its transverse components are independent of the isometry direction, with the dual circle component determined by a scalar correction equation. The framework is then applied to several examples, including the Hopf T-dual of $S^3$, Schwarzschild spacetime, and the Nappi-Witten plane wave with exact NS-NS torsion, where we analyze the transformed Killing-Yano forms explicitly. Finally, we give a generalized-geometric and double-field-theoretic interpretation of emergent Killing 1-forms in the T-dual geometry, showing how they arise from generalized Killing data in the original duality frame.

The content of the following pages was part of a special topics lecture given at the Australian National University during the first semester of 2026. That was a very nice expericence, I really enjoyed giving this 12 weeks (2 hours a week) lecture. It is meant to be self contained, starting with results on Fourier transform and Sobolev spaces. As a toy model, before treating the Navier-Stokes system, we focus on the non linear heat equation where the non linearity is polynomial. Ultimately, we prove existence and uniqueness of mild solutions of the Navier-Stokes equations in critical spaces. This script contains some exercises and the text of the mid-semester exam as well as the final exam.

As an application of the generalized principal bundle theory to covariant Lagrangian field theories, we aim at the development of an instance of generalized gauge theories, with the prospect of a unifying language for Yang-Mills theories and General Relativity. After reviewing the basic definitions of Lie group fiber bundles and generalized principal bundles, we provide horizontal lift and local characterizations of Lie group fiber bundle connections and generalized principal connections. Subsequently, we consider in the framework of classical field theories the kinematics and dynamics sides of generalized principal connections, which lead to a proposed notion of generalized Yang-Mills theories. Finally, we show how vector bundles are examples of generalized principal bundles and that a generalized principal connection on a vector bundle is an affine connection given in terms of basic soldering forms. We are able to recover, under appropriate assumptions, the Vielbein formulation of (vacuum) General Relativity in this setting, hinting at a (generalized) gauge theory of gravity.

The global and local $\mathcal{G}$-covering number $c_{\mathrm{g}}^{\mathcal{G}}(H)$ and $c_{\mathrm{l}}^{\mathcal{G}}(H)$ encode how well the edges of a graph $H$ can be covered with graphs from a graph class $\mathcal{G}$: in the global setting, we minimize the number of graphs from $\mathcal{G}$ required, in the local setting how often a vertex is hit by the graphs of the cover. Within this work we consider for $\mathcal{G}$ the graph classes $\mathcal{B}$ of all bipartite and $\mathcal{B}_{\mathrm{c}}$ of all complete bipartite graphs. We give a tight lower bound on $c_{\mathrm{l}}^{\mathcal{B}}(H)$ in terms of the fractional chromatic number of $H$, thereby giving a local analogue of a result by Harary, Hsu and Miller. Answering a question by Fishburn and Hammer, we show that it is NP-hard to determine $c_{\mathrm{l}}^{\mathcal{B}_{\mathrm{c}}}(H)$. Further, we provide a finite and monotone graph class $\mathcal{G}$ such that $c_{\mathrm{g}}^{\mathcal{G}}(H)$ can be computed in constant time for every graph $H$ while determining $c_{\mathrm{l}}^{\mathcal{G}}(H)$ is NP-hard. This yields a natural example to a question raised by Knauer and Ueckerdt.

We study higher moments of the divisor function evaluated at polynomial expressions in the Fourier coefficients of a non-CM newform. The logarithmic exponent appearing in our estimates depends only on the number of irreducible factors of the polynomial and remains unchanged under a Sato–Tate restriction. The proof combines an effective Chebotarev theorem, or an effective Chebotarev–Sato–Tate theorem, with the arithmetic of joint cycle types and a mean value estimation for multivariable multiplicative functions with Frobenian support.

The mammalian kidney concentrates urine using a mechanism with no analogue in current neural architectures: the countercurrent multiplier. Two anti-parallel flows joined at a hairpin recirculate a weak magnitude-bounded local pump into a large axial gradient achieving a four-fold concentration increase from a single-effect gradient that never exceeds 200 mOsm at any point. We formalize this mechanism as a differentiable sequence operator the Countercurrent Multiplier (CCM) layer and study it as an alternative to residual iterative refinement.

In this note, we introduce a generalisation of multiplicative chaos measures which is both non-Gaussian and non-Abelian. The renormalisation procedure takes as inputs an irreducible unitary representation of a compact connected Lie group, together with a Kac-Moody unitarising measure at some level $\kappa$, and outputs a random measure on the circle with values in the space of positive definite Hermitian endomorphisms of the representation space. So far, our construction is valid for the range of $\kappa$-values corresponding to the $L^2$-phase. The proof follows the usual route in the theory of multiplicative chaos, relying on an exact formula for the one-point function and a bound on the two-point function at colliding points. These expressions are derived using the rich algebraic structure of the theory: namely, we establish a one-dimensional version of the Knizhnik-Zamolodchikov equations.

We prove that any homogeneous family $\sigma_3,\sigma_5,\ldots$ in $\mathfrak{grt}_1(\mathbb K)$ with nonzero coefficients of $x^{2k}y$ in $\sigma_{2k+1}$ generates a free Lie subalgebra, building on the work of Brown.

