Papers

Simulating elastic wave propagation in heterogeneous media presents significant mathematical and computational challenges, since resolving fine-scale material variations requires extremely fine spatial discretizations, while mixed stress–displacement formulations typically lead to large saddle-point systems that are not well suited for efficient time integration. Moreover, conventional multiscale reductions generally produce nontrivial coarse mass matrices, thereby requiring additional linear solves at every time step and diminishing the advantages of explicit schemes. To address these difficulties, we develop a time explicit multiscale method based on a multipoint stress control volume discretization. The central innovation is a unified spatial–temporal reduction strategy that simultaneously removes two dominant algebraic bottlenecks: the complex global fine-scale mixed solve and the repeated coarse-scale mass solve arising in conventional multiscale time stepping. More specifically, the stress and auxiliary rotation variables are eliminated locally, leading to a symmetric positive definite reduced stiffness operator, upon which a multiscale space is constructed using a density-weighted local spectral problem. The associated projected mass bilinear form preserves the physical mass inner product and gives an identity mass matrix under a density-orthonormal auxiliary basis, leading naturally to an explicit central-difference scheme. We establish discrete energy stability under a coarse-scale CFL condition and derive convergence estimates for both the density-weighted displacement and the locally recovered stress. Numerical experiments in heterogeneous media validate the theoretical results and demonstrate the effectiveness of the proposed method for elastic wave propagation.

This paper aims to develop new and efficient distributed algorithms for solving a class of monotone inclusions, $0 \in \sum_{i=1}^n (G_ix + T_ix)$, over a connected network of $n$ agents, where the single-valued operator $G_i$ and the possibly multivalued operator $T_i$ remain private to agent $i$. Existing distributed algorithms for this problem class are primarily non-accelerated, and their exact convergence rates in the original primal space are largely unexplored. To bridge this gap, we propose two Decentralized Fast Fixed-Point-based algorithms, ND-DFFP and NI-DFFP, which integrate Nesterov-type acceleration with primal-dual techniques under two prominent settings: (i) \textit{Lipschitz continuity of $G_i$ and maximal monotonicity of $G_i+T_i$}; and (ii) \textit{co-coercivity of $G_i$ and maximal monotonicity of $T_i$}. While ND-DFFP utilizes a homogeneous network-dependent stepsize, NI-DFFP reformulates the problem into a three-operator inclusion to decouple the network topology, enabling heterogeneous network-independent stepsizes. Under appropriate assumptions, we establish an $\mathcal{O}(1/k)$ convergence rate for the consensus error and an $\mathcal{O}(1/k)$ rate for both the restricted gap function and the squared forward-backward splitting residual, with the latter two metrics evaluated at the network-average iterate or its projection onto the effective domain. Finally, numerical experiments on distributed bilinear matrix games and a virtual power plant problem demonstrate the competitive performance and computational efficiency of our methods over recent decentralized baselines in the literature.

We study boundary transmission and excursion geometry in the planar Lorentz mirror model through finite-volume numerical experiments. Cylinder penetration quantiles grow approximately linearly with circumference over the simulated range, with a systematic upward drift in the normalized quantiles when vacancies are present. Estimated slab-transmission probabilities yield exponential confinement bounds with quantified statistical confidence at the tested widths. Planar excursions conditioned to reach a distant boundary exhibit transverse ranges proportional to the exit radius, while their repeated-visit fractions vary strongly with mirror density. Endpoint-label collision probabilities substantially exceed a general gluing lower bound, although they account for only part of the observed same-column crossing probability. Finally, rerouting at repeated vertices preserves the exit while substantially shortening the sampled paths. These findings characterize finite-volume transport and excursion geometry without determining an asymptotic scaling exponent or resolving infinite-plane localization.

