Papers

We establish the complete integrability of the defocusing Calogero–Sutherland derivative nonlinear Schrödinger equation \begin{equation}\label{CS-abstract}\tag{CS} i\partial_t u +\partial_x^2 u + 2i u \partial_x\,\Pi\!\left(|u|^2\right)=0,\qquad x\in\mathbb T, \end{equation} by constructing a nonlinear Fourier transform \begin{equation*} \Phi : u\in L^2_+(\mathbb T)\mapsto\ (\zeta_n(u))_{n\geq 0}\in \ell^2(\mathbb N_0),\qquad \zeta_n(u)=\sqrt{\gamma_n(u)}\;e^{\,i\arg\langle u\mid f_n\rangle}, \end{equation*} built from the eigenvalues $(\lambda_n)$, the spectral gaps $\gamma_n:=\lambda_n-\lambda_{n-1}-1\geq 0$ and the eigenfunctions $(f_n)$ of the Lax operator $L_u$. We prove that $\Phi$ is a norm-preserving homeomorphism. In these coordinates, the flow becomes the explicit rotation $\zeta_n(t)=e^{-i\omega_n t}\zeta_n(0)$ for all $n$, whose frequencies $\omega_n$ depend only on the conserved spectrum $(\lambda_n)$. As consequences, every orbit of \eqref{CS-abstract} is precompact in $L^2_+(\mathbb T)$, and $t\mapsto u(t)$ is Bohr almost periodic. For finite-gap data, we obtain quasi-periodicity in time and a sharp criterion for genuine periodicity.

We consider sparsely observed functional data generated by a latent Gauss-Markov process modeled through a linear stochastic differential equation (SDE). Motivated by an application to tumor growth data, we allow for outcome-dependent dropout, where sampling terminates once the most recent observation exceeds a prescribed threshold. Such observation schemes arise naturally in longitudinal studies and violate the fundamental missing completely at random assumption commonly imposed in the analysis of partially observed functional data. We develop a likelihood-based estimation framework that, in contrast to existing moment-based methods, avoids the bias induced by outcome-dependent dropout. We establish convergence rates for the proposed estimators and show that they are mini-max optimal up to logarithmic factors, with the diffusion coefficient admitting a faster rate than the drift coefficients under a random initial condition for the SDE. An application to tumor growth data further demonstrates the advantages of a fully probabilistic functional data model for tasks beyond second-order inference, including prediction bands, first-passage times, and tumor-age estimation.

In this paper, we study modulated $(C,\alpha)$-ergodic averages with operator-valued weights and their subsequential versions in the setting of semifinite von Neumann algebras. Our framework combines $(C,\alpha)$-type modulation with operator-valued weights and extends both scalar-weighted and operator-valued weighted ergodic theories. We establish mean ergodic theorems and maximal inequalities for these averages on noncommutative $L_p$-spaces. We then study bilateral almost uniform and almost uniform convergence in noncommutative Orlicz spaces, including convergence along subsequences of density one. A key ingredient in our approach is the introduction of the $p$-convexity condition on the underlying Orlicz function. This condition enables us to transfer appropriate maximal estimates from noncommutative $L_p$-spaces to establish bilateral uniform equicontinuity in measure at zero on the corresponding noncommutative $p$-convex Orlicz spaces. Together with the noncommutative Banach principle, this yields the desired pointwise convergence results. As an additional consequence of our $p$-convexity argument, we extend Rota's theorem to a larger class of noncommutative Orlicz spaces associated with the $p$-convex Orlicz functions considered in this article.

We study the functional equation $ f \circ p = g \circ q,$ where $ f,$ $g,$ $p,$ $q$ are branched covering maps between closed surfaces. In particular, we extend results on semiconjugate holomorphic maps between compact Riemann surfaces to this broader setting. As an application, we present an alternative proof of the fact that every quotient of a torus endomorphism has a parabolic orbifold, along with some extensions of this result.

