Papers

We derive sharp $\ell^p(\mathbb{Z})$ bounds for the $k$th derivative of the discrete uncentered maximal operator applied to characteristic functions $f:\mathbb{Z}\to\{0,1\}$ in the cases $k=0,1,2$. When $k=1,2$ these are the first sharp bounds for derivatives of a Hardy-Littlewood maximal function in continuous or discrete settings when $1<p<\infty$. We also establish several lower bounds for $k\geq 3$ and for general functions $f:\mathbb{Z}\to\mathbb{R}$.

Following Alon, Hanneke, Holzman, and Moran (FOCS 2021), we define a partial concept class (PCC) as a family of partial functions \(f: V\to\{0,1,\ast\}\); equivalently, its concepts partition the ground set into black ($f^{-1}(1)$), grey ($f^{-1}(\ast)$), and white parts ($f^{-1}(0)$). Its VC dimension is defined by shattering sets on which the value $\ast$ is not taken. We study two geometric PCCs in real Banach spaces, both with a margin \(\delta>0\): expanded half-spaces, where the grey part is a strip of width at least \(\delta\) adjacent to a half-space, and expanded balls, where the grey part is an annulus of width \(\delta\) around a unit radius ball. Our main results are dimension-free upper bounds on the VC dimension of the PCC of expanded balls in \(L_p\parenth{\mu}\), \(1\le p<\infty\), including the non-Euclidean and algorithmically particularly relevant case \(\ell^d_1\). These bounds depend on the margin and on the radii, but not on the ambient dimension or the underlying measure space. These are extensions of the work of Bourneuf, Charbit, and Thomassé (FOCS 2025) who studied the PCC of expanded balls in Euclidean space, that is, $\ell_2^d$. We also prove lower bounds on the VC dimension that match the upper bounds in terms of the margin parameter $\delta$. Finally, we derive a Dense Neighborhood Lemma in \(L_p\)-spaces, again extending the known Euclidean results. Our method relies on the linearization of the distance through a map into a space of non-trivial Rademacher type, and then the use of a balanced signed-sum estimate, or a no-dimensional Radon theorem. The arguments rely on ideas from functional analysis that are clearly explained for the non-expert in that field.

We completely characterize the $L^p$-boundedness and the weak-type (1,1) estimate of radial weighted Bergman projections on the unit disk. Our result, in particular, confirms a conjecture proposed by Peláez and Rättyä in 2021 and thereby settles a longstanding problem in the area that was formally posed by Dostanić in 2004. Consequently, we establish the dichotomy that a radial weighted Bergman projection is bounded either only for $p=2$, or for all $p\in(1,\infty)$.

We analyze a variant of the cubic-regularized Newton method for nonconvex optimization. This variant is parameter-free in that it requires no prior knowledge of problem-dependent parameters. Under the generalized smoothness condition $\|\nabla^3 f(x)\| \leq L_0 + L_1 \|\nabla f(x)\|$, we derive an oracle complexity bound for finding an $(\varepsilon, \delta)$-second-order stationary point. This assumption is weaker than the generalized smoothness conditions used in existing analyses of second-order methods, while the complexity bound improves upon existing guarantees for parameter-free second-order methods. In particular, when $L_1 = 0$, the bound matches the optimal dependence on $L_0$ as well as on $\varepsilon$, $\delta$, and the initial function value gap, up to additive logarithmic terms. To establish this bound, we derive Taylor-type inequalities and prove their equivalence to the generalized smoothness condition.

The transmission of a vertex $v$ in a connected graph $G$ is the sum of distances from $v$ to all vertices in $G$. A transmission irregular (TI) graph is a connected graph in which any two distinct vertices have different transmissions. We extend the concept of transmission to edges by defining the transmission of an edge as the sum of the transmissions of its two endpoints. A connected graph can now be called edge transmission irregular (ETI) if any two distinct edges have different transmissions. We show that almost all graphs are not ETI and then investigate several related order realizability problems involving chemical ETI graphs. In particular, we prove that for every $n \ge 15$, there exists a subcubic tree of order $n$ that is both TI and ETI.

