Papers

In this paper we investigate the relation between the bonded braid monoid BBn and the group of pure braids $P_n$, realised by an algebraic analogue of the topological tangle insertion, namely by replacing a bond by (a power of) a pure braid generator. To do so, we provide appropriate presentations for $P_n$ using cyclic relations and clasp generators, and extend them to presentations of $B_n$. We construct the monoid epimorphism $\Phi: BB_n \to Bn$, mapping bonds to pure braid generators, which is the main result of the paper. Using the isomorphism from the singular braid monoid $SB_n$ to the tight bonded braid monoid $BB_n$, we extend $\Phi$ to a new type of epimorphism $s \Phi: SB_n \to B_n$. The epimorphisms $\Phi$ and $s \Phi$ are extended to families of epimorphisms $\Phi_p$, $s \Phi_p$, $p \in \mathbb{Z}$.

Zero-knowledge proofs (ZKPs) enable the verification of a statement without revealing any information beyond its validity and constitute a fundamental primitive in cryptography and information theory. However, existing constructions rely on computational assumptions and are predominantly confined to bipartite settings, leaving their information-theoretic realization in bipartite or network scenarios largely unexplored. Here we develop a framework for zero-knowledge verification based on the indistinguishability of quantum states under operational constraints. Exploiting the fundamental limitations imposed by local operations, we show that a verifier is inherently restricted from extracting information about the underlying state while retaining the ability to verify correctness. We construct explicit protocols for multiparty quantum networks that achieve information-theoretic security, ensuring that no subset of collaborating parties can gain knowledge beyond the validity of the statement, independent of their joint computational power.

This declaration calls for action to address the challenges posed by the use of artificial intelligence within mathematics research. Technological developments have repeatedly transformed the practice of mathematics. Recent artificial intelligence technologies, including symbolic and neural methods for the generation and formalization of mathematics, may already have initiated a significant chapter in this long history. Among researchers, artificial intelligence has produced a wide range of reactions: enthusiasm for its potential to yield new discoveries; intimidation by the pace of developments; indifference to these rapid changes; and concern for the implications, both for mathematics and in wider society. Mathematicians have a choice about whether and how to adopt artificial intelligence in the conduct of their research. They also have a responsibility to ensure the continued flourishing of the discipline. This Declaration calls upon mathematicians to exercise this responsibility, and provides recommendations for individuals, institutions, government, and industry. Although we adopt the perspective of mathematical research, much of what we write applies equally to other aspects of mathematics. This includes work in the broader mathematical sciences, education, mentoring, publishing, funding, science policy, and use of mathematics in the wider world.

We study statistical limit theorems for time-series generated by multidimensional random fields that are finitary factors of iid random processes. Based on the decay of the tail of the coding radius, we establish the Central Limit Theorem with explicit rates of convergence, as well as the weak convergence to a Brownian sheet, for suitably regular functions. Our results can be applied to a variety of classical models in statistical mechanics, such as the Ising model, the hard-core model, and proper colourings.

STFT phase retrieval aims to reconstruct a signal from spectrogram magnitudes. While two compactly supported windows theoretically guarantee uniqueness, a complete algorithmic framework remains lacking. This paper systematically constructs numerical schemes for multi-window STFT phase retrieval. We theoretically provide a nonuniqueness counterexample for the single-window case. We then propose a two-window reconstruction algorithm based on Shannon interpolation and analytic square-root extraction. To overcome the instability of the global sign choice, we introduce a third window and a robust $K$-sign method that determines the sign locally. We also adapt SCR algorithm to the STFT setting. For the $K$-sign method, we establish deterministic stability on a finite contiguous block of frequency samples, with the contribution of the complementary tail explicitly included in the error bound. Numerical experiments validate the effectiveness of all schemes.

We give a computer-assisted proof of Frenkel's root-multiplicity bound for the rank-three hyperbolic Kac–Moody Lie algebra $A_1^{++}$. For every root $\alpha$, we prove $\dim g_\alpha\le p(1-(\alpha,\alpha)/2)$, where $p$ is the ordinary partition function. The proof combines exact affine characters with a coefficientwise logarithmic majorant derived from parabolic homology. After Weyl reduction, analytic estimates establish the inequality on infinite regions of large depth or large level. The remaining finite region is verified by computer. Together with the known low-level formulas, the assembly of these estimates establish the conjecture for every root of $A_1^{++}$.

