数学前沿

We prove a Fuglede-type quantitative isoperimetric inequality for the Pansu sphere, the conjectured isoperimetric set in the sub-Riemannian Heisenberg group $\mathbb{H}^1$. We show that, among suitable volume-preserving normal perturbations of the Pansu sphere that are small in a weighted $C^2$-norm, the difference between the sub-Riemannian perimeters of the perturbed and unperturbed regions is bounded below by a positive constant times the square of their $L^1$-distance. The proof is based on a Riemannian approximation argument with uniform estimates.

We bound the error term in the counting function of cubic extensions of $\mathbb{Q}$ by $\mathcal{O}_\epsilon(X^{\theta+\epsilon})$, with $\theta < 2/3$, improving upon results due to Bhargava, Taniguchi, and Thorne. The proof proceeds by relating a certain discriminant-reducing sieve, due to Bhargava, Shankar, and Tsimerman, to averages of Artin characters associated with the Artin $L$-function $\zeta_K(s)/\zeta(s)$, with $K$ a cubic field. Conditional on the Generalised Riemann Hypothesis (GRH) for Dedekind zeta functions $\zeta_K(s)$, we obtain further savings and show that the error term is $\mathcal{O}(X^{2/3-1/51+\epsilon})$. Moreover, for the analogous smooth counting problem, we obtain a bound $\mathcal{O}(X^{1/2+\epsilon})$ for the error term, conditionally improving results of Shankar, S\"odergren, and Templier. Our arguments can also handle finitely many splitting conditions. The methods we use to study cubic fields are natural extensions of methods which are used to study the one-level density associated with the above family of Artin $L$-functions. In this direction, we improve the admissible support $[-\sigma,\sigma]$, from the previously known admissible value $\sigma < 2/5$, to $\sigma < 1$, conditional on the GRH. This also proves that at least $75\%$ of these $L$-functions are non-vanishing at the central point, conditionally improving upon results of Shankar, S\"odergren and Templier. Finally, one may study similar questions over a rational function field $\mathbb{F}_q(T)$, with $q$ coprime to $2$ and $3$. Here, we obtain a proportion of non-vanishing of at least $75\%$ for the analogous family of $L$-functions. Furthermore, we obtain a bound $\mathcal{O}\left(X^{1/2+\epsilon}\right)$ for the error term in the counting function of cubic function fields, improving previous results of the author. Over function fields, our results are unconditional.

We compute the local and global Denef--Loeser topological zeta functions of the determinant on the space of symmetric matrices. It verifies the strong monodromy conjecture of the symmetric determinant function. The computation relates to the geometry of complete quadrics and wonderful embeddings.

Focusing on reaction-diffusion models with an interface region dividing the domain, we study the stability of systems as the permeability parameter approaches zero or infinity under Kedem-Katchalsky boundary conditions imposed on the membrane or interface region. These boundary conditions establish that the flux across the interface manifold is proportional to the density jump across that region, with the constant of proportionality defined as the permeability parameter. We analyze both limiting behaviors: when permeability vanishes and when infinite flux is permitted. This analysis is carried out for both the well-known symmetric case, where permeability is identical in both directions across the membrane, and the asymmetric case, where it depends on the direction of the flux. For the asymmetric case, we obtain novel and surprising results that reveal a richer variety of behaviors in this class of interface problems and that differ from the scalar models.

We study the calibration of continuous-time martingale models to observed derivative prices via a dynamic martingale optimal transport problem that penalizes deviations from the volatility of a prescribed reference model. Using $q_x$-Bass martingales, we approximate this problem through a sequence of discrete-time problems and prove that the continuous-time interpolations of their optimizers are asymptotically optimal. Building on the theory of $q_x$-Bass martingales, we develop numerical methods for the discrete-time problems that do not require full marginal distributions at calibration maturities.