Let $M^n$ $(n\geqslant3)$ be a closed minimal submanifold in the unit sphere $\mathbb S^{n+m}$ $(m\geqslant2)$ with flat normal bundle, and let $S$ denote the squared norm of its second fundamental form. We prove an explicit second-gap rigidity theorem for $S$. More precisely, if $S$ is constant and \[ 0\leqslant S\leqslant n+\delta, \] where $\delta$ is an explicit constant satisfying $\delta\geqslant \frac{n}{87}$, then either $S\equiv0$ and $M$ is a totally geodesic sphere, or $S\equiv n$ and $M$ is a Clifford torus contained in a totally geodesic $\mathbb S^{n+1}\subset\mathbb S^{n+m}$. %We observe that the flat-normal-bundle assumption is necessary here. The flat-normal-bundle condition is essential in the general higher-codimensional setting: without it, the corresponding rigidity statement already fails in dimension two. This theorem provides positive evidence for Chern's conjecture in higher codimension.

In this paper we establish Hardy-Rogers and Jungck type fixed point theorems in perturbed metric spaces, where the observed distance $D$ is separated from the exact metric $d$ by a nonnegative perturbation $P$. Rather than the uniform absorption of $P$ by $d$ required under domination, we examine a weaker demand, imposed only on the pairs of points that appear with a contractive coefficient. We show that without some such condition, and without a continuity hypothesis on $T$, a perturbed Banach contraction on a complete perturbed metric space may fail to have a fixed point. We further prove Ulam-Hyers stability, well-posedness, and data dependence results in which residuals are measured in the observed distance $D$, with all constants explicit, and we derive a priori error estimates for the Picard and Jungck iterations computable from observed data. The Jungck type theorem is established for weakly compatible pairs.

Let $K$ be a global function field of characteristic $p>0$ and $\ell\neq p$ be a prime number. We study Selmer groups over a $\mathbb{Z}_\ell$-extension $K_\infty/K$. For a lisse $\mathbb Z_\ell$-sheaf we prove that the Pontryagin dual of the associated Selmer group is a finitely generated torsion module over the Iwasawa algebra and has $\mu$-invariant equal to zero. This gives a positive-characteristic, prime to $p$, analogue of Greenberg's $\mu=0$ conjecture. Our result applies in particular to abelian varieties, fine Selmer groups, and adjoint representations. We also prove an analogue of the weak Leopoldt conjecture in this context over $K_\infty$, and deduce that the framed deformation ring of a residual representation is a formal power series ring. The same conclusion holds for the unframed deformation ring if the residual representation has no non-scalar endomorphisms.

We develop a new framework for the analysis of co-radiant sets based on a new notion of regularity. This framework extends the classical theory beyond co-radiant sets admitting a norm-base and provides a unified setting for separation and optimization. We introduce two new tools: inner approximating sequences and the radial profile function. The former provides a sequential approach to regularity and is the key ingredient in the proof of a general separation theorem for regular co-radiant sets, extending the existing separation theory. The latter is defined for arbitrary co-radiant sets and yields new characterizations of the existence of norm-bases, together with sequence-based characterizations of regularity under a solidness assumption on the associated cone. It also gives rise to a scalar function leading to scalarization results for approximate efficiency without additional structural assumptions on the underlying co-radiant set. Finally, we provide a variational interpretation of the separation residual by establishing radial-depth and metric error bounds. These yield sufficient conditions for approximate radial feasibility in equilibrium problems and variational inequalities, and an exact penalization principle on the zero level set of the residual.

Can a population of people not individually inclined to harm others nonetheless produce harmful collective outcomes, purely because of the institutional structure they inhabit? Social scientists have long argued yes, but existing accounts are largely qualitative and provide no precise condition distinguishing safe institutions from unsafe ones. We develop a threshold cascade model in which agents have positive activation thresholds, harmful behavior is irreversible, and the institution exerts both standing pressure and peer influence along a weighted network. We give a necessary and sufficient condition, checkable from the institution's structure and its members' thresholds, for resistance to any shock up to a given size. The criterion extends to signed influence, in which some peer effects counteract harm, and yields a convex optimization formulation for least-cost repair. It also reveals a sharp frontier between functionality and safety. An institution can coordinate its members and remain safe if and only if the exposure that coordination creates stays below the weakest member's net threshold. A further tension arises when coordination requires responsiveness to peer influence, which can make it impossible to prevent the most exposed group from cascading. We then analyze a mean-field model of two groups differing in how easily their members are pushed into harm. When one group is unstable in isolation but the system is stable under full mixing, disproportionate within-group influence creates a sharp homophily threshold beyond which the harm-free state becomes unstable. In the model, identical treatment of both groups does not generally equalize their cascade robustness.