For every group $\Gamma$, we define a homomorphism $d^\Gamma : \mathrm{Comm}(\Gamma) \to \mathbb{Z}^{(\mathcal{FS})}$ from the abstract commensurator $\mathrm{Comm}(\Gamma)$ to the free abelian group $\mathbb{Z}^{(\mathcal{FS})}$ with basis the collection $\mathcal{FS}$ of isomorphism classes of finite simple groups. We investigate the homomorphism $d^\Gamma$ when $\Gamma$ is a finitely generated free group $F$. We explicitly describe the image of $d^F : \mathrm{Comm}(F) \to \mathbb{Z}^{(\mathcal{FS})}$, which is a free abelian group of infinite rank, and we show that the kernel is the monolith of the group $\mathrm{Comm}(F)$. We deduce in particular that every proper quotient of $\mathrm{Comm}(F)$ is abelian. We use this to study the commensurator of a cocompact lattice in the automorphism group of a regular tree, which can be seen as a subgroup of $\mathrm{Comm}(F)$. We show that the image of this group under $d^F$ is again a free abelian group of infinite rank, showing in particular this group is not virtually simple.

This paper is a companion to one in which we prove there is a discontinuous phase transitions in the $k$-creation process with fast stirring in $d=1$ when $k \ge 2$. Dickman and Tomé (1991) introduced models on $Z$ in which $k$ consecutive occupied sites give birth at rate $\lambda$ and individual particles die at rate 1. Here, we show that without stirring the models with $k\ge 2$ have qualitative properties much like the contact process, which is the case $k=1$. The critical value can be charracterized by the speed of interface when the process starts from the initial configuration $(-\infty,0]$. The process dies out at the critical value. In the supercritical phase the complete convergence theorem holds which implies there is only one nontrivial stationary distribution, and convergence to the limit occurs exponentially rapidly. In the subcritical phase, the process dies out exponentially fast starting from any finite set.

We deform the complex structure of compact simply-connected Bismut flat manifolds, obtaining new complex manifolds that admit pluriclosed metrics but do not admit Bismut flat metrics. This produces genuinely new examples of pluriclosed manifolds while enriching our understanding of the deformation theory of pluriclosed metrics. Moreover, our main result is relevant for the search of pluriclosed solitons and non-trivial Bismut Hermitian Einstein metrics in all dimensions.

Let $\mathcal G$ be a split semiabelian variety of dimension $n \ge 2$ and let $\mathcal C \subset \mathcal G$ an irreducible curve, where everything is defined over $\overline{\mathbb{Q}}$. We prove that, under the natural hypothesis that $\mathcal C$ is not contained in any proper algebraic subgroup of $\mathcal G$, the set of points $P$ of $\mathcal C$ lying in an algebraic subgroup $H$ which intersects the curve tangentially at $P$ is finite. This generalizes a result for curves in 2-dimensional tori by Marché and Maurin. We apply these results to square-freeness of certain geometric divisibility sequences.

Let $\mathcal{C}$ be a spherical fusion category with global dimension $D$. The Galois conjugates of the dimension character index a block of the $S$-matrix of the Drinfeld center, which we call the dimension block. We prove that for every character $\chi$ of $\mathrm{Gal}(\mathbb{Q}(D)/\mathbb{Q})$, the twisted sum $\sum_\sigma\chi(\sigma)/\sigma(D)$ has absolute value at most $e^{-1}\mathfrak{f}(\chi)^{-1/2}$, where $\mathfrak{f}(\chi)$ is the conductor of $\chi$ and $e$ is the order of the dimensional grading group of $\mathcal{C}$. As an application, we determine the possible global dimensions of spherical fusion categories below $\sqrt{5}$, extending the known classification up to $4\sqrt{3}/5$ of V.\ Ostrik and P.\ Etingof.

We study the Dai–Li conjecture concerning the pointwise monotonicity of the energy density along the $\mathbb C^*$-flow of stable $\mathrm{SL}(n,\mathbb C)$ Higgs bundles. We prove that the conjecture holds in rank two: for every stable $\mathrm{SL}(2,\mathbb C)$ Higgs bundle, the energy density is pointwise nondecreasing along the $\mathbb C^*$-orbit. In contrast, we show that this phenomenon is genuinely rank-dependent. For every rank $n\geq 3$, we construct stable $\mathrm{SL}(n,\mathbb C)$ Higgs bundles for which the energy density fails to be monotone along the $\mathbb C^*$-flow.