In this paper, we classify the Conway symbols of the Euclidean symmetry groups of frieze patterns of Dynkin types $A_n$, $B_n$, $C_n$, $D_n$, $F_4$ and $G_2$. First, we show that the Conway symbol of a Conway–Coxeter frieze is determined by the bilateral and the central symmetry of the underlying triangulation. Secondly, we show that a frieze pattern of non-simply-laced type has the symbol ${*}\infty\infty$ or $\infty\infty$ according as its first interior row is palindromic or not, with the single exception of type $C_3$, which realises four symbols. Thirdly, we show that the same dichotomy holds for type $D_n$ with $n \ge 5$, the symbol being read off the tagged triangulation of the once-punctured polygon attached to the frieze together with the value on its puncture edges, and that type $D_4$ realises five symbols.

We consider boundary stabilization of a heavy chain hanging from a moving trolley with no tip load. Because the tension vanishes at the free end, the wave speed vanishes there; in Riemann coordinates the model becomes a degenerate $2\times2$ hyperbolic system in which the coupling is singular and the free-end reflection is generated in the domain rather than by a boundary condition. We construct a Volterra backstepping transformation that maps this system to the same chain with uniform damping of an arbitrarily prescribed rate and an elastic restraint at the trolley, yielding exponential convergence of displacement, velocity, and strain to zero. The singular kernel equations are solved by selecting their bounded Frobenius branch at the free end, which replaces the missing boundary datum, and the four kernels are generated by a globally convergent power series. The transformation is boundedly invertible on the energy space, with inverse obtained by reversing the prescribed decay rate. The result extends the radial backstepping structure developed for parabolic equations on disks and balls to a degenerate hyperbolic system.

We prove explicit upper bounds for the number of eigenvalues of a one-dimensional time-frequency localization operator in an open transition window. An abstract residual-energy principle combines a rank-N approximant with the Hilbert-Schmidt energy of its remainder and retains the strict integer correction imposed by the open threshold. For the Fourier kernel, centered least-squares polynomials, optimized moments, and Chebyshev-Bessel truncations improve or match the diameter-Taylor certificate; on interval blocks the moment envelope gains the exact factor 1/(2N+1). Finite measurable partitions yield best-polynomial matrix and residual-energy bounds, together with two-level threshold allocations that exploit spatial or Fourier-side orthogonality and remain stable under large empty gaps. A complementary defect channel gives a componentwise variational Schatten envelope for finite interval unions. For arbitrary finite-measure hard windows, exact cross-boundary and symmetric-difference formulas give strict transition counting and a certified near-one cluster that can be subtracted from direct counts. We also treat integrable soft masks, a root-sum-square core-tail decomposition, and unbounded windows controlled by moments. An explicit unbounded finite-measure set has all polynomial moments and infinite fractional translation perimeter for every order in (0,1], yet admits an O(log(1/epsilon)/log log(1/epsilon)) certificate. Finally, kernels with finitely many bounded separated phase interactions and finite separated amplitude rank admit factorial, least-squares, Chebyshev-Bessel, span-compressed, and anisotropic-degree bounds. For bounded hard Fourier windows, the final hybrid is no larger than the corresponding earlier certificates proved here; no universal comparison with all regular-domain estimates is asserted.

Long-range expansion is a combinatorial graph property introduced to construct metric spaces with strong quantitative obstructions to low-distortion embeddings. Such extremal constructions are motivated by fundamental questions in metric geometry and nonlinear functional analysis. However, the only known constructions of long-range expander sequences are random regular graphs; the problem of constructing explicit sequences, especially with logarithmic girth, remains open. We resolve this question in strong form by proving that the Ramanujan condition implies long-range expansion. In particular, classical Ramanujan constructions (such as LPS graphs) give explicit long-range expander sequences with logarithmic girth. We also clarify that long-range expansion implies spectral expansion, and both implications are strict. While long-range expansion, the Ramanujan property, and spectral expansion are strictly separated, it is a priori not clear how robust these gaps are. We next compare them from a dynamics perspective: how do the different notions of expansion affect the mixing rate of random walks? Our second main result is that, from a dynamics perspective, LRE and the Ramanujan property are equivalent. We show that the random walk on any LRE sequence has the same cutoff location and Gaussian limit profile as on a Ramanujan graph. This incidentally offers intermediate progress between cutoff for Ramanujan graphs—proven by Lubetzky and Peres—and the long-standing conjecture of cutoff for transitive spectral expanders.