This paper is concerned with the existence theory of isometric immersions of surfaces with negative Gaussian curvature into the three-dimensional Euclidean space. We reformulate the Gauss–Codazzi equations, i.e., the partial differential equations for isometric immersions, into hyperbolic conservation laws for the flows of Chaplygin gas with nonzero source terms. Then, employing theories of invariant regions and compensated compactness, we establish for any given $p \in [2,\infty[$ the existence of $W^{2,p}_{\rm loc}$-isometric immersions of various general families of metrics over infinite strips $\mathbb{R}\times [0,T]$ with arbitrarily large $T$. Such metrics include those of various classical minimal surfaces: helicoid, catenoid, and Enneper surfaces, as well as metrics in isothermal coordinates or of the “reciprocal-type”. In our fluid dynamical formulation of the isometric immersion problem, we specialise in the case that the two Riemann invariants for the associated hyperbolic conservation law remain bounded and of distinctive signs, and obtain $L^p$-solutions to the initial-boundary value problem via the method of relative entropy with respect to a nonstationary ODE background. The isometric immersions constructed in this paper have second fundamental forms belonging to $L^p_{\rm loc}\setminus L^\infty_{\rm loc}$.

We consider a Cauchy-Dirichlet problem for a heat equation with variable coefficients in non-divergence form. The initial values are assumed to be homogeneous, while the Dirichlet boundary values are non-constant. In this setting, we derive an asymptotic formula for the short-time behavior of the solution, which extends the celebrated Varadhan formula. We stress the fact that the boundary values are allowed to vanish or change sign. The formula is obtained by combining the original ideas of Varadhan with those of Evans and Ishii, pertaining to the theory of viscosity solutions, and some further remarks. In passing, we prove the elliptic counterpart of the formula. This concerns the slow-diffusion behavior of the solutions of the resolvent equation. Furthermore, for the case of the classical Laplace operator, we prove asymptotic formulas for the heat content and the mean value of the solution of the resolvent equation on spheres touching the boundary of the domain. These extend to the case of non-constant boundary values, certain formulas previously obtained by the second author and S. Sakaguchi. The formulas involve the principal curvatures at the touching point and the Dirichlet boundary values. We then use our formulas to describe time-invariant surfaces for the heat equation, in presence of non-constant Dirichlet boundary values. We give two symmetry results in case of non-negative Dirichlet boundary values and a non-existence result, when those values change sign.

Circulant digraphs are Cayley digraphs over finite cyclic groups and constitute a fundamental class of objects in algebraic graph theory. Extending the classification of locally-primitive circulant graphs \cite{JZ-2026}, we completely determine all locally-quasiprimitive circulant digraphs. Our main theorem shows that a connected locally-quasiprimitive circulant digraph is isomorphic to one of the following: the complete graph \(\K_n\), the complete bipartite graph \(\K_{n/2,n/2}\), the graph \(\K_{n/2,n/2}-\frac{n}{2}\K_2\) (with \(n/2\) odd), the cycle \(\C_n\), the directed cycle \(\vec \C_n\), a normal circulant digraph of prime valency, the lexicographic product \(\vec \C_m[\overline{\K_b}]\), or the tensor product \(\vec \C_m\times \K_b\) with \(\gcd(m,b)=1\).

In this paper, using a limiting approach, we establish a new type of weighted Carleman inequality in all dimensions $n\geq 2$ and classify all extremal functions. In particular, when $n=2$, we prove that our inequality is equivalent to a sharp norm inequality in the Bergman space. In even dimensions, we further establish a sharp weighted Huber isoperimetric inequality on the unit ball, which generalizes Huber's original result \cite[Ann. Math., 1954]{Huber} and may be regarded as a sharp counterpart of Y. Wang's isoperimetric inequality in the unit ball \cite[Adv. Math., 2015]{Wang}.

This paper presents a comprehensive analysis of directional derivatives and error bounds for the merit function $\theta(x)=\sup_{a\in A}\bigl(-\Delta_C(F(x)-F(a))\bigr)$ associated with the vector optimization problem $\operatorname{Min}_C\{F(x):x\in A\}$, where $\Delta_C$ is the oriented distance function. We first prove that $\theta$ is concave and Lipschitz continuous on the whole space and derive its dual representation via the weak$^*$ compact convex set $K=\overline{\operatorname{co}}^{w^*}(S(C^+))$. At a weakly efficient solution $\bar x$, we obtain the explicit formula $\theta'(\bar x;d)=\min_{y^*\in W(\bar x)}\langle y^*,F(d)\rangle$ with $W(\bar{x})=\{y^*\in K:F^*y^*\in -N_A(\bar{x})\}$, characterize the zero-directional-derivative cone, and prove that, under a local error bound condition, the tangent cone to the solution set is $T_{E_w}(\bar x)=T_A(\bar x)\cap T_{\widehat A}(\bar x)=\{d\in T_A(\bar x):\theta'(\bar x;d)=0\}$. We establish the equivalence of thirteen distinct global error bound conditions, including characterizations via linear regularity, the global slope, an asymptotic condition, and perturbation stability. A central result shows that the global error bound property for $\theta$ on the feasible set $A$ is characterized by a uniform negativity condition on the unit-sphere minimal directional derivative, namely $\sup_{x \in A \setminus E_w} \varphi(x) < 0$. We also determine the optimal local error bound constant precisely as $1/\varphi(\bar x)$ when $\varphi(\bar x)>0$, and provide a counterexample demonstrating that an additional directional condition is essential when $\varphi(\bar x)=0$. These results provide a complete bridge between the directional derivative of the merit function and the geometry of the solution set, offering fundamental tools for the convergence analysis of algorithms in vector optimization.