We study nonsmooth convex-concave saddle-point problems over compact convex sets, assuming access to stochastic subgradients of the payoff function. We develop single-loop projection-free algorithms that use linear minimization oracles over the primal and dual domains. Unlike prior projection-free approaches that rely on smoothing, our methods are purely subgradient-based and handle nonsmoothness directly. This design makes the framework modular: when a linear minimization oracle is not used over a specific domain, the corresponding update can be replaced by a standard projection oracle without changing the single-loop structure. Thus, our framework covers both fully projection-free and hybrid oracle configurations. We prove anytime strong saddle gap guarantees under standard unbiased stochastic oracle assumptions with bounded variance. Our algorithms achieve an $O(\epsilon^{-2})$ iteration complexity, matching projected stochastic subgradient methods. We also provide matching lower bounds for the corresponding oracle models, establishing minimax optimality. Our results show that, for nonsmooth stochastic minimax problems, the computational advantages of linear minimization oracles need not come at the expense of statistical or oracle efficiency.

For an integer solution of $y^2=x^3+k$ with $x,y\ge 1$ and $k\ne 0$, the integers $x^3$, $|k|$ and $y^2$ form a triple in which one term is the sum of the other two. When $\gcd(x,y)=1$ this triple is coprime, and the $abc$ conjecture predicts that its quality is asymptotically at most $1$. Following Müller, Taktikos and de Weger, we factor this quality through the product $P=xy|k|$ into the approximation gain $G_a=\log\max(x^3,y^2)/\log P$ and the power gain $G_p=\log P/\log\rad(xy|k|)$, which is at least $1$. We show that the weak form of Hall's conjecture, which asks that $|k|>x^{1/2-\delta}$ for every $\delta>0$ outside finitely many solutions, is equivalent to the asymptotic bound $G_a\le 1$. More generally, for $6/11\le\kappa<6/5$, an asymptotic bound $G_a\le\kappa$ gives $|k|>x^{3/\kappa-5/2-\delta}$ for every $\delta>0$ outside finitely many solutions, and for $3/4\le\kappa<6/5$ the two statements are equivalent. Since the approximation gain never exceeds the quality, an $abc$ inequality with exponent $1\le\kappa<6/5$ gives the same bound for primitive solutions. The case $\kappa=1$ extends to all solutions and recovers the classical implication from the $abc$ conjecture to weak Hall. We also relate small values of $|k|$ to large partial quotients of $\sqrt x$.

Motivated by recent work on conjugacy languages in groups, we introduce a general framework for studying languages of representatives associated to equivalence relations on finitely generated semigroups. After developing this notion in full generality, we particularize it to Green's relations and obtain the corresponding Green languages, which form the central objects of this article. For free inverse monoids, we describe these languages via Munn trees and establish several structural relations among them, including characterizations of the $\mathcal{D}$-, $\mathcal{R}$-, $\mathcal{L}$-, and $\mathcal{H}$-languages. We then investigate the formal language-theoretic complexity of these Green languages. For free inverse monoids of rank at least two, we show that $\mathcal{H}$-Geo is context-free and co-context-free but not regular, while $\mathcal{R}$-Geo, $\mathcal{L}$-Geo, and $\mathcal{D}$-Geo are neither context-free nor co-context-free. In contrast, in the monogenic case, $\mathcal{D}$-Geo is regular and the remaining Green languages are deterministic context-free. We further relate these results to the ShortLex language and the growth of free inverse monoids, proving in particular that $(\text{FIM}_X,X)$ is growth tight. The article concludes with a collection of open problems and directions for future research.

Matrix Hilbert series retain local elementary-divisor data that their determinants discard. For locally finite positively graded elementary algebras whose vertex simples are of type $FP_\infty$ and whose matrix Hilbert series is rational, we construct from possibly infinite minimal resolutions a rational Euler matrix and its signed local Smith profile at $x=1$. The profile refines pole-order growth for perfect modules and is invariant under shift-compatible graded Morita and perfect-derived equivalences. Every finite integer profile occurs, while a wholly negative profile forces exponential corner growth. This extends the nonnegative local Smith framework for twisted Calabi–Yau algebras to a setting in which the Euler matrix may itself have poles.

