Papers

We study the nonlinear phenomenon of the asymptotics of higher-order derivatives of the Gram function $t_u$. We also prove the uniform distribution and continuous uniform distribution modulo one of $t^n_u$ for $n = 1, 2, \dots$. We use AI to formalize our proofs in Lean.

We study linear Stieltjes control systems in which actuation is concentrated at finitely many temporal events, and we ask what happens when one of those events progressively loses its atomic weight in the associated Stieltjes measure. For constant coefficients, ordinary finite-atomic reachability is governed by a truncated Krylov space and is therefore insensitive to the positive values of the event weights. A different picture emerges once trajectory regularity is imposed. In the supercritical super-$g$-Hölder regime, we minimize the largest event-normalized state jump and characterize exactly the terminal directions for which this cost remains uniformly bounded as a prescribed event weight tends to zero. The characterization combines what can be reached before the vulnerable event, a drift-cancellability condition at that event, and the directions that remain available afterwards. Ordinary reachability of the deleted-event system can therefore be strictly larger than bounded-cost reachability in the singular limit. We also separate two mechanisms of divergence: loss of ordinary reachability after deletion and failure of the intermediate compatibility condition. For compatible targets, the minimum cost converges to a constrained deleted-event problem. A dimensionless position–velocity benchmark motivated by pulsed proximity maneuvers shows the distinction explicitly.

Let $\mathcal{S}_c$ be the subgroup of $Aut(\mathbb{C}^2)$ generated by affine maps and shear automorphisms with constant Jacobian. We determine the possible conformal types of the rank-one limit set of an invariant recurrent Fatou component when the induced dynamics has infinite order. In particular we show that, under this infinite-order hypothesis, the limit set of a finite composition of Hénon maps (polynomial or transcendental), can only be conformally equivalent to a disk or an annulus. Thus we exclude the remaining three conformal types, i.e. $\mathbb{D}^*$, $\mathbb{C}^*$ and $\mathbb{C}$, from the classification result by Arosio, Benini, Fornæss and Peters. We also prove that the set of fixed points of a finite composition of Hénon maps (polynomial or transcendental) contains no entire curves.

We study the asymptotic response time tail in the M/G/n queue with heavy-tailed (regularly varying) job sizes. For this setting, strongly tail optimal policies are known only at lower load, where n-1 servers suffice for stability. At higher load, the tail optimal scheduling has been open. This paper presents DynaSPLIT, the first strongly tail optimal policy at high load. Together with prior work, this result shows that strong tail optimality is achievable across the entire stability region of the M/G/n with heavy-tailed job sizes.

We show how the methods of delta geometry can be used to bound the degree of points in the Zilber–Pink locus of a subvariety of an abelian variety, if the varieties are defined over a number field and we restrict to points with bounded ramification at a fixed prime.

We develop a Sobolev theory for the kinetic Fokker-Planck equation subject to a singular Morrey-form $L^\infty+M^q_K$ force field which can produce local blow-ups due to strong attraction to a submanifold. Our approach is based on the fractional representation for the Duhamel series of the equation and on establishing kinetic Adams-type estimates. Accordingly, we construct weak solutions of Brownian-driven kinetic SDEs in a critical Morrey set-up, obtain kinetic Krylov-type estimates and a strongly continuous Feller propagator, and establish uniqueness criteria for a large class of weak solutions.

Let $G$ be a split connected semi-simple group over a local non-archimedean field $F$. In this paper we study the action of Hecke operators on the Schwartz space of half-densities on the moduli stack $\bpo$ of $G$-bundles on ${\mathbb P}^1$ endowed with a trivialization at $0$ and $\infty$. This space is a representation of $G(F)\times G(F)$ which contains the regular representation of $G(F)$. We show that this inclusion is an isomorphism on the cuspidal part. In addition we show that every irreducible square-integrable representation $\pi$ of $G(F)$ defines a Hecke eigenvalue. As a byproduct we show that every irreducible square-integrable representation of $G(F)$ with an Iwahori fixed vector defines a ${\mathbb C}$-point on the Calogero-Moser space associated to $G$. We also show that for $G=PGL(N)$ the eigenvalues of certain "simplest non-trivial" Hecke operators are essentially equal to the values of the character of $\pi$ on regular elliptic elements. These results are parallel to some of the results of our previous publication where some analogous statements are proved in the case when $F$ is a finite field.

