Roughly speaking, the Kakeya Conjecture asks to what extent lines which point in different directions can be packed together in a small space. In $\R^2$, the problem is relatively straightforward and was settled in the 1970s. In $\R^3$ it is much more difficult and was only recently resolved in a monumental and groundbreaking work of Hong Wang and Joshua Zahl. This note describes the origins of the Kakeya Conjecture, with a particular focus on its classical connections to Fourier analysis, and concludes with a discussion of elements of the Wang--Zahl proof. The goal is to give a sense of why the problem is considered so central to mathematical analysis, and thereby underscore the importance of the Wang--Zahl result.

We survey progress on the Kakeya conjecture in Euclidean space, with an emphasis on developments that have occurred since the previous surveys by Wolff and Katz-Tao.

For a class of $\mathbb{R}^d$-ations and $\mathbb{Z}^d$-actions on the $n$-dimensional torus $\mathbb{T}^n$, we characterize their unique ergodicity and establish a theorem of Weyl type. This result allows us to establish an isomorphism between the Banach algebra of quasi-periodic functions with spectrum in a given $\mathbb{Z}$-module and the Banach algebra of periodic functions on a torus. This, in return, allows us to give a very simple proof of Hausdorff-Young inequalities for Besicovitch almost periodic functions. The regularity of the parent function of a quasi-periodic function is also studied.

Let $f:\mathbb{R}^n \to \mathbb{R}$ be a continuously differentiable convex function with its minimizer denoted by $x_*$ and optimal value $f_* = f(x_*)$. Optimization algorithms such as the gradient descent method can often be interpreted in the continuous-time limit as differential equations known as continuous dynamical systems. Analyzing the convergence rate of $f(x) - f_*$ in such systems often relies on constructing appropriate Lyapunov functions. However, these Lyapunov functions have been designed through heuristic reasoning rather than a systematic framework. Several studies have addressed this issue. In particular, Suh, Roh, and Ryu (2022) proposed a constructive approach that involves introducing dilated coordinates and applying integration by parts. Although this method significantly improves the process of designing Lyapunov functions, it still involves arbitrary choices among many possible options, and thus retains a heuristic nature in identifying Lyapunov functions that yield the best convergence rates. In this study, we propose a systematic framework for exploring these choices computationally. More precisely, we propose a brute-force approach using symbolic computation by computer algebra systems to explore every possibility. By formulating the design of Lyapunov functions for continuous dynamical systems as an optimization problem, we aim to optimize the Lyapunov function itself. As a result, our framework successfully reproduces many previously reported results and, in several cases, discovers new convergence rates that have not been shown in the existing studies.

This note shows that the three theorems presented in J. Math. Anal. Appl. 556 (2026), 130199, whose proofs, in their present formulation, are purely formal, follow from elementary calculus.

Let $d\ge3$ and $\mathbb{F}_q^{d}$ be the $d$-dimensional vector space over a finite field of order $q$, where $q$ is a prime power. Fix a slice $\pi=\{x_d=\lambda\}$ of the unit sphere $S^{d-1}=\{x\colon ||x||=1\}$ and let $X_\pi$ be the set of lines through the origin meeting $\pi\cap S^{d-1}$. For $E\subset\mathbb{F}_q^{d}$ and $N\ge1$, we study the exceptional sets $$\mathcal{T}_1(X_\pi,E,N)=\bigl\{V\in X_\pi:\ |\pi_V(E)|\le N\bigr\},\qquad \mathcal{T}_2(X_\pi,E,N)=\bigl\{V\in X_\pi:\ |\pi_{V^\perp}(E)|\le N\bigr\},$$ on their respective natural ranges of $N$. Using discrete Fourier analysis together with restriction/extension estimates for cone and sphere type quadrics over finite fields, we obtain sharp bounds (up to constant factors) for $\lvert \mathcal{T}_1\rvert$ and $\lvert \mathcal{T}_2\rvert$, with separate treatment of the special slices $\lambda=\pm1$ and of the isotropic slice $\lambda=0$. The bounds exhibit arithmetic-geometric dichotomies absent in the full Grassmannian: the quadratic character of $\lambda^{2}-1$ and the parity of $d$ determine the size of the exceptional sets. As an application, when $|E|\ge q$, there exists a positive proportion of elements $\mathbf{y}\in X_\pi$ such that the pinned dot-product sets $\{\mathbf{y}\cdot \mathbf{x}\colon \mathbf{x}\in E\}$ are of cardinality $\Omega(q)$. We further treat analogous families arising from the spheres of radii $0$ and $-1$, and by combining these slices, recover the known estimates for projections over the full Grassmannian, complementing a result of Chen (2018).