We study the global fluctuations of the height function associated with the Airy line ensemble. Using its determinantal structure and a steepest-descent analysis of the extended Airy kernel, we prove that, after a suitable rescaling, the centered height function converges to an explicit pullback of the Gaussian free field. The convergence holds in the sense of joint moments of linear statistics against compactly supported continuous test functions.

We propose a four-field auxiliary reformulation of a time-discrete Cahn-Hilliard step. The construction is motivated by the scalar trace structure underlying two-dimensional Rafetseder-Zulehner decompositions and represents the scalar quantity $-\Delta c$ in the form $2p+div\,u$ through an auxiliary scalar field $p$ and an auxiliary vector field $u$. The resulting auxiliary problem is a mixed second-order Stokes/elasticity-type system, while the evolution equation for the phase field retains its standard mass-conserving gradient-flow structure. We derive a continuous four-field formulation that is equivalent to the classical convex-splitting mixed time step. We also state a conforming finite element discretization and prove one-step spatial estimates for the phase-field variables together with a stable auxiliary block estimate containing an explicit weak-Laplacian recovery defect. The numerical experiments verify the expected phase-field convergence in a manufactured setting, mass conservation, energy decay, and projected consistency of the auxiliary block.

It is well known that the McKean-Vlasov stochastic differential equation with a symmetric double-well potential exhibits a continuous phase transition. In contrast, for an asymmetric double-well potential, the system undergoes a discontinuous phase transition, in which an invariant measure abruptly appears in the shallower well and subsequently splits into two as the temperature decreases. In this work, we systematically investigate the types of phase transitions driven by non-convex confining potentials and cooperative interactions in $\mathbb{R}^n$. Our main results consist of a pitchfork bifurcation theorem characterizing continuous phase transitions and a saddle-node bifurcation theorem governing discontinuous phase transitions. In addition, despite the dimension-dependent nature of phase transitions, we are able to extend Dawson's criterion for phase transition points – originally formulated for quadratic interactions in $1$-dimensional space – to $n$-dimensional space. This extension directly relates the critical parameter and bifurcation direction to the eigenvalue and eigenvector of the critical covariance matrix, respectively. Finally, we apply our theoretical results to the aforementioned double-well models, to a class of four-well models in two-dimensional Euclidean space that exhibit multiple phase transitions, and to an attractive Gaussian interaction model.

We consider the ring M of column-finite matrices with integer coefficients. We prove that M-modules form an additive closed symmetric monoidal abelian category that essentially contains complete metrizable linearly topologized abelian groups as a full subcategory. We also introduce and discuss the notion of an M-ring. In the end, we show that the category of M-modules (resp. M-rings) is actually equivalent to the category of light solid abelian groups (resp. light solid rings) of Clausen and Scholze, which should come as no surprise to specialists. However, we believe that our more straightforward approach may offer a fresh perspective on their theory.

We study the geometry of reset distributions for absorbed Markov processes with geometric resetting, working at an abstract level that isolates the structural mechanism underlying reset-neutral invariance. We show that the spectral duality endows the simplex of reset distributions with a non-trivial spectral response geometry: the simplex $\Delta_{m-1}$ carries a foliation by level sets of the coupling functional $C$, organized around a critical manifold $\Sigma$. Under four structural conditions (S1)-(S4) on the coupling functional, we establish the existence and explicit characterization of the separatrix $\Sigma$, derive the invariant value $C^* = 1/(1+\sqrt{K})$, identify a projective structure in the spectral coefficients, and prove a global sign principle in the two-site case. In the canonical realization, the biased random walk with multi-site geometric resetting, numerical evidence shows that $\mathrm{sgn}(\partial_\gamma C) = \mathrm{sgn}\langle\pi-\pi^*,\psi(\gamma)\rangle$, where $\psi(\gamma)$ is the linear response functional at the separatrix; this global orientation does not, however, follow from (S1)-(S4) alone, and is dynamical rather than algebraic information. This is the third paper in a program connecting stochastic resetting with spectral theory and information geometry.

This article revisits earlier work by the last author together with Kay Magaard. We correct mistakes in results about the Sylow 2- and 3-structure of groups that act with fixity 3 and one result about possible components, we improve some results and we add details.