Let $F\in S_{2k}^{new}(M,\chi^2)$ be a Hecke newform, and $D$ a fundamental discriminant with $(-1)\chi(-1)D>0$. Via the Shimura Correspondence, we get associated modular forms of half-integral weight, and Waldspurger showed that for such $g=\sum c_nq^n \in S_{k+1/2}(4M',\chi;F)$, if $c_{|D|}(g)\neq 0$, then $|c_{|D|}(g)|^2 \propto |D|^{k-1/2}L(F\otimes \overline{\chi}\otimes \chi_D,k)$. Kohnen refined this in the case that $f\in S_{2k}^{new}(N)$ for $N$ odd and squarefree, making explicit $|c_{|D|}(g)|^2$ for $g \in S^+_{k+1/2}(4N;f)$. However, $c_{|D|}(g)=0$ unless $\left(\frac{D}{p}\right)=w_p(f)$ for all $p|N$. In this paper, we produce more generally $g \in S^+_{k+1/2}(4\ \mathrm{lcm}(N,N_1^2),\chi;f\otimes \chi)$, where $\chi$ is a character of odd squarefree conductor $N_1$, using a family of generalised Shimura/Shintani maps $\Psi_\chi^D,{\Psi_\chi^D}^*$ due to Kojima and Tokuno. Extending their work (and Kohnen's), we compute its Petersson norm, and give an explicit formula for all the $|c_m(g)|^2$. The key technical ingredient is an explicit set of representatives for $L^0_{M_1,M}(M_1^2D^2)/\Gamma_0(M)$ of binary quadratic forms. We also use a new multiplicity-one result based on work of Kohnen, Ueda, and Waldspurger. In the case $N_1|N$, we reconstruct forms previously considered by Baruch-Mao and Prasanna, which change Kohnen's condition to $\left(\frac{D}{p}\right)=-w_p(f)$ for all $p|N_1$, thus relating twisted $L$-values to squares of Fourier coefficients for all classes of discriminants.

Leader-follower gradient iterations, including bilevel and two-time-scale methods, can exhibit large finite-time excursions despite asymptotic stability. For cascaded iterations with block matrix J = [[A,0],[C,D]], symmetric diagonal blocks of norm at most gamma < 1, and coupling norm at most c, we prove an exact, dimension-free minimax formula: at every horizon t, the worst-case amplification equals the spectral norm of [[gamma^t,0],[c t gamma^(t-1),gamma^t]], attained by an aligned two-dimensional quadratic. Consequently, the worst transient is on the order of 1 + c/(1-gamma), and eigenvalues alone cannot predict finite-time gain. Under martingale noise, the aligned system has stationary follower variance sigma^2 c^2 (1+gamma^2)/(1-gamma^2)^3, establishing cubic inverse-gap scaling. For quadratic followers, the coupling is linked to follower sensitivity R by C = (I-D)R. When the leader is isotropic and the follower map is positive semidefinite, amplification is at most 1 plus the operator norm of R, independently of the spectral gaps. A look-ahead Newton update eliminates follower memory and is minimax-optimal among linear updates that track the leader exactly. Experiments validate the bounds and show that accelerated or inexact follower solvers can exceed the sensitivity bound when their effective maps have negative eigenvalues. Code and experiment scripts: https://github.com/Axym-Labs/transient-amplification

For a directed graph with a chosen root, the arborescence polynomial sums the products of edge weights over all spanning trees directed toward that root. In a finite graph cover, each lifted edge inherits the variable of its base edge. At corresponding roots, the ratio of the cover's arborescence polynomial to the base polynomial is known to be an integer polynomial. We prove that every coefficient of this ratio is nonnegative, resolving the positivity conjecture of Pylyavskyy. The key to the proof is to allow fibers of unequal size and replace the permutations defining a cover by arbitrary maps. For these more general systems, we prove a stronger coefficient-positivity theorem for a determinant derived from the graph Laplacian. The enlarged class is closed under two reductions: contraction of an edge over a base loop and elimination of a base vertex. Both yield nonnegative recurrences for the relevant coefficients in terms of smaller systems, allowing an inductive proof. We also pose the problem of finding a direct combinatorial interpretation of the coefficients.

We study the initial boundary value problem for the compressible Navier--Stokes--Poisson equations without damping in the three-dimensional half-space and in smooth exterior domains. The velocity satisfies the no-slip boundary condition, and the electric potential satisfies the homogeneous Neumann boundary condition. For small perturbations of a constant state in $H^3$, with a small initial electric field in $L^2$, we establish global existence and uniqueness of strong solutions. The uniform estimates combine density dissipation, tangential and normal derivative estimates, and a modified energy expressed through the Helmholtz projection. In exterior domains, boundary normal coordinates and localized elliptic estimates control the geometric terms. We also obtain algebraic decay in the full energy norm. In the half-space, finite tangential negative Sobolev norms of order $-s$, with $\frac12\leq s<1$, yield the rate $(1+t)^{-s/2}$. In exterior domains, integrable initial density and velocity perturbations yield the same form of estimate for $0<s<\frac14$. The exterior decay proof uses whole-space negative-order estimates for localized variables and the time integrability of the dissipation.