Traveling waves are fundamental objects in reaction–diffusion systems; however, spatial heterogeneities can distort or inhibit their propagation. Determining whether a given heterogeneity results in propagation failure remains challenging, particularly in systems with coupled components. In this paper, we present explicit constructions of stationary coupled upper and lower solutions tailored to specific heterogeneities for a one-dimensional FitzHugh–Nagumo system. By leveraging the mixed quasimonotone structure of the system, these barriers yield sufficient conditions for the failure of propagation for families of trapped initial conditions. The stationary constructions establish directional signal blocking in FitzHugh–Nagumo neuron models with localized depressed dynamics or abrupt geometric variations, and the resulting stationary barriers persist for sufficiently small recovery diffusion. Additionally, we introduce explicit almost stationary barriers, which provide conditional bounds on the expansion of componentwise threshold regions in homogeneous media.

A scalar hyperbolic PDE, actuated at one boundary, is coupled with a fast PDE in two spatial dimensions: transport in the axial coordinate and in an internal coordinate, and possibly diffusion in the internal coordinate. Backstepping designs exist for coupled hyperbolic systems, for their ensembles, and for their continua, but in all of these the second variable of the fast subsystem carries no transport or diffusion; when it does, no design is available, and from a scalar input no exact control of the two-dimensional state is to be expected. The pair is stabilized here by time-scale separation: in the quasi-steady limit the two-dimensional subsystem collapses into a spatial Volterra operator inside a one-dimensional reduced plant, backstepping applies there, and the two-dimensional subsystem is left to be a stable boundary layer. One theorem establishes exponential stability of the pair for every time-scale ratio below an explicit threshold, uniformly in the internal diffusion coefficient down to the pure-transport case.

We study the arithmetic constraints that two-term Rogers dilogarithm relations impose on rank-two Nahm systems with mixed denominator steps. Monomial complement equations give a uniform conversion to quadratic data in every algebraic degree. Within a specified ten-record simplest-cubic family, positive coupling, positive definiteness, and integer-valued exponents leave, for m > 2, exactly the steps (1,3), (1,13), and (1,31). At the smallest compatible modulus 31, the required support cannot be a multiplicative subgroup coset. Quadratic examples include known product identities and two systems for which the first radial correction excludes every individual modular sum with rational linear terms. For the index-13 system, we prove a two-term Rogers identity and recover its arguments from the matrix of a previously proved two-dimensional Fricke transformation. Its cusp growth agrees with the resulting saddle action, and an independent calculation reproduces all three leading Fricke amplitudes exactly. The associated two-summand product evaluations remain conjectural; the small rank and dimension contrast with the classical rank-five, six-component Andrews-Gordon construction at the same modulus. A mixed-signature cubic family illustrates the limits of the arithmetic restriction

We establish new pointwise convergence results for weighted ergodic averages along sequences of the form \( (\lfloor a(n) \rfloor)_{n \in \mathbb{N}}, \) where $a(x)$ is a subpolynomial function in a Hardy field. For example, we establish pointwise convergence of logarithmic averages along sequences of the form $(\lfloor n^k + \log^{c} n \rfloor)_{n \in \mathbb{N}}$, where $k \in \mathbb{N} \cup \{0\}$ and $c > 0$. This result should be juxtaposed with the fact that either for $k=0$ or for $k \geq 2$ and for sufficiently small $c>0$ (depending on $k$), the standard ergodic averages along these sequences fail to converge pointwise. We also obtain pointwise joint ergodicity results for multiple weighted ergodic averages along slow Hardy field functions. For example, it follows from our results that for $c> 0$ and for any $f, g \in L^{\infty} (\lambda)$, \begin{equation*} \lim_{N \rightarrow \infty} \frac{1}{\log N } \sum_{n=1}^{N} \frac{1}{n} f(T_b^{\lfloor \log^c n \rfloor}x) \, g(T_G^{\lfloor \log^c n \rfloor} x) = \int f \, d \lambda \cdot \int g \, d \mu_G \quad \text{for almost every } x \in [0,1], \end{equation*} where $T_b:[0,1] \rightarrow [0,1]$ is the times-$b$ map defined by $T_b x = bx \, \bmod \, 1 $ and $T_G:[0,1] \rightarrow [0,1]$ is the Gauss map defined by $T_G(x) = \frac{1}{x} \bmod \, 1$ for $x \ne 0$ and $T_G (0) =0$. Here $\lambda$ is the Lebesgue measure on $[0,1]$ and $\mu_G$ is the Gauss measure on $[0,1]$ given by $\mu_G (A) = \frac{1}{ \log 2} \int_A \frac{1}{1+x} dx$ for any measurable set $A \subset [0,1]$.