We consider a Whitham-Boussinesq type system with surface tension arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water, which is characterized by a nonlinearity parameter $0<\epsilon\le 1$, a shallow water parameter $0<\mu\le 1$ and a nonnegative surface tension parameter $\beta$. In this paper, we establish well-posedness of the associated Cauchy problem for $\beta>1/3$ on a time scale of order $ ( \beta_\ast^{ 1/4} \mu^{ 1/4} h_0 \epsilon^{-1} )^{4/3}$ in one dimension and of order $( \beta_\ast^{ 1/4} \mu^{ 1/4} h_0 \epsilon^{-1})^{2-}$ in two dimensions, where $\beta_\ast= \min(|3\beta-1|, 1/2)$, assuming a non-cavitation condition with a parameter $h_0>0$. In addition, we established well-posedness for the classical Whitham equation with either $\beta=0$ or $\beta>1/3$ on the timescale of order $ ( \beta_\ast^{ 1/4} \mu^{ 1/4} \epsilon^{-1} )^{4/3}$. These results explicitly capture how the existence time depends on all the parameters involved. Our proofs rely on dispersive and Strichartz estimates combined with energy estimates.

We study complex geodesic actions and heat-kernel resurgence on isotropic Heisenberg groups. For real endpoints with $r>0,y\ne0$, the nonzero finite singularities reached by analytic continuation of the minimizing-saddle Borel germ are exactly the finite complex action differences. For nonresonant endpoints, we construct row-finite directional Stokes operators and compute the positive-ray Stokes matrix and its logarithm explicitly; the primitive alien support can be strictly smaller than the full Borel singular support. Near the vertical axis, we obtain uniform two-term large-order asymptotics from the nearest Borel singularity. Cubic resonances admit uniform $A_2$ confluence formulas. We also identify the combination of resurgent sectors selected by the original Fourier contour. Positive Borel summation is exact at horizontal endpoints. At nonzero isotropic complex endpoints, no finite normal trajectories exist, but the signed vertical action lattice is the complete limiting spectrum. The action and Stokes structures are independent of the dimension on isotropic $H^n$.

We study a circular variant of the $s$-choice parking model: $n$ cars park on $m=n+1$ spots arranged on a circle, each car carrying an anchor and $s-1$ clockwise increments at least $d$ apart with return gap at least $d$; a car tries its choices in order and then sweeps clockwise from its last choice. On the circle every car parks and exactly one spot remains empty. Exploiting rotational symmetry in the spirit of Pollak's proof of the count $(n+1)^{n-1}$, we prove that the empty spot is exactly equidistributed, which yields the closed formula $m^{n-1}\binom{m-sd+s-1}{s-1}^{n}$ for the number of preferences leaving any prescribed spot empty. This appears to be the first closed product formula in the multi-choice parking landscape. We further prove a refinement: within every class of preferences with prescribed increments, the empty spot is still exactly equidistributed, which explains the product structure of the formula and, for $s=2$, $d=1$, the appearance of the classical count $(n+1)^{n-1}$ as a factor of $(n+1)^{n-1}n^{n}$. The admissible tuples are enumerated through Kaplansky's lemma on circular selections, and all results are verified by exhaustive computer enumeration.

Let $E$ be the exterior algebra on $n$ generators over a field, and let $F$ be a graded free $E$-module with $r$ generators, of degrees $d_1 \le \dots \le d_r$. We determine the convex hull of the set of Hilbert functions of the quotients $F/M$, where $M$ runs over the monomial submodules of Amata and Crupi. The hull is the Minkowski sum of $r$ shifted copies of one and the same simplex: the Kozlov simplex, which Kozlov proved in 1997 to be the convex hull of the $f$-vectors of simplicial complexes on $n$ vertices. For $r=1$ the statement is Kozlov's theorem, so this is a rank-$r$ generalization of it. We give a self-contained proof, including a short proof of Kozlov's theorem from the local Lubell-Yamamoto-Meshalkin inequality, and an explicit facet description of the Kozlov simplex that appears not to be recorded in the literature. Along the way we isolate the class of submodules for which the argument works, the ideal-direct-sum modules, and show by example that the Hilbert function of a graded submodule outside that class need not be a shifted sum of $f$-vectors at all. The combinatorial content is stated separately, in language that uses no algebra, as a result on $r$-vectors of simplicial complexes and their $ff$-vectors. Finally we describe the vertex structure of the Minkowski sum – which sums of vertices of the summands survive as vertices – proving the answer for the all-ones leg vector at every rank and for the Kozlov leg vector at rank two.