We identify the $(\infty,1)$-category presented by the dagger Bergner, dagger Joyal, and dagger Rezk models of Parts I and II with that of univalent flagged dagger $(\infty,1)$-categories. We first construct a realization–nerve adjunction between reversal simplicial anima and Hermitian flags. We prove that its nerve restricts to an equivalence between flagged dagger $(\infty,1)$-categories and dagger Segal anima, under which univalence corresponds to unitary completeness. We then construct the adjunctions for univalentization and Hermitian completion and identify univalentization with unitary completion.

This work studies certified reach-avoid control of stochastic hybrid systems in which a stochastic differential equation governs continuous flow and a controlled stochastic kernel governs guard-triggered resets. The input appears only in the reset kernel, where it determines the post-jump distribution. Certificates for such systems have so far assumed spontaneous or controller-chosen jump times, a finite horizon, or discrete-time dynamics; we certify, over the infinite horizon, that the target is reached before the unsafe set with at least a prescribed probability. First, we introduce reach-avoid contact certificates, whose flow and jump conditions are imposed only on a sublevel corridor, and prove that any such certificate with a compatible reset policy meets the specification; a pointwise safety filter extends this guarantee from verification to control by constraining arbitrary nominal controllers. Next, we prove that executions are non-Zeno and bound the expected settling time and reset count. We frame certificate search as a linear program over any twice-differentiable dictionary followed by independent verification. We illustrate our approach via comparisons where the certified region is a subset covering 93.6% of the ground truth and where our certificate-based safety filter reaches the goal while reducing collisions from 68 to zero across 200 episodes of a dribbling ball problem.

We study the rough differential equation $$ \frac{du_t}{dt}= F(u_t) \zeta_t, \ \ u(0) = u_0 \in \mathbb{R}^d, $$ where $F : \mathbb{R}^d \to L(\mathbb{R}^d, \mathbb{R}^d)$ is a sufficiently regular vector field, and the noise $\zeta$ is a distribution in $\mathscr{C}^{\alpha-1}(\mathbb{R})$ for $\alpha \in (\frac{1}{4}, \frac{1}{3})$. We construct solutions to this equation at lower regularity than previous work of Gubinelli-Imkeller-Perkowski and Prömel-Trabs using ideas from the high-order paracontrolled viewpoint of Bailleul-Bernicot. The key tool in our work is the development of a harmonic-analytic framework for proving a set of iterated commutator estimates which are necessary for the solution theory. We also show the procedure of enhancement when the noise $\zeta = dZ$, where $Z$ is a realization of fractional Brownian motion with Hurst parameter $H \in (\frac{1}{4}, \frac{1}{3})$.

We prove that, for any two integers $2\le m<n$, a bounded operator $A$ on a complex Hilbert space satisfying \[ A^{*k}A^k=(A^*A)^k, \qquad k\in\{m,n\}, \] is quasinormal. More generally, we show that the inequalities \[ A^{*m}A^m\ge(A^*A)^m \quad \text{and} \quad A^{*n}A^n\le(A^*A)^n \] already imply quasinormality. Together with the known single-exponent counterexamples, this completely resolves the reduced Embry problem for bounded operators: the identities $A^{*k}A^k=(A^*A)^k$, $k\in S\subseteq\{1,2,\ldots\}$, characterize quasinormality if and only if $S$ contains at least two distinct integers greater than one. The proof combines Hansen's inequality and the transformer inequality for weighted operator geometric means with an extremal argument for optimal power bounds.