We present Serre-type generators and relations for the derived Hall algebra of a graded gentle one-cycle algebra when a certain integral invariant is not $0$ or $\pm 1$.

We prove that deciding whether the weak partition connectivity (WPC) of an unweighted hedgegraph is at most a given integer is NP-complete, resolving an open question about its computational complexity. Hardness holds even for connected full-support split systems, where each hedge consists of two nonempty, vertex-disjoint hyperedges covering all vertices. We also obtain hardness on connected edge-colored simple graphs in which each color class is a spanning forest with exactly two nontrivial trees. On both classes, exact computation of fractional WPC is NP-hard, whereas hedge connectivity is computable exactly in polynomial time. A binary matrix formulation yields the reduction and a deterministic polynomial-time exact algorithm under a linear intersection condition on row supports after normalization and column compression. In particular, both integer and fractional WPC are computable in polynomial time when these supports have size at most two, whereas NP-hardness holds already when their size is bounded by three. For every fixed $b$, we give deterministic and randomized efficient polynomial-time approximation schemes on hedgegraphs in which each hedge has at most $b$ connected components, counting isolated vertices. Both schemes return one partition that simultaneously approximates integer and fractional WPC. Unless $\mathrm P=\mathrm{NP}$, neither objective admits a deterministic FPTAS, even on the two hard instance classes.

A result of Futer, Kalfagianni, and Purcell implies that an upper volume bound for all link complements in the 3-sphere cannot depend solely on the braid index. In this paper, we introduce the Lorenz braid index and generalise the bunch algorithm to provide a general upper volume bound for all link complements in the 3-sphere. Such an upper bound is a quadratic polynomial in the Lorenz braid index. In addition, we construct an explicit family of hyperbolic Lorenz knots for which the classical braid index and the Seifert genus both tend to infinity, while the Lorenz braid index remains bounded.

Amberg and Kazarin proved that a finite group is solvable if the normalizer of every cyclic subgroup of prime power order has a solvable supplement. We substantially relax this hypothesis by requiring it only for cyclic $2$-subgroups. This condition, denoted by $\mathrm{SSN}_2$, sharply restricts the nonabelian composition factors of the group to the family $\PSL_2(q)$, where $q\geq7$ is a prime power satisfying $q\equiv3\pmod4$. Conversely, this family is precisely the nonabelian finite simple groups that satisfy $\mathrm{SSN}_2$. Consequently, a finite group satisfying $\mathrm{SSN}_2$ is solvable if and only if it has no section isomorphic to one of these groups.

Let $\mathcal G\to S$ be a semiabelian scheme over a smooth complex algebraic base, with abelian relative dimension $g$ and toric rank $r$. We prove that a dominating irreducible subvariety not contained in a proper relative special subvariety has Zariski-dense fibrewise torsion if and only if its mixed Betti rank is $2(g+r)$. In particular, its dimension is at least $g+r$. For every prime, primary torsion is then dense even among points of maximal rank. The exceptional loci include generalized Ribet loci. Over algebraic numbers, we prove that generic small points force full mixed rank whenever the abelian Betti rank is maximal. This yields a transfer from abelian relative Bogomolov dimension theorems to arbitrary toric extensions, together with toric height gaps above abelian torsion and order-linear gluing-height bounds for nodal Pell equations. The proof combines algebraic quotients of marked $1$-motives, quadratic nef compensation, a central-monodromy calculation, and algebraicity of mixed Betti strata.