An antimagic labelling of a graph G = (V,E) is a bijection from E to {1,2,...,|E|}, such that all vertex-sums are pairwise distinct, where the vertex-sum of each vertex is the sum of labels over edges incident to this vertex. A graph is antimagic if it has an antimagic labelling. Hartsfield and Ringel in 1990 stated the celebrated conjecture: Every connected graph other than K_2 is antimagic. We prove the conjecture for join graphs with at least three vertices.

Let k be an integer from 1 to n, let f be a positive smooth function on the unit sphere, and consider the normalized flow X_t = -f({nu})|X|^{alpha} sigma_k({kappa}) {nu} + beta X, beta = C_n^k. In the supercritical range alpha > k+1, we prove the flow exists for all time and converges exponentially in C^{infty}, from every smooth strictly convex initial hypersurface enclosing the origin, to the unique smooth strictly convex solution of f(x)r^{alpha} sigma_k({kappa}) = beta u. The convergence is driven by the relative residual p = (log u)_t = beta - f(x)r^{alpha} {sigma}_k({kappa})/u, whose L^{infty} norm is nonincreasing and decays with the explicit exponent beta(alpha-k-1); the residual estimates are proved before, and independently of, the curvature estimates. Existence of a strictly convex solution of the stationary equation in this range, for arbitrary positive angular data, was previously obtained by Bryan-Ivaki-Scheuer through an expanding-type flow started from a barrier; the contribution here is the convergence of the normalized contracting flow from arbitrary initial data, with an explicit exponential rate, together with uniqueness of the limit.

The complex spacing ratio is the complex displacement from a reference eigenvalue to its nearest neighbor divided by the corresponding displacement to its next-to-nearest neighbor. Its statistics provide a useful diagnostic of spectral correlations and nonintegrability in open quantum systems. Here, starting from the exact joint eigenvalue probability densities, we derive finite-$N$ complex-spacing-ratio distributions for the Gaussian ensembles of non-Hermitian random matrices in classes AI$^{\dag}$ and AII$^{\dag}$, realized by complex symmetric and complex self-dual random matrices, respectively. In class AII$^{\dag}$, we obtain an exact algebraic expression for arbitrary $N$ and explicitly evaluate the distributions and representative moments for $N=3, 4, 5, 6$. In class AI$^{\dag}$, although the joint density retains a noncompact integral over nonunitary eigenvector degrees of freedom, we analytically derive a normalized one-dimensional integral representation for $N=3$ and determine the asymptotic behavior, including a logarithmic correction to the cubic level repulsion and a nonanalytic contribution to the angular density. We further confirm these analytical results through direct numerical diagonalization of non-Hermitian random matrices.

We settle in the negative the longstanding question whether every Bézout domain is an elementary divisor domain by constructing a Bézout domain over which an explicit $2\times 2$ matrix has no Smith normal form. The obstruction is topological and is detected by the Möbius line bundle.

Taxi-based mobile sensing has emerged as a cost-efficient paradigm for large-scale urban environmental monitoring. In practice, both passive and active sensing strategies are adopted by ride-hailing platforms. Passive sensing, conducted during passenger-serving trips, is constrained by stochastic and spatially imbalanced ride demand, leading to limited and uneven coverage. Active sensing, executed by vacant taxis, provides greater control over sensing operations but incurs additional operational costs, and is thus typically treated as a supplementary strategy. To address the inefficiencies of the conventional "passive-first, active-second" paradigm, which may delay the monitoring of critical Points of Interest (POIs) and increase system-wide costs, we propose a Distributionally Robust Optimization (DRO)-based framework for active mobile sensing. First, we develop an enhanced A*-based drive-by sensing routing policy that integrates vehicle-task matching while capturing both global routing efficiency and local sensing opportunities. Second, we formulate the active sensing problem as a DRO model that explicitly accounts for uncertainty in ride demand through an ambiguity set of probability distributions, enabling robust decision-making for vacant taxi routing and vehicle-task matching. The proposed framework is evaluated on both static and dynamic mobile sensing settings using real-world data. Computational results demonstrate that our approach achieves better performance in terms of sensing coverage, operational cost, and robustness compared to benchmark strategies, highlighting the value of integrating distributional robustness into taxi-based sensing operations.

