数学前沿

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}

We obtain bounds on the Kullback--Leibler divergence to equilibrium for mass-action chemical reaction networks (CRNs) with equilibrium. The associated trajectory-dependent bounds are characterized in terms of the singular values of the stoichiometric matrix, convexity parameters, and time-integrated activities via deformed-exponential-type functions. We further extend these bounds within a generalized gradient flow framework. We highlight the biological relevance of this framework: the resulting bounds apply to quasi-steady-state regimes, where long transients and plateau-like behavior are common and functionally important. We illustrate the framework using a catalytic CRN exhibiting plateaus, where the bounds provide an a posteriori characterization of slow relaxation induced by local convexity and provide a bound-based approach to quantifying relaxation in CRNs.

We investigate the semilinear wave equation with potential on weighted graphs. We establish sufficient conditions for the nonexistence of global-in-time solutions. Both nonnegative and sign-changing solutions are considered. In particular, the proof for sign-changing solutions relies on a novel technique for this type of result.

Erd\H{o}s asked about the least possible diameter of an $n$-point set in Euclidean space when any two pairwise distances differ by at least $1$. We consider the uniform-in-dimension variant of his question, in which the dimension $d$ may grow with $n$ and the error term is independent of $d$, and answer it in the negative by an explicit construction. For every prime power $q$, we construct a set $\mathcal X_n$ of $n=q+1$ points in $\mathbb R^{q^2+q}=\mathbb R^{n^2-n}$ such that the $\binom{n}{2}$ pairwise distances are mutually at least $1$ apart, yet $$\operatorname{diam}(\mathcal X_n)\le \Bigl(1-\frac{1}{\pi^2}+o(1)\Bigr)n^2.$$ The construction combines a Singer cyclic difference set with a weighted product of $N=\binom n2$ regular $m$-gons, where $m=q^2+q+1$. The weights are chosen so that the resulting distances increase while the gaps between consecutive distances decrease. After normalizing the smallest gap to $1$, all distinct pairwise distances are separated by at least $1$, while the diameter has asymptotic constant $1-1/\pi^2<1$.

We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and regularity theorems for Weingarten surfaces in H^3, which satisfy (1-a)K = a(2-H), for an a < 0, and have a specified boundary curve at infinity. These surfaces are shown to be closely connected to conformal mappings of domains in S^2 into the unit disk and provide Riemannian interpretations for some conformal invariants associated to such mappings. This paper was originally written in 1984, before I learned to use TeX, and was typed by one of the secretaries in the Princeton Math Department. It was more or less, my first original work after my dissertation. For some reason, I was not able to get this paper published in a timely manner. The results and perspective in this paper have proved to be useful to a variety of people, some of whom asked me to render the article into TeX and post it to the arXiv. I had been seriously thinking about doing this, when Martin Bridgeman sent me a transcription of my original article into TeX. I am extremely grateful to him for the effort he has put into this project. The paper is now formatted in a more or less modern AMS-article style, but for lots of additional punctuation, a few corrections and some minor stylistic changes, the content has been largely reproduced as it originally was. Remarks about the 'state-of-the-art' in hyperbolic geometry are obviously way out of date, as there has been enormous progress in many aspects of this still rich subject.

We construct a comparison morphism $\operatorname{HH}^{\mathrm{Rog}}\to\operatorname{HH}^{\mathrm{Ols}}$ between Rognes' logarithmic Hochschild homology and Olsson's logarithmic Hochschild homology which generalizes the comparison between Gabber's log cotangent complex $\mathbb{L}^{\mathrm{Gab}}$ and Olsson's log cotangent complex $\mathbb{L}^{\mathrm{Ols}}.$ In particular, Olsson's comparison is recovered as the first associated graded piece of our comparison morphism $\operatorname{HH}^{\mathrm{Rog}}\to\operatorname{HH}^{\mathrm{Ols}}.$ In the development of this comparison, we also prove a descent result for the cotangent complex generalizing a result of Bhatt, and construct a notion of HKR completed Hochschild homology for a general morphism of derived stacks which is functorial in the morphism and generalizes existing constructions of Hochschild homology when the base is affine.

We characterize metric measure spaces that satisfy intrinsic parabolic Harnack inequalities for doubly nonlinear equations of slow diffusion type. The considered parabolic Harnack inequalities are intrinsic in the sense that the waiting time depends on the solution. Our main results shows that in a metric measure space with a doubling measure, such an intrinsic parabolic Harnack inequality holds if and only if a Poincar\'e inequality holds. We thus show that estimates in scale and function dependent intrinsic geometries can characterize scale and location invariant functional inequalities.

We investigate a predator-prey model with constant prey harvesting and a generalized Holling type III functional response, focusing on the biological and dynamical consequences of allowing the parameter $b$ to be negative. In this regime, predation efficiency declines at high prey densities, potentially reflecting group defense. This change permits an additional coexistence equilibrium and introduces new bifurcation mechanisms affecting population persistence. We establish the existence of a codimension-three Bogdanov--Takens bifurcation, derive a codimension-three nilpotent-saddle unfolding, and investigate a codimension-two generalized Hopf bifurcation and the emergence of multiple limit cycles. Our analysis show that moderate harvesting weakens the protection conferred by high prey abundance; excessive harvesting leaves too few prey to sustain predator growth; and low harvesting preserves strong group defense, requiring high predator density for population persistence.

In this paper, we introduce the notion of stable imaginary elements in quantum groups and use crystal basis theory to construct explicit one-parameter families of such elements for all finite types. We show that all of Leclerc's examples of imaginary elements are stable imaginary. Using the generalized quantum Schur--Weyl duality, we also construct explicit examples of corresponding imaginary modules over quantum affine algebras of untwisted affine types.

We study martingality and strict local martingality in a broad class of stochastic and path-dependent Volterra volatility models and explain the underlying mechanisms. Our martingality criteria allow rough kernels and rely on non-accessibility of an explosive direction or one-sided growth bounds on the Girsanov drift. For smooth kernels, we derive sufficient conditions for strict local martingality from an explosion criterion allowing multiplicative noise, coupled factors and general initial input curves. Our results recover classical criteria and yield new results for Volterra and path-dependent models.