We introduce a class of piecewise deterministic Markov processes with resetting, motivated by self-exciting models of cyber attacks. Under assumptions reminiscent of ruin theory, we prove the existence and uniqueness of an invariant distribution and derive its density explicitly, thereby establishing ergodicity of the process. We then formulate an associated long-run average control problem in which resetting acts as the intervention mechanism. For exponentially distributed jump sizes, the model becomes analytically tractable, allowing an explicit characterization of the invariant distribution and of the optimal intervention policy.

In this paper, we obtain several characterizations of harmonic Zygmund type spaces on the real unit ball $\mathbb{B}$ of $\mathbb{R}^n$. First, we characterize the harmonic Zygmund space $\mathcal{HZ}^\alpha$ ($0<\alpha\leq2$) in terms of Zygmund type conditions. Our result extends Theorem 3.4 of [M. Aljuaid and F. Colonna, On the harmonic Zygmund spaces, Bull. Aust. Math. Soc. 101 (2020), 466–476.] to the setting of the real unit ball $\mathbb{B}$. Second, we establish an analogue of the Holland-Walsh characterization of the harmonic Zygmund space $\mathcal{HZ}$. Finally, an integral characterization of $\mathcal{HZ}^\alpha$ is also discussed. All obtained results can be viewed as counterparts of the known results for Bloch spaces.

We construct the planar Linares–Otto–Tempelmayr Hopf algebra, thereby filling the missing planar noncommutative multi-index corner in the square relating the LOT, Butcher–Connes–Kreimer, and Munthe-Kaas–Wright Hopf algebras. Starting from the free associative algebra on a weighted alphabet $\mathbb Z_{\ge -1}\times A$, we define an insertion-type product yielding a post-Lie structure on the Lie algebra generated by the linear span $V(A)$ of weight $-1$ monomials whose proper left prefixes all have nonnegative weight, and the Guin–Oudom construction then produces the planar LOT Hopf algebra. We introduce a planar tree fertility map from decorated planar rooted trees to monomials in $V(A)$, prove that it is a linear isomorphism, and obtain a natural Hopf algebra isomorphism with the Munthe-Kaas–Wright Hopf algebra. We further derive an explicit coproduct formula in terms of left-admissible cuts, establish the extraction-contraction coproduct, and construct a word symmetrization operator compatible with the classical tree symmetrization operator.

The goal of this paper is to determine how the family of ideals of star configurations intersects with the class of geometrically vertex decomposable ideals. The main result of this paper shows that the answer is subtle since the geometrically vertex decomposability property of an ideal is not invariant under a linear change of variables, and thus the answer will depend upon the choice of the linear forms that define the ideal of the star configuration. We also show that the ideal of a star configuration is a Knutson ideal precisely when it is a geometrically vertex decomposable ideal.

We show that the second integral homology group of the link of an isolated compound Du Val (cDV, for short) singularity of type $cA_n$ is either trivial or a torsion-free abelian group. Consequently, by a result of Smale, it follows that the link is either $S^5$ or a connected sum of finitely many copies of $S^2\times S^3$. We also determine the rank of the second integral homology group of the link of a singularity of type $cD_n$ with $n>4$ under the assumption that the singularity is Newton non-degenerate. Furthermore, we focus on the weighted homogeneous case and determine the homology group of the link, including its torsion subgroup, under the assumption that the singularity is a Thom-Sebastiani sum of singularities of Brieskorn-Pham, cyclic, or chain type.

We consider a general class of first order ODE systems for the evolution of $N$ interacting particles (in Euclidean space $\mathbb{R}^d$) in a mean-field regime. The class of interactions treated includes singular interactions of inverse power type up to power $d+1$, attractive or repulsive, and not necessarily deriving from a potential – unlike, for instance, the modulated energy method. We introduce a new method to prove quantitative convergence of the discrete system to solutions of the mean-field equation. It relies on studying the evolution of a metric encoding a multiscale control of the difference between the empirical measure and its limit, via mollification by heat kernels. We prove that the desired convergence holds (i) up to the maximal time of existence of the smooth solution to the limiting equation if the singularity is sub-coulombic in any dimension, or coulombic in dimensions 1 and 2 (where, to do so, we introduce a notion of weak solution to the ODE system), or (ii) for short time in the case of Coulomb singularity in dimension 3 and above and (iii) up to a short $N$-dependent timescale for super-coulombic interactions in all dimensions. The latter two results are demonstrated to be optimal as we prove that collisions occur within the same timescale for a class of attractive interactions.