We study the following prescribed scalar curvature problem $$-\Delta u=K(y)u^{2^*-1},\ \ u>0 \,\ \ \text{in}\ \ \mathbb{R}^N, \tag*{$(*)$}$$ where $2^*=\frac{2N}{N-2}$. For $N\geq4$, we construct a positive $K(y)$ with maximum set $\{|y|=1\}$ and critical order of flatness $m=N-2$ at $|y|=1$, such that problem $(*)$ admits only finitely many positive solutions in $D^{1,2}(\mathbb{R}^N)$. To this end, we prove compactness of the solution set of $(*)$ by blow-up analysis and then apply Sard's theorem to choose $K$ for which each solution is nondegenerate and hence isolated, so that the solution set of $(*)$ is finite. Our results give a negative answer to Wei--Yan's conjecture in [J. Wei and S. Yan, JFA, 2010].

In this short note, we obtain upper bounds on the chromatic numbers of spheres using recent improvements in bounds for Euclidean space. We show that $\chi (S^n_{R}) \leq (\xi_R + o(1))^n$, where $\xi_R = \frac{3\sqrt 3}{2}(1-1/(4R^2))$ and $\chi(S^n_R)$ denotes the chromatic number of the $n$-dimensional sphere of a fixed radius $R$.

This paper presents a counterexample to the optimality conjecture for the trace distance based optimal state transformation problem proposed by Coutts et al. [Quantum 5, 448 (2021)]. The conjecture posits that for the trace distance based optimal state transformation problem, applying the sign function to the error matrix uniquely determines a dual certificate for every optimal solution. By constructing a counterexample in 2-dimensional Hilbert spaces, we disprove this conjecture.

Let $G$ be a finite group acting on a connected open Riemann surface $X$ by holomorphic automorphisms and acting on a Euclidean space $\mathbb R^n$, $n\ge 3$, by orthogonal transformations. In a previous paper, the second author gave a necessary and sufficient condition for the existence of a $G$-equivariant conformal minimal immersion $F:X\to\mathbb R^n$, and he asked whether such $F$ can be chosen to be a proper map. Assuming that this condition holds, we give an affirmative answer when $X$ is a bordered Riemann surface and $G$ acts freely on the boundary of $X$. This contributes to the Sullivan conjecture concerning the existence of proper hyperbolic minimal surfaces in $\mathbb R^3$ of finite topology. Under the same conditions, we also construct a $G$-equivariant proper conformal minimal immersion of $X$ to the unit ball of $\mathbb R^n$ which is complete in the induced metric, thereby opening a new direction in the Calabi--Yau theory regarding the study of complete bounded minimal surfaces. Analogous results hold for null curves in $\mathbb C^n$.

For a closed, oriented 3-manifold $M$, we show that any finite subgroup of its mapping class group preserves an isotopy class of filling sphere system on $M$. This is an isotopy version of the "equivariant sphere theorem" of Meeks-Yau. As part of our analysis, we establish a bijection between simplices of the Guirardel--Levitt outer space for $\pi_1(M)$ and filling sphere systems in $M$, generalizing an argument of Hatcher. We also compute the image of $\mathrm{Homeo}(M)\to\mathrm{Out}(\pi_1M)$ for any such $M$.

In this paper, we study integral operators $$\begin{equation*} T_\alpha f(x)=\int_{\mathbb{R}^{n}}K(x,y) f(y)dy, \end{equation*}$$ with kernels $K(x,y)= k_1( x- A_1y)...k_m( x-A_my),$ where $k_i(x)=\frac{\Omega_i(x)}{|x|^{n/q_i}}$ and $\Omega_i: \mathbb{R}^n\to \mathbb{R}$ are homogeneous functions of degree zero, satisfying a size and a Dini condition, $A_{i}$ are certain invertible matrices, and $\frac n{q_1}+\dots+\frac n{q_m}=n-\alpha,$ $0\leq \alpha <n.$ We obtain the $H^{p}_{w^p}(\mathbb{R}^{n})-L^{q}_{w^q}(\mathbb{R}^{n})$ boundedness of these operators, for a class of Muckenhoupt weights $w$, satisfying the condition $$\begin{equation*} w(A_ix)\leq cw(x), \end{equation*}$$ $a.e.x\in\mathbb{R} ^n$, $1\leq i\leq m$.

In this note we propose a basic $L^2$-based approach for studying the global and local energy equalities of the incompressible 3D Navier-Stokes equations in the standard energy class on $\mathbb{T}^3 \times (0,T]$. Motivated by L. Onsager's principle of least dissipation of energy (1931), we give a new sufficient condition for the energy equalities in terms of the limit of a sequence of minimizers. In particular, we show that the equalities are attained if this limit is non-vanishing. We observe that the indeterminacy in the vanishing case is reminiscent of turbulence-driven energy transfer dominating other transport processes.