We consider the Neumann Laplacian on a three-dimensional periodic network of thin rods with edge thickness of order $h$ and vertex diameter of order $h^{2/3}$. We extend the fixed-graph resonant reduction to obtain a finite-$h$ periodic graph approximation with generalised norm-resolvent error $O(h)$, uniformly in the Floquet quasi-momentum. An exact gauge transformation identifies the phase-corrected graph fibres with the canonical lattice Floquet fibres, giving the same rate for the global operators. Collapsing the contact points while retaining the ambient edge tangents gives an approximation with error $O(h^{2/3})$. The proof uses a Ryzhov boundary triple and a Schur–Feshbach reduction. The estimates yield quantitative approximation of Bloch bands and stability of open gaps on bounded energy intervals. For the simple-cubic lattice, we refine the known resonant dispersion calculation to obtain a complete band-gap decomposition and $O(h)$ control of the persistence of each fixed finite collection of gaps in the thin structure.

We study the behavior of saddle connection directions on translation surfaces from the point process viewpoint. We define a direction point process for saddle connections of length at most $R>0$ on every translation surface. Given a general ergodic $\mathrm{SL}(2,\mathbb R)$-invariant measure $\mu$ on a stratum of translation surfaces, we also define a slope point process associated with $\mu$. By utilizing the dynamics and geometry of strata, we show that for almost every translation surface, the direction point processes, rescaled by $R^2$, converge in distribution to the slope point process as $R\to\infty$. Using Palm distribution theory, we prove the existence of limiting joint gap distributions and obtain new regularity results for gap distributions associated with the general $\mu$. Finally, we study how these point processes and joint gap distributions behave as the genus $g$ tends to infinity. In the principal stratum, we show that the slope point processes, rescaled by $g^2$, converge in distribution to a Poisson point process. By passing to the corresponding Palm distributions, we show that the rescaled slope gaps converge jointly to independent exponential random variables, which is in stark contrast to the known non-exponential gap distributions in a fixed genus.

We prove an improved fractal Weyl upper bound for resonances of infinite-area convex cocompact hyperbolic surfaces, expressed in terms of the Hausdorff dimension of the limit set and the Fourier decay exponent of the Patterson–Sullivan measure. This bound also yields a sharper upper bound for the essential spectral gap for these surfaces.

Fast Fourier Transform (FFT)-based preconditioners provide an efficient approach for approximating inverse operators in large-scale linear inverse problems when the underlying operator structure can be exploited in the Fourier domain. Optical Fourier transforms (OFTs), the analog of digital Fourier transforms, exploit light propagation through optical systems and offer the potential for greater throughput and energy efficiency than conventional digital FFT implementations. However, practical optical systems are inherently imperfect and introduce a variety of errors into the computed output. Thus, OFTs can only provide approximations to their digital counterpart, and the implications of these inexact transforms for iterative solvers remains largely unexplored. This work investigates the feasibility of OFT-based preconditioning of iterative Krylov subspace methods for the solution of discrete inverse problems. We digitally simulate the optical Fourier transform using the FFT and incorporate sources of error expected in practical optical systems into the simulation. Numerical experiments suggest that given sufficient optical system quality, iterative methods utilizing OFT-based preconditioners can maintain reconstruction quality and convergence comparable to those with fully digital FFT-based preconditioners.

Oda's smoothness conjecture asserts that every smooth lattice polytope has the integer decomposition property. The three-dimensional case of this conjecture was recently proven by AI. Here we give a much shorter proof of this result.

Learning symmetries through data augmentation offers an alternative to architectural equivariance, but does not guarantee that a network respects those symmetries. Measuring learned equivariance is especially difficult for intermediate representations, whose group actions are not specified. In a finite discrete setting, we show that equivariance requires representations of original and transformed inputs to determine one another exactly, yielding a test based on normalized mutual information (NMI). For high-dimensional activations, we use centered kernel alignment (CKA) as a scalable test, with an exact guarantee for orthogonal feature transformations. Across architectures and tasks, augmentation usually increases transformation consistency, showing that learned equivariance can be probed directly from representations without recovering the hidden transformation rules that relate them.