We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.

We study Gaussian approximation for probability measures obtained by normalizing one-dimensional profile functions, with one-dimensional marginals of Euclidean balls as the principal example. We first establish quantitative concentration estimates near the set of maximizers. For profiles with a unique nondegenerate maximizer, we give sufficient conditions under which centering at the maximizer and rescaling according to the local quadratic approximation of the logarithm of the profile yield densities that converge in $L^1(\mathbb{R})$ to the standard Gaussian density. We then specialize to one-dimensional marginals of Euclidean balls in $\mathbb{R}^n$. Writing $N=n-1$, we consider two standardizations of the marginal distribution: one determined by the logarithmic curvature at the maximizer, and the other by the exact standard deviation. For each standardization, we identify the first-order correction, of order $N^{-1}$, to the standard Gaussian density in $L^1(\mathbb{R})$. These expansions also determine the corresponding first-order corrections to the probabilities of symmetric intervals and the leading terms of the total variation distances from the standard Gaussian distribution. In particular, under exact-variance standardization, the total variation distance is asymptotic to an explicit positive constant times $N^{-1}$. Consequently, the previously known $O(N^{-1})$ bound for approximation by the Gaussian distribution with the same variance is sharp in order.

We prove moderate deviation principles and laws of the iterated logarithm in the sparse regime for geometric and topological functionals associated with $k$-point connected components. These limit theorems are established at both the measure-valued and vector-valued levels, for Poisson and binomial point processes. As applications, we derive corresponding results for component counts in random geometric graphs and Čech complexes and for counts of Morse critical points.

For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize $\alpha.$ Steady-state convergence (SSC) concerns the limit of the scaled stationary distribution as $\alpha \downarrow 0.$ Existing SSC theory requires i.i.d. or additive noise and global differentiability of the mean operator, and yields suboptimal rates. We develop a unified SSC theory for constant-stepsize contractive SA driven by Markovian, multiplicative noise, covering both locally differentiable and locally nondifferentiable mean operators. A key methodological contribution is a multi-step universality framework that progressively reduces the original stochastic recursion to tractable auxiliary dynamics while preserving its steady-state limit. Under local quadratic linearization at the fixed point, we obtain a Gaussian approximation of the scaled steady state at the optimal rate $O(\sqrt{\alpha})$ in Wasserstein-2 distance, which further gives finite-time Gaussian approximations for the raw iterates. In the locally nondifferentiable regime, we establish a general SSC result and show that the leading-order asymptotic bias can be of order $\sqrt{\alpha}$, in contrast to the $\alpha$-order bias in the smooth regime. We apply the theory to Markovian linear SA and asynchronous Q-learning, neither of which is covered by prior results. We further propose a bias-reduction scheme for Q-learning that requires no knowledge of the local smoothness regime, validated by numerical experiments.

We study poor irreducible holomorphic symplectic manifolds, namely those containing no rational curves and no codimension-one subvarieties. We show that, for such manifolds, several natural cones in \(H^{1,1}(X,\mathbb R)\) coincide, giving strong rigidity consequences. In particular, poor elliptic irreducible holomorphic symplectic manifolds admit no holomorphic foliations. Using this rigidity property, we give a criterion for detecting monodromy birationally minimal classes from the roots of Riemann–Roch polynomials on any irreducible holomorphic symplectic manifold. To each primitive negative class we associate a polynomial depending only on the deformation type and on its Beauville–Bogomolov–Fujiki square. Under a simple-root assumption on the positive real roots, the position of the number \(1\) among these roots implies that the class is MBM whenever it is of type \((1,1)\). This gives a uniform root-theoretic sufficient condition for the existence of rational curves on irreducible holomorphic symplectic manifolds. We then describe poor elliptic irreducible holomorphic symplectic manifolds in terms of their deformation spaces and show that every connected component of the moduli space contains a poor irreducible holomorphic symplectic manifold of positive Picard rank.