Autocatalysis, the ability of a chemical subsystem to sustain its own constituents when supplied with sufficient food molecules, has been closely related to the origin of life on Earth. Emerging from Wilhelm Ostwald's considerations about an explicit autocatalytic reaction, different notions of autocatalysis have been developed over the years. The two most prominent are reflexively autocatalytic F-generated sets (RAFs) and stoichiometric autocatalysis. After having shown that each RAFs is, under reasonable conditions, in general stoichiometrically autocatalytic, we examine here the relationship between the two notions of minimality: irreducible RAFs and autocatalytic cores. To this end, we overcome the obstacle that RAFs and stoichiometric autocatalysis have been formalized in distinct systems of chemical reactions, i.e., catalytic reaction systems (CRS) and chemical reaction networks (CRNs), respectively. We show that reactions in a CRS constitute equivalence classes of reactions of the corresponding CRNs w.r.t. their specific catalyzations. Using the fact that CRN and CRS can be canonically identified whenever each CRS reaction is associated with a single catalyzation, we demonstrate that the Kőnig graph of a monocatalyzed, irreducible RAF is composed of strong blocks devoid of food and waste species that are separated by reaction vertices, each of which contains an autocatalytic core. In fact, a single irreducible RAF can, in general, contain exponentially many autocatalytic cores.

We prove a rigidity theorem for closed Einstein manifolds of dimension $n\ge6$ under two lower bounds on the curvature operator of the second kind. More precisely, for $L_1, L_2\ge0$ and some real number $\alpha>1$ in a suitable range, we consider \[ \lambda_1\ge-L_1\bar\lambda, \qquad \frac1\alpha\sum_{j=1}^{\alpha}\lambda_j \ge-L_2\bar\lambda. \] Here $\lambda_1\le\cdots\le\lambda_N$ are the eigenvalues of the curvature operator of the second kind $\mathring R$, $N=(n-1)(n+2)/2$, and $\bar\lambda=N^{-1}\sum_{j=1}^N\lambda_j$. Let $\theta(n, \alpha)$ be the constant appearing in \cite[Theorem~1.1]{CW26}; see (2). In the parameter range considered here, $L_2>\theta(n, \alpha)$, so the second condition is strictly weaker than the corresponding condition in \cite{CW26}, while the first condition and that condition do not imply each other. Under an additional explicit relation between $L_1$ and $L_2$, we prove that the manifold is either flat or a spherical space form.

We prove Kahn's flow conjecture, which is a strong form of Chvátal's conjecture on the largest intersecting subfamily of a downset.

In this paper, we study the Dirichlet eigenvalue problem for the logarithmic Hardy operator, defined as the logarithmic Laplacian with a critical logarithmic potential, in a bounded Lipschitz domain containing the origin. We first establish a logarithmic Hardy inequality and use it to identify the associated energy space, showing that the embedding into \(L^2\) is compact when the perturbation parameter exceeds the endpoint value \(-1\), but fails to be compact at that endpoint. For parameters above \(-1\), we develop a complete variational spectral theory, prove that the problem admits a discrete sequence of eigenvalues tending to infinity, characterized by successive minimizations over orthogonal complements, whose eigenfunctions form a complete orthonormal basis of \(L^2\). We further establish a uniform lower bound for the first eigenvalue, a scaling property under domain dilation that determines a critical radius for the sign of the first eigenvalue and leaves the eigenvalue gaps invariant.