For each $1\le p\le\infty$ and $j=1,2$, let $AC^p(\Omega_j)$ denote the Banach space of complex-valued absolutely continuous functions on a closed unit interval $\Omega_j=[x_j,x_j+1]$. We equip $AC^p(\Omega_j)$ with the $p$–norm $\|f\|_{AC,p}$, and the order $\ge_{AC}$ defined by $f(x_j)\ge0$ and $f'\ge 0$ a.e. Set $$S(AC^p(\Omega_j))^+=\{f\in AC^p(\Omega_j):\|f\|_{AC,p}=1,\ f\ge_{AC}0\}.$$ We prove that, for each $1\le p\le\infty$, every surjective isometry $S(AC^p(\Omega_1))^+\to S(AC^p(\Omega_2))^+$ extends uniquely to a complex–linear isometric order isomorphism from $AC^p(\Omega_1)$ onto $AC^p(\Omega_2)$. As an application, we obtain a corresponding extension theorem for surjective phase–isometries.

A Hopf monoid structure on extended generalized permutahedra (EGP) was recently introduced by M.Aguiar and F.Ardila. We investigate the existence of a cointeracting bialgebra structure on EGP's. We show that a suitable notion of cointeraction exists, not in the classical comodule sense, but via the framework of measuring algebras. The comodule-type map assigns to each polyhedron the sum of pairs of face and tangent cone at the face. EGP's and affine cone EGP's form the cointeracting bimonoids in species with EGP as a third measuring structure. EGP's are in bijection to extended submodular functions. For an EGP, we also describe explicitly the submodular functions of its faces and tangent cones. The braid fan and its relation to preorders play a key role in this description.

We obtain a Wei-type duality between the footprint bound and the dual footprint bound for the generalized Hamming weights of an evaluation code. This duality applies between the Andersen-Geil and Feng-Rao bounds as well. We also prove that the footprint and dual footprint bounds cannot be used to guarantee the asymptotic goodness of a family of evaluation codes.

We study the density patch problem for the two-dimensional inhomogeneous incompressible Navier–Stokes system with vacuum, for initial data consisting of a Lipschitz density patch and a divergence-free velocity field in $L^2(\mathbb{R}^2)$. We establish uniqueness of solutions at the natural energy level, thereby concluding the global well-posedness for $L^2$ data. Furthermore, we prove a log-Lipschitz estimate for the velocity field, extending the classical result of Chemin-Lerner to the inhomogeneous setting. As a consequence, the associated flow belongs to $L^\infty_t C_x^{1-\varepsilon}$. for any $\varepsilon \in (0,1)$, ensuring that the patch boundary remains a continuous curve of Hausdorff dimension $1$, thus preserving its initial dimension for all time.

We prove the existence of spectral submanifolds in nonlinear partial differential equations (PDEs) describing forced-damped continuum vibrations. Our results cover structural vibrations subject to generalized structural damping. Based on these results, backbone curves and forced response curves can be rigorously extracted from PDEs, without a priori discretization. On two examples involving an elastic beam and a thin plate, we show that backbone and forced response curves can even be calculated by hand directly from the PDEs.

In this paper, we prove a Second Main Theorem type inequality for holomorphic curves into the complex projective plane intersecting a Fermat-type curve.

These are lecture notes for the first part of a short course that I taught jointly with Kaibo Hu as part of the thematic program "Differential Complexes: Theory, Discretization, and Applications" at the Erwin Schrödinger Institute (ESI) in Vienna. We give an introduction to differential forms and the construction of Berstein-Gelfand-Gelfand (BGG) complexes on open domains in R^n, focusing on the role of representation theory of semisimple Lie groups and Lie algebras in the construction.

We extend to arbitrary disconnected reductive $p$-adic groups the duality on the category of smooth finite-length complex representations defined by Aubert, as well as its cohomological analog defined by Bernstein, and prove various properties of these functors, such as uniqueness, preservation of irreducibility, compatibility with parabolic induction and restriction, and a character formula. As a sample application, we obtain a definition of the Steinberg representation for a disconnected reductive $p$-adic group, compute its character, and discuss its twisted endoscopic properties.