We introduce a verification framework to numerically analyze inexact model predictive controllers (MPCs) in the constrained non-linear discrete-time setting. Rather than modifying the controller so that guarantees hold by construction, we treat the controller as given. In particular, we focus on two types of inexact controllers: (a) one whose input is extracted from a primal-dual point satisfying the Karush-Kuhn-Tucker (KKT) conditions of the non-convex MPC problem, and (b) one whose input is obtained by linearizing the dynamics and solving a convex quadratic program. The main idea of our verification framework is to formulate an optimization problem that searches over the worst-case initial state within a given set and control inputs consistent with the inexact controller to maximize a carefully-chosen performance metric. Using this framework, we show how to certify (i) the worst-case suboptimality gap of a single MPC problem, (ii) the worst-case closed-loop suboptimality gap over a given number of dynamical system iterations, (iii) closed-loop stability, and (iv) feasibility of the closed-loop system. Through numerical examples, we showcase the ability of our framework to precisely quantify both types of suboptimality, and to test the stability and feasibility of the inexact controllers.

Let $\mathcal{H}_k$ denote the set of normalized holomorphic Hecke cusp forms of weight $k$ for the full modular group $ \mathrm{SL}_2(\mathbb Z)$. For each $f\in\mathcal{H}_k$, let $\lambda_f(n)$ denote the corresponding eigenvalue of the normalized Hecke operator $T_n$. We prove nontrivial upper bounds for \[\sum_{f \in \mathcal{H}_k}\omega_f \left|\sum_{n \leq x} \lambda_f(n)\right|^{2q},\] where $k$ is a positive even integer, $1\leqslant x\leqslant k$, $0\leqslant q \leqslant 1$ and $ \omega_f = \frac{\Gamma(k-1)} {(4\pi)^{k-1}\langle f,f\rangle} = \frac{2\pi^2} {(k-1)L(1,\mathrm{Sym}^2 f)}. $ Our estimates match the conjecturally sharp upper bound for $1\leqslant x\leqslant k^{0.99}$. In particular, whenever both $x$ and $k/x$ tend to infinity with $k$, we obtain \[\sum_{f \in \mathcal{H}_k}\omega_f \left|\sum_{n \leq x} \lambda_f(n)\right|=o(\sqrt{x}).\] We also prove upper bounds for low moments of sums of Hecke eigenvalues associated with Hecke–Maass cusp forms for $\mathrm{SL}_2(\mathbb Z)$. We study these sums through the probabilistic model proposed by Cogdell–Michel. We also determine the order of magnitude of low moments of this probabilistic model, which is equivalent to determining the order of magnitude of low moments of Hecke eigenvalue sums in the limit.

We prove a sharp cut-off resolvent estimate for the Laplacian on non-compact Riemannian manifolds whose ends satisfy the Cardoso–Vodev assumption and whose geometry near the trapped set agrees with that of an even asymptotically hyperbolic manifold with the corresponding resolvent estimate. As applications, we show the global Strichartz and spectral projection estimates on negatively curved asymptotically hyperbolic surfaces, without assuming the evenness at infinity.

A hypergraph Laplacian was introduced to investigate the structure of networks written as hypergraphs. It was shown that the behavior of solutions to an evolution equation associated with the hypergraph Laplacian quite resembles that of solutions to the classical heat equation. Hence by replacing the Laplacian in PDEs with the hypergraph Laplacian, we might be able to introduce various diffusion models on hypergraphs (discrete domains), whose behavior of solution is similar to PDEs. In this paper, we consider a system of equations obtained by replacing the Laplacian in the original Cahn–Hilliard equation with the hypergraph Laplacian. Due to the nonlinearity and multivaluedness of the hypergraph Laplacian, it is difficult to apply methods for the original Cahn–Hilliard equation to our problem. To cope with these difficulty, we shall introduce a new proof by using properties of the hypergraph Laplacian in this paper.