This paper investigates the $L_p$-dual mixed Euclidean-Gauss Minkowski problem associated with the $L_p$-dual mixed Euclidean-Gaussian curvature measure. The solvability of this problem is equivalent to that of a class of Monge-Ampère type equations on the unit sphere. Under smooth assumptions, We establish the existence of solutions for $p<1$, $q<0$, and $q<p$ and uniqueness of solutions for $p\geq q$ to the aforementioned Monge-Ampère type equations, respectively. Furthermore, we obtain a complete solution concerning the existence part of this problem when $p<1$ and $q<0$.

In the 1950s and 1960s, Agrawal introduced a class of experimental designs that later became known as triple arrays. Gordeev, Markström and Öhman proposed near triple arrays by relaxing all three intersection properties of triple arrays, allowing two values concentrated around the average intersection size, as well as two consecutive values for the replication numbers. They completely resolved the existence of near triple arrays with three rows, showing that there exists a $(3\times c,v)$-near triple array if and only if $v\geq c\geq 3$ except for $(c,v)\in\{(3,6),(4,6),(5,8)\}$. In this paper, we further investigate the existence of near triple arrays with four rows and prove that there exists a $(4\times c,v)$-near triple array if and only if $v\geq c\geq 4$ except for $(c,v)\in\{ (4,9),(5,7),(5,10),(6,8),(7,9),(10,12),(11,13)\}$.

We study finite left-nested Jordan products of idempotents in finite-dimensional simple Jordan algebras over a field $\mathbb{F}$ of characteristic different from $2$. Over an algebraically closed field, the structure theorem reduces the problem to the scalar algebra, spin factors, the full matrix Jordan algebras $M_n(\mathbb{F})^+$, the symmetric matrix algebras $H_n(\mathbb{F})$, the symplectic Hermitian algebras $H_{2n}(\mathbb{F},\mathrm{sp})$, and the split Albert algebra. We determine exactly which elements are representable: except in $H_2(\mathbb{F})$, these are all nonscalar elements together with $0$ and $1$, whereas in $H_2(\mathbb{F})$ the nonzero square-zero elements are also not representable. We also study the minimal word length of representable elements. For $M_n(\mathbb{F})^+$ over an arbitrary field, the maximal word length grows logarithmically in $n$, and for $n=2$ we determine the exact length of every nonscalar matrix. The same logarithmic growth holds for $H_{2n}(\mathbb{F},\mathrm{sp})$. The spin-factor and symmetric-matrix results hold over quadratically closed fields, whereas the full matrix, symplectic Hermitian, and split Albert results hold over arbitrary fields of characteristic different from $2$.

We establish sharp sampling and interpolation criteria for the radial–angular Gaussian mixed-norm Fock spaces $\mathcal F_\alpha^{p,q}$, where $\alpha>0$ and $0<p,q\le\infty$. The natural point-evaluation scale leads to the normalized restriction $R_{\alpha,\Lambda}^{p,q}f = \bigl(f(\lambda)e^{-\frac{\alpha}{2}|\lambda|^2}(1+|\lambda|)^{\frac{1}{q}-\frac{1}{p}}\bigr)_{\lambda\in\Lambda}$. With this normalization, the critical density is the classical value $\frac{\alpha}{\pi}$ throughout the Banach and quasi-Banach ranges. For an arbitrary locally finite set $\Lambda$, sampling is equivalent to $D_{\mathrm{sep}}^{-}(\Lambda)>\frac{\alpha}{\pi}$, together with relative separation when $p<\infty$; no relative-separation condition is required when $p=\infty$. Here $D_{\mathrm{sep}}^{-}$ is the supremum of the lower Beurling densities over separated subsets of $\Lambda$. Interpolation is equivalent to separation and $D^{+}(\Lambda)<\frac{\alpha}{\pi}$. The same criterion characterizes interpolation for the little endpoint $f_\alpha^{p,\infty}$, and in both settings the normalized restriction map admits a bounded linear right inverse. The sufficiency arguments are based on rapidly localized Hilbert dual and Lagrange atoms together with a weighted localized synthesis theorem on the full mixed-norm scale. For necessity, we prove that lower stability of a rapidly localized matrix on a weighted annular mixed sequence space implies lower stability on $\ell^2$. The proof first passes to $\ell^\infty$ by translated polynomial cutoffs and a commutator estimate, and then uses the $p$-independence theorem for the Sjöstrand class. Applied through a fixed Hilbert sampling lattice, this reduces the strict density conditions to the classical Hilbert Fock sampling and interpolation theorems.