Large-scale antenna arrays are essential to future wireless networks but impose significant hardware and digital-processing burdens. The microwave linear analog computer (MiLAC) offers a promising architecture for addressing these challenges. In a MiLAC, beamforming is realized entirely in the analog domain through a reconfigurable multiport microwave network comprising tunable admittance elements. However, existing studies assume that these elements are continuously tunable, whereas practical implementations can support only finitely many configurable states. This paper investigates finite-resolution MiLAC-aided multiuser beamforming under lossless and reciprocal constraints with prescribed connectivity patterns. Each tunable susceptance is selected from a symmetric uniform finite-resolution codebook. We formulate an online problem that jointly optimizes the discrete susceptances and power allocation for a prescribed codebook, and an offline problem that designs the codebook based on channel statistics. By exploiting the odd-uniform codebook structure, we express each multilevel susceptance as a weighted sum of two-level variables and develop exact continuous penalty models for both the online beamforming and offline codebook design problems. We prove that, for sufficiently large penalty parameters, these penalty models are globally equivalent to their original discrete counterparts. Building on these formulations, we develop efficient ADMM-based algorithms for solving the two problems. Numerical results demonstrate that the performance of an unquantized fully-connected MiLAC can be approached with substantially reduced connectivity and finite resolution. In particular, a 3-bit stem-connected MiLAC achieves more than 95% of the sum-rate performance of the unquantized fully-connected MiLAC while requiring only 22% of its tunable components.

For vertex-transitive Ramanujan graphs with logarithmic girth, a simple random walk bridge of length of order $\log N$, where $N$ is the size of the graph, has a maximum distance that changes from order $\sqrt{\log N}$ to order $\log N$ in a bounded critical window. We prove this by separating bridges whose lifts to the regular tree close from those whose lifts do not. A uniform two-term return estimate determines the probabilities of these two cases and the real-valued critical center. In the same $O(1)$ window, the normalized range has a two-point limiting law whose mixture weights vary nontrivially across the window.

We prove that the weak Haagerup constant is multiplicative with respect to direct products, namely $\Lambda_{\mathrm{WH}}(G \times H)=\Lambda_{\mathrm{WH}}(G) \Lambda_{\mathrm{WH}}(H)$ for any locally compact groups $G$ and $H$. Knudby previously proved the submultiplicative inequality and asked whether equality always holds.

The Katok examples on $S^2$ are induced by Randers metrics obtained by perturbing the round metric by the standard rotational Killing field: allowing a scaling factor of the round metric gives a two-parameter family of Randers metrics $F_{\alpha,\beta}$, parametrised by positive real numbers $(\alpha,\beta)$. We compute the set of all $(\alpha,\beta)\in (0,2+\sqrt{3})^2$ for which $D^*(S^2,F^*_{\alpha,\beta})$, the unit codisc bundle with respect to $F^*_{\alpha,\beta}$, symplectically embeds into the round codisc bundle $D^*S^2$. This set has the structure of an infinite staircase. We also establish a dictionary between embeddings of these codisc bundles, singular $A_1$-ellipsoid embeddings, and $\mathbb{Z} _2$-equivariant ellipsoid embeddings, showing that these embedding problems are equivalent. Furthermore, we prove the analogous results for the $\mathbb{R} P^2$ case and discuss a broader family of examples for which this dictionary applies.

We study the stochastic first-order oracle complexity of finding $\epsilon$-stationary points of the primal function in smooth nonconvex-strongly-concave minimax optimization. For sufficiently small $\epsilon$, we establish lower bounds of $\Omega(\kappa L\Delta\sigma^2\epsilon^{-4})$ under the bounded-variance assumption and $\Omega(\kappa^{3/2}\bar L\Delta\sigma\epsilon^{-3})$ under the additional assumption of averaged smoothness. Here, $L$ and $\bar L$ denote the smoothness and averaged-smoothness constants, respectively, $\Delta$ is the initial primal gap, $\sigma^2$ bounds the oracle variance, and $\kappa=L/\mu$ or $\bar L/\mu$ in the respective settings, where $\mu$ is the strong-concavity parameter. Our bounded-variance lower bound improves the dependence on the condition number from $\kappa^{1/3}$ in previous lower bounds to $\kappa$, while our averaged-smoothness lower bound is the first of its kind. In both settings, the resulting lower bounds match existing upper bounds in their dependence on $\kappa$ and $\epsilon$. Our proofs are based on a unified quadratic lifting construction that transfers a hardness instance for stochastic nonconvex minimization to unconstrained minimax optimization while preserving the required variance and smoothness properties.