The polynomial Hardy–Littlewood inequalities bound coefficient norms of homogeneous polynomials on $\ell_p$ balls by their supremum norms, with constants independent of the dimension. Their $p=\infty$ endpoint is the Bohnenblust–Hille inequality. A longstanding obstruction to connecting their constants is the polarization loss in the usual multilinear approach. We overcome this obstruction by proving $D_{m,p_m}\le(1+o(1))D_m$ whenever $p_m/m^2\to\infty$, where $D_{m,p}$ and $D_m$ are the optimal complex polynomial Hardy–Littlewood and Bohnenblust–Hille constants. At the opposite boundary, we determine the optimal endpoint constant exactly, $D_{m,m}=m$. We then prove the full first-order asymptotic $D_{m,p_m}\sim m$ whenever $p_m\ge m$ and $p_m-m\to0$, and the exact polynomial growth $D_{m,p_m}=m^{1+o(1)}$ on the wider scale $p_m-m=o((m/\log m)^{1/3})$.

We prove global existence, both to the future and to the past, for a family of scalar quasilinear wave equations in two space dimensions for sufficiently small $C_c^\infty$ initial data. These equations have cubic leading quasilinear nonlinearities of the form $u^2\partial^2u$ and satisfy a sign condition that allows the cubic null condition to fail. To explain the sign condition, we derive the geometric reduced system for quasilinear wave equations with cubic leading nonlinearities in two space dimensions and introduce a notion of geometric weak null condition. We prove that the geometric weak null condition holds for the model scalar equation if and only if the sign condition holds. At the level of the geometric reduced system, the sign condition prevents the corresponding characteristic curves from intersecting in finite time. We also recover the geometric reduced system from the global solutions in our main theorem. If the model scalar equation satisfies the sign condition but fails the cubic null condition, and if $u$ is a nonzero global solution to the model equation, we show that, for $r=|x|$, the $L^\infty$ norm of $t^{\frac{1}{2}}\partial_r^2u$ in a region where $|r-t|\ll t$ tends to infinity as $t\to\infty$. Thus, its asymptotic behavior differs from that of a solution to the linear wave equation $\Box \psi=0$ with $C_c^\infty$ data.

We consider smooth symmetric bimodal maps with Fibonacci combinatorics. We prove that, when the critical order is sufficiently large, the common $\omega$-limit set of the two critical points is a wild Cantor attractor. Using a full-family argument, we first prove the existence of smooth symmetric Fibonacci bimodal maps with any prescribed critical order $\ell>3$. The proof of the existence of the wild attractor is based on a two-to-one semi-conjugacy which reduces the generalized renormalization of the bimodal map to a Fibonacci box mapping with one critical point.

We develop Geometric Invariant Theory for affine superschemes under the action of reductive algebraic supergroups, formulating the theory in terms of coordinate Hopf superalgebras in order to accommodate anticommuting and nilpotent variables. Within this algebraic setting, we construct affine superquotients and establish their basic structural properties under suitable hypotheses. We then prove a supergeometric analogue of the Hilbert-Mumford criterion, giving a numerical test for the semistability and stability of points under natural constraints on the supergroup coaction. We show that this numerical criterion can be used to construct the GIT superquotient of these superschemes, and we give examples illustrating how the supergeometric GIT superquotient differs from its classical counterpart.

We introduce level structures on tropical abelian varieties and give a modular interpretation of $A_g[m]^{\mathrm{trop}}$, the tropicalization of the moduli space $\mathcal{A}_g[m]$ of principally polarized abelian varieties with level $m$ structure. We study the case of abelian surfaces in greater depth. The link of $A_2[m]^{\mathrm{trop}}$ is an explicit simplicial complex whose vertices are the primitive vectors of $(\mathbb{Z}/m\mathbb{Z})^4$ up to sign. As a topological space, this link is homotopic to a wedge sum of closed orientable surfaces and circles; we compute the number of each of these and the genera of the surfaces. We deduce the weight zero compactly supported rational cohomology of $\mathcal{A}_2[m]$, completing a calculation of Oda-Schwermer from 1990.