We reconsider the theory of probabilistic formal languages generated by n-gram models and by probabilistic context-free grammars (PCFGs). The expected hierarchy of probabilistic grammars is established by proving that every probabilistic language generated by an n-gram model is also generated by some PCFG, while some probabilistic languages generated by PCFGs cannot be generated by any $n$-gram model. We introduce the notion of fully connected PCFGs, namely PCFGs in Chomsky normal form where every production rule only involving non-terminals has non-zero probability. Our main result shows that any probabilistic language generated by an $n$-gram model differs from any probabilistic language generated by a fully connected PCFG. Therefore, the class of probabilistic languages generated by $n$-gram models is not a subset of the class generated by fully connected PCFGs.

In this paper, we establish two classes of refined sum formulas for finite multiple zeta values of level two, a variant of finite multiple zeta values recently introduced by M. Kaneko et al. One is an explicit formula for the sum of level-two finite multiple zeta values over indices in which one prescribed component is odd and all the remaining components are even. The other is an Ohno–Zagier type relation for level-two finite multiple zeta values, which gives a generating function for sums of fixed weight, depth and height in terms of depth-one values. As applications, we derive sum formulas for fixed weight and depth and for several special heights.

Let $G$ be a split reductive group defined over a field of characteristic bigger than the Coxeter numbers of its simple factors. We identify the Harish–Chandra centers of the associated Kac–Moody vertex algebras and enveloping algebras at noncritical levels with algebras of functions on moduli spaces of connections for the Langlands dual group $\check{G}$. Namely, given a noncritical level $\kappa$ for $G$, consider the associated Kac–Moody vertex algebra $V_\kappa(\mathfrak{g})$ defined on a formal disc $\mathscr{D}$, and its Harish–Chandra center of arc group invariants $$V_\kappa(\mathfrak{g})^{\mathscr{J}G} \subset V_\kappa(\mathfrak{g}).$$ Write $\check{\kappa}$ for the dual level for $\check{G}$, $\check{\kappa}^p - \check{\kappa}$ for its image under the Artin–Schreier map, and $\operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\check{\kappa}^p - \check{\kappa}}$ for the moduli space of $(\check{\kappa}^p - \check{\kappa})$-opers on the Frobenius twisted disc $\mathscr{D}^{(1)}$. We establish a canonical isomorphism $$\operatorname{Spec} V_\kappa(\mathfrak{g})^{\mathscr{J} G} \simeq \operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\check{\kappa}^p - \check{\kappa}}.$$ There is a similar identification for the Harish–Chandra center of the filtered complete enveloping algebra at level $\kappa$, where instead of $\mathscr{D}$ we need to consider the punctured disc $\mathscr{D}^\times$. The presence of these Harish–Chandra centers is a genuinely new phenomenon for loop groups at noncritical level in positive characteristic: these centers are nontrivial, unlike for loop groups at noncritical level in characteristic zero, and in particular are not the reductions mod $p$ of the characteristic zero Harish–Chandra centers, unlike for finite dimensional reductive groups or loop groups at critical level.

In this paper, we study strictly weakly integral fusion categories of rank 6 and give a complete classification of their Grothendieck rings. We obtain twelve candidate Grothendieck rings. Eight of them can be realized by fusion categories, while the remaining four are ruled out by necessary obstructions to categorification. Together with previously known results, this completes the classification of Grothendieck rings of strictly weakly integral fusion categories of rank at most 6.