We characterize the lower and upper attainability of the Wiener bound (also known as Voigt-Reuss bound) for singularly distributed conductive material mixtures. For the lower attainability we consider mixtures in which high-conductance materials support on sets having finite one-dimensional Hausdorff measures. We show that, under a mild coercivity condition, the kernel of the effective tensor of the mixture is equal to the orthogonal complement of the homotopy classes of closed paths in the supporting set. This shows that a periodic planar network has positive definite effective tensor, i.e., it is resilient to fluctuations, if and only if the network is reticulate. We provide a geometric characterization of the upper attainability by applying a transformation from varifolds to matrix-valued measures. We show that this transformation leads to an equivalence between two distinct notions from material science and geometric measure theory respectively: conductance maximality and area criticality. Based on this relation we show a pointwise dimension bound for mixtures that attain the upper Wiener bound by applying a fractional version of the monotonicity formula for stationary varifolds. This dimension bound illustrates how the maximality condition constrains the local anisotropy and the local distribution of conductance magnitudes. Both the lower and upper attainability results have potential novel applications in modeling leaf venation patterns.

Let $A, B$ be subsets of $(\mathbb{Z}/p^r\mathbb{Z})^2$. In this note, we provide conditions on the densities of $A$ and $B$ such that $|gA-B|\gg p^{2r}$ for a positive proportion of $g\in SO_2(\mathbb{Z}/p^r\mathbb{Z})$. The conditions are sharp up to constant factors in the unbalanced case, and the proof makes use of tools from discrete Fourier analysis and results in restriction/extension theory.

Let $(\mathcal X, d,\mu)$ be an RD-space, and let $\rho$ be an admissible function on $\mathcal X$. We establish necessary and sufficient conditions for the boundedness of a new class of generalized Calderón-Zygmund operators of log-Dini type on the Hardy space $H^1_\rho(\mathcal X)$, introduced by Yang and Zhou. Our results extend and unify some recent results, providing further insights into the study of singular integral operators in this setting.

Let $E$ be a subset of the affine plane over a finite field $\mathbb{F}_q$. We bound the size of the subgroup of $SL_2(\mathbb{F}_q)$ that preserves $E$. As a consequence, we show that if $E$ has size $\ll q^\alpha$ and is preserved by $\gg q^\beta$ elements of $SL_2(\mathbb{F}_q)$ with $\beta\geq 3\alpha/2$, then $E$ is contained in a line. This result is sharp in general, and will be proved by using combinatorial arguments and applying a point-line incidence bound in $\mathbb{F}_q^3$ due to Mockenhaupt and Tao (2004).

We consider restriction analogues on hypersurfaces of the uniform Sobolev inequalities in Kenig, Ruiz, and Sogge and the resolvent estimates in Dos Santos Ferreira, Kenig, and Salo.

We consider eigenfunction estimates in $L^p$ for Schrödinger operators, $H_V=-\Delta_g+V(x)$, on compact Riemannian manifolds $(M, g)$. Eigenfunction estimates over the full manifolds were already obtained by Sogge \cite{Sogge1988concerning} for $V\equiv 0$ and the first author, Sire, and Sogge \cite{BlairSireSogge2021Quasimode}, and the first author, Huang, Sire, and Sogge \cite{BlairHuangSireSogge2022UniformSobolev} for critically singular potentials $V$. For the corresponding restriction estimates for submanifolds, the case $V\equiv 0$ was considered in Burq, Gérard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp}. In this article, we will handle eigenfunction restriction estimates for some submanifolds $\Sigma$ on compact Riemannian manifolds $(M, g)$ with $n:=\dim M\geq 2$, where $V$ is a singular potential.

This expository essay accompanied the author's presentation at the Séminaire Bourbaki on 01 April 2023. It describes the breakthrough work of Du--Zhang on the Carleson problem for the Schrödinger equation, together with background material in multilinear harmonic analysis.

In this note, we establish several interpolation inequalities in $\mathbb R^n$ in the Lebesgue spaces and Morrey spaces. By using the classical Calderon--Zygmund decomposition, we will reprove that $L^{p}(\mathbb R^n)\cap\mathrm{BMO}(\mathbb R^n)\subset L^{q}(\mathbb R^n)$ for all $q$ with $p

We introduce new function spaces $\mathcal{L}_{W,s}^{q,p}(\mathbb{R}^{n})$ that yield a natural reformulation of the $\ell^{q}L^{p}$ decoupling inequalities for the sphere and the light cone. These spaces are invariant under the Euclidean half-wave propagators, but not under all Fourier integral operators unless $p=q$, in which case they coincide with the Hardy spaces for Fourier integral operators. We use these spaces to obtain improvements of the classical fractional integration theorem and local smoothing estimates.

The book develops the fundamental ideas of the famous Kac-Rice formula for vectorvalued random fields. This formula allows to compute the expectation and moments of the measure, and integrals with respect to this measure, of the sets of levels of such fields. After a presentation of the historical context of the Kac-Rice formula, we give an elementary demonstration of the co-area formula. This formula replaces the change of variable formula in multiple integrals and a direct application of this formula gives the Kac-Rice formula for almost all levels. We emphasize the necessity of having the formula for all levels, because for some applications one needs, for example, the formula for level zero.