We develop a multilevel Monte Carlo framework for coercive elliptic problems with random diffusion and reaction coefficients on polytopic meshes, with particular emphasis on the approximation and inter-level coupling of random coefficients generated from spatially represented latent fields. The main analytical contribution is a Strang-type error decomposition in a DG norm that separates the forward discretisation error from the error introduced by approximating the random coefficients. On this basis, we study two coefficient-generation and multilevel-coupling strategies, a spectral sampler based on a stationary Gaussian latent field defined on a fixed Cartesian reference grid, and an SPDE-based Mat\'ern sampler in which levels are coupled through projected white-noise loads on agglomerated hierarchies. Both fit into the same coefficient-aware analytical framework and couple adjacent levels directly through a common latent random input. The theory is supported by numerical experiments on general Voronoi and agglomerated polytopic meshes, demonstrating the predicted forward and coefficient-representation errors, decay of multilevel corrections, strong inter-level coupling, and the resulting work-accuracy behaviour relative to single-level Monte Carlo.

For fractional difference energies with two power weights and distinct inner and outer exponents, we determine regimes in which the constant generated by the homogeneous power is optimal and regimes in which it loses optimality. In the range $1<p<q$, we prove sharp Hardy inequalities on nontrivial weight intervals for every sign of $D=d+2\beta-sp$, including a two-sided interval in the balanced case $D=0$. For $q\ge2$, these intervals extend strictly beyond an elementary pointwise scalar criterion. For $1<p<q<2$, we determine the optimal power exponent in the scalar supporting-line inequality and obtain a larger explicit sharpness interval. The proof pairs opposite logarithmic displacements and controls the nonlinear deficit by a quadratic reference energy of the transformed function. Its translation estimate allows a positive contribution at bounded scales to absorb negative contributions at remote scales. Concentrating tests prove strict loss of homogeneous sharpness near both finite-energy endpoints, and angular variations give concrete bounds for the first sharpness threshold. We extend the method to flat singular sets and regional half-space energies, retaining the distinction between ambient finite-energy conditions and finiteness of the homogeneous candidate. We also treat the complementary order $p>q$, including flat-subspace inequalities and a complete alternative proof for point singularities. All sharpness sequences belong to the specified smooth test classes, and every positive optimal constant extends to their completion in the difference norm.

Inverse source problems in divergence form consist in finding a vector field with prescribed support S, whose divergence is the Laplacian of some observed potential. In this paper, we assume the unknown vector field is a vector-valued measure, and we study the corresponding least square inversion problems, regularized by penalizing the total variation, without discretizing the criterion nor the unknown. We prove that this problem has a unique minimizer in the case where S is a slender set; i.e., it has zero Lebesgue measure and each connected component of its complement has infinite Lebesgue measure. The proof relies on a refinement of Smirnov's decomposition of divergence-free measures [62] which is of independent interest.

Let $\pi(H)$ be the Tur\'an density of an r-uniform hypergraph $H$ and let $H_k^r$ denote the $r$-uniform hypergraph on $r+1$ vertices with exactly $k$ edges, where $1\le k\le r+1$. Sidorenko~(JCT-B, 2024) proved that $\pi(H_3^r)\ge (1.7215-o(1))r^{-2}$ as $r\to\infty$ and $\pi(H_k^r)\ge (C_k+o(1))r^{-(1+1/(k-2))}$ for fixed $k$ as $r\to\infty$. Clemen~later improved the first bound to $\pi(H_3^r)\ge cr^{-2}\sqrt{\log r}$ for some constant $c>0$. In this article, we prove the following results. \begin{itemize} \item For any fixed $\varepsilon>0$, there is a constant $c_\varepsilon>0$ such that $$\pi(H_3^r)\ge \frac{c_\varepsilon}{r(\log r)^{2+\varepsilon}}.$$ %$\pi(H_3^r)\ge 1/(r(\log r)^{2+o(1)})$. Together with the known upper bound $\pi(H_3^r)\le1/r$, this implies $\pi(H_3^r)=r^{-1+o(1)}$. \item For every $3\le k\le r+1$, let $s=\min\{k-2,r-k+2\}$. Then $$\begin{equation*} 0\le \frac{k-2}{r}-\pi(H_k^r) \le \frac{128}{r}\left(\sqrt{s\log\frac{er}{s}}+\log\frac{er}{s}\right). \end{equation*}$$ This estimate yields several asymptotically sharp results for $\pi(H_k^r)$. For example, $\pi(H_k^r)=(1+o(1))(k-2)/r$ when $\log(er/(k))=o(k)$. \end{itemize}