We prove that an operator-norm separated family of matrices satisfies $\mathbb{E}\sup_{A\in T} G^{T}AG' \geq ca\log |T|$, where G,G' are independent standard Gaussian vectors and a is the separation. The main information estimate concerns arbitrary separated coisometries: conditional entropy is bounded by a source-dependent operator energy times $\log|T|$, up to an additive quadratic term in the common row dimension. An adaptive Gaussian experiment proves this estimate by charging actual information increments to one weighted posterior-entropy potential. Convex separation and a Gaussian covering estimate then yield a bounded-radius result. To reach the general case, we first choose an operator scale preserving the Sudakov ratio, apply the known Hilbert-Schmidt minoration, and recompute a common Gaussian block compression at the retained entropy. This ordering preserves the normalization needed by the coisometry argument.

We study second- and higher-order methods for solving smooth monotone variational inequalities (MVI). Monteiro and Svaiter (SIAM J. Optim., 2012) showed that a second-order method, NPE, converges at a rate of $\mathcal{O}(T^{-1.5})$. For convex-concave minimax optimization, a subclass of MVI problems, Chen, Liu, Luo, and Zhang (COLT 2025) recently improved this rate to $\tilde{\mathcal{O}}( T^{-1.75})$. However, the result has a substantial gap compared to the lower bound of $\Omega(T^{-2.5})$ established by Chen et al. (2026). In this paper, we propose a novel second-order method that achieves the optimal rate of $\mathcal{O}(T^{-2.5})$. Our algorithm also extends to higher-order methods: for any integer $ p \ge 1$, we obtain a $p$th-order method with a convergence rate of $\mathcal{O}(T^{-(3p-1)/2})$, matching the known lower bounds and therefore establishing optimal rates across all orders.

We construct gauge-origami systems on the toric Calabi–Yau fourfold $\mathcal{C}\times\mathbb{C}$ from its brane brick model. Brick matchings and phase boundaries determine elementary D6- and D4-brane framings, respectively, realizing the magnificent-four, tetrahedron-instanton, and spiked-instanton sectors in a unified framed quiver quantum mechanics. We define their partition functions by Jeffrey–Kirwan residues and describe the fixed points in terms of colored BPS crystals. General gauge-origami configurations are obtained by combining these elementary sectors through their common fractional D0-brane quiver. This construction provides a first step toward gauge origami on general toric Calabi–Yau fourfolds.

One novel monotonous finite volume element (MFVE) scheme is put forward for convection dominant diffusion problem. The main contributions of this paper include four aspects. Firstly, one upwind sub-control volume, called upwind volume, is introduced for the discretization of the convection term similar to the upwind element, which leads to the upwind property. Secondly, one first-order linear discrete operator for the convection term in the balance equation is directly obtained by numerical integration in upwind volume, which is different from the classical finite volume methods.Thirdly, one asymptotic expansion of gradient functions in the dual element of each node is derived by one extension interpolation function. Then, one second-order nonlinear discrete operator for the convection term is constructed by the asymptotic expansion and error estimation of its FVE solution, where some perturbed coefficients are skillfully put into the expansion, which serves as a high-order correction. Fourthly, one novel two-order MFVE scheme, also accompanied by another one-order MFVE, is designed for convection-dominated diffusion problems together with one positivity-preserving finite volume element (PFVE) discrete diffusive operator. The L2 norm of the error of the approximate solution is derived. Finally, numerical results validate theoretical conclusions.

The classical semiparametric convolution theorem characterizes the limiting distributions of regular estimators of pathwise differentiable parameters. We extend this analysis to directionally pathwise differentiable functionals. Regularity along a subspace or convex cone forces the directional derivative to agree there with a bounded linear functional, and every such estimator has a Gaussian convolution factor determined by that functional. We characterize the relevant subspaces, compare their efficiency bounds, and identify the influence functions of efficient estimators. These comparisons show when a stronger regularity requirement increases the variance bound. Under an additivity condition, the Gaussian limit experiment separates into a linear estimation problem and a nonlinear one. Their minimax risks add under squared error; for more general losses, the Gaussian component enters through convolution of the loss. Applications to treatment values, instrumental-variable bounds, and calibration give explicit efficient influence functions, compare existing procedures, and establish feasible attainment in finite-stratum models.