In 2005, Tuval et al. introduced a continuum model for the interaction of aerobic bacteria, oxygen, and an incompressible fluid. For the associated three-dimensional chemotaxis-consumption systems, global classical regularity for large data has remained a longstanding question, even after omitting the nonlinear convection term from the fluid equation. We answer this question and prove global classical solvability of the chemotaxis–Stokes Cauchy problem in \(\mathbb{R}^3\), with linear cell diffusion and bilinear signal consumption, under the following data assumptions. The initial density and signal are smooth, bounded, and nonnegative, with finite density mass and second moment and finite signal Fisher energy; the divergence-free initial velocity belongs to \(H^4\). No smallness assumption is imposed, and a positive constant signal background is allowed. The solution remains bounded on every finite time interval. The main new ingredient is a localized entropy estimate uniform under density normalization, even when the resulting consumption coefficient becomes arbitrarily large. A terminal representation of the signal separates two complementary mechanisms: consumption excludes normalized density concentration at positive-signal points, while a localized weighted estimate improves density integrability at zero-signal points. Together, these mechanisms yield decay of a scale-invariant density quantity and allow a local $\varepsilon$-regularity criterion to be applied without a separate smallness assumption on the signal gradient.

For systems with steep gradients, sharp interfaces, or severe spatio-temporal coupling, Physics-informed neural networks (PINNs) suffer from spectral bias, geometric inflexibility, and boundary constraint conflicts, which undermine accuracy and convergence. To overcome these issues, we propose a geometry-adaptive and constraint-enhanced PINN (GAC-PINN). The framework comprises four components: a gradient-driven adaptive grid mapping (AGM) for diffeomorphic point concentration with Jacobian regularization, an adaptive bandwidth hard-constraint ansatz with spatially-varying boundary transition widths, a Gaussian Fourier feature mapping as a spectral preconditioner to further enhance high-wavenumber representation, and an operator-aware router that automatically selects the appropriate hard-constraint construction based on whether the governing PDE contains temporal derivatives. An AGM callback mechanism and a three-stage training strategy ensure stable coordination. Benchmarks including the viscous Burgers equation, a sharp-peaked 2D Poisson problem, and the Allen-Cahn phase-transition equation show that GAC-PINN attains relative (L^2) errors of ((1.747\pm 0.450)\times 10^{-4}), ((2.868\pm 0.947)\times 10^{-5}), and ((1.756 \pm 0.712)\times 10^{-3}), respectively, consistently outperforming the baselines. Ablation studies further reveal that AGM alone yields a substantially lower error than residual-based adaptive refinement (RAR), while RAR becomes beneficial only when combined with FFM, demonstrating a context-dependent module interaction. Convergence analysis verifies rapid error reduction and saturation with increasing resolution, establishing a practical adaptive framework for high-fidelity simulation of problems with localized sharp features in applied mechanics and computational physics.

For a finite group $G$, let $\lambda(G)$ denote the minimum number of orbits on the elements of a finite lattice $L$ with $\operatorname{Aut}(L)\cong G$. Babai and Goodman conjectured that $\lambda(G)$ is bounded by an absolute constant. We prove that $\lambda(G)\leq 50$ for every finite group $G$, thereby confirming their conjecture. Moreover, the lattice can be chosen to have a regular orbit. The main algebraic ingredient is a decomposition of a generating set of an arbitrary finite $2$-group into an elementary abelian part and two sets in which no quotient of distinct elements is an involution.

We prove a sharp Szemerédi-Trotter estimate \[\mathcal{I}(A,\mathcal{L})\lesssim \frac{|A||\mathcal{L}|}{p}+|A|^{2/3}|\mathcal{L}|^{2/3}+|A|+|\mathcal{L}|\] for prime finite field $\mathbb{F}=\mathbb{F}_p$ by a new polynomial decomposition theorem. As applications, we first prove the sharp Furstenberg set estimate in $\mathbb{F}^2$. Secondly, we improve sum-product estimate \[\max\{|A+A|,|A\cdot A|\}\gtrsim\min\{(p|A|)^{1/2},|A|^{5/4}\},\quad A\subset\mathbb{F}.\] Finally, we improve the Fourier restriction estimate $R^*(2\to\alpha)$ holds for $\alpha>\frac{10}{3}$ in $\mathbb{F}^3$ when $p\equiv 3\mod 4$.