We disprove Laugesen's conjecture that the ratio of the first two Robin eigenvalues decreases with the positive boundary parameter. On any prescribed compact interval of positive parameters, the ratio is strictly increasing on a suitable connected polyhedral domain in every dimension at least two; in the plane, the domain can be a simply connected polygon. In contrast, we prove strict decrease on every rectangular box. Both results follow from a comparison of the relative growth of interval eigenvalues.

A tree poset $T$ is a poset whose Hasse diagram is a tree. Bukh proved that if a family $\mathcal F\subseteq 2^{[n]}$ contains $(h(T)-1+\varepsilon)\binom{n}{\lfloor \frac{n}{2}\rfloor}$ sets, then $\mathcal F$ contains a weak copy of $T$, where $h(T)$ is the height of $T$, the number of elements in a longest chain of $T$. Several strengthenings and generalizations of this result have been obtained. We prove the following robust variant. For a tree poset $T$ and $x\in T$, the $b$-blow-up $T(x,b)$ rooted at $x$ is the tree poset that we obtain from $T$ by replacing every element $u$ by $b^d$ new elements, where $d$ is the distance $d(x,u)$ in the Hasse diagram of $T$ and an edge $uv$ with $v$ being closer to $x$ is replaced by edges such that every new copy of $v$ is joined to $b$ new copies of $u$ such that these new copies form pairwise disjoint sets for the copies of $v$. We prove that for any tree poset $T$, $x\in T$, and $\varepsilon>0$ there exists $\delta$ such that if $\mathcal F\subseteq 2^{[n]}$ contains $(h(T)-1+\varepsilon)\binom{n}{\lfloor \frac{n}{2}\rfloor}$ sets, then $\mathcal F$ contains a weak copy of $T(x,\lfloor \delta n\rfloor)$. This settles a conjecture of Treglown and the author. As applications, we derive the known asymptotic counting and random versions of Bukh's theorem from this stronger embedding result, and obtain new maximal anti-Ramsey results for tree posets.

We present a spectral signing algorithm solving the Komlós problem with a constant discrepancy in polynomial time. Given a matrix $A\in\mathbb{R}^{m\times n}$ whose columns have Euclidean norm at most $1$, the algorithm finds a vector $\varepsilon\in\{-1,1\}^n$ satisfying $\|A\varepsilon\|_\infty\le C$, where $C$ is an absolute constant. By minimizing a cubic spectral potential, our spectral signing algorithm updates the fractional coloring toward Boolean signs with time complexity $O((mn^9+n^{10})\log(2+m+n))$.

Let $\pi:\X\to\D$ be a proper holomorphic submersion with connected fibers, and suppose that $X_t$ is biholomorphic to a fixed smooth Moishezon manifold $S$ for every $t\ne0$. If $\kappa(S)\ge0$, we prove that $X_0\simeq S$.

Let $I_{\inf}(a):=\lim_{n\to\infty}I_{n}(a)=\inf_{n\ge1}I_{n}(a)$ be the asymptotic isoperimetric profile for product of a weighted Riemannian manifold satisfying $\mathrm{CD}(0,\infty)$ and, more broadly, for product of a nonsmooth subspace with density that can be approximated by a sequence of densities satisfying $\mathrm{CD}(0,\infty)$. We establish the following identity: \[ \lim_{a\downarrow0}\frac{I_{\inf}(a)}{a\sqrt{2\ln(1/a)}}=\sqrt{K_{\mathrm{LS}}}, \] where $K_{\mathrm{LS}}$ is the optimal log-Sobolev constant. The same argument also yields the large-deviation and moderate-deviation asymptotics for isoperimetry.