Modern mixed-integer programming (MIP) solvers repeatedly solve linear programming (LP) relaxations, but the LP state transferred from the simplex algorithm to the branch-and-bound tree is typically treated as an artifact of perturbation, presolve history, and numerical trajectory. This is harmless in pure linear programming, where alternative optimal bases certify the same bound. In MIP, however, those bases can expose distinct fractional structures, active rows, reduced costs, branching candidates, and reoptimization states. The inherited LP state is therefore an initial condition for the integer search. We exploit this initial condition to mitigate computational performance variability. The proposed framework generates alternative LP states on or near degenerate optimal faces, evaluates them within an auxiliary solver environment, and either selects a single state or executes a portfolio of candidate states in parallel. A handcrafted synthetic instance isolates the relationship between the initial LP state, the fractional profile, the branching trajectory, and the resulting node-count distribution: root-state steering reduces the weighted variability score~(WVS) by 67% over 1,000 paired seeds and produces no node-count regressions relative to default SCIP. On a separate 130-instance MIPLIB 2017 validation set with up to 50 paired seeds per instance, the top-five policy improves solve completion from 3,792 to 3,894 runs, reduces the shifted geometric mean of the recorded node count from 2,574.7 to 1,198.0, and lowers the median WVS by 9% on instances reliably solved by the baseline. These results demonstrate that the branch-and-bound phase inherits a specific LP state rather than merely an LP bound, and that the active management of this state systematically mitigates computational performance variability.

The Frobenius problem asks, given two positive, relatively prime integers $a$ and $b$, what is the maximum integer that cannot be expressed as a linear combination of $a,b$ using non-negative coefficients? This problem is resolved. However, if we add the restriction that the coefficients themselves must also be relatively prime, we create an open problem with connections to the Stern and Farey sequences and the Jacobsthal function. We prove that under this new restriction, a maximum inexpressible integer still always exists, and explore its values and properties as functions of $a,b$.

This paper initiates the study of arithmetic properties of closed points of arbitrary degree on curves. We show that a pointless curve of rank $0$ can have only finitely many cyclic degree $\ell$ points for all sufficiently large primes $\ell$. More generally, we show that the residue fields of points on curves of genus at least $2$ and rank $0$ are often biased away from total ramification. The proof technique illuminates a natural intermediate conjecture between the Manin–Mumford conjecture and the Zilber–Pink conjecture for self-products of abelian varieties. This intermediate conjecture implies analogous results for other splitting profiles.

Standard semidefinite programming (SDP) solvers working in double precision can become unreliable on difficult problems, even when their output appears plausible. We study this issue for SDP problems arising from high-order moment-SOS hierarchies. Three polynomial optimization problems are analysed by comparing standard double-precision computations with reference solutions obtained using the extended-precision solver \mbox{Loraine\,64x4}. We observe two distinct patterns of numerical failure. In false convergence, the computed bounds form a smooth sequence but approach the limit at the wrong rate. In stagnation, the sequence develops a plateau or oscillations and no longer reflects the convergence of the underlying hierarchy. We introduce an empirical effective condition number based on the positive spectral components of the primal and dual solution matrices. It serves as a warning indicator rather than an error bound, and its growth accompanies numerical deterioration in our examples. We also examine the imbalance between primal and dual spectral conditioning and a Cholesky inverse residuum as potential diagnostic indicators. The results show that solver termination messages and apparently regular convergence curves may be insufficient to establish the reliability of high-order moment-SOS bounds.

The higher Green's functions $G_{k/2}(z,w)$ on the modular curve are real-valued eigenfunctions of the hyperbolic Laplacian in each variable, with logarithmic singularities along the diagonal. Gross and Zagier conjectured that their values at CM points, after multiplication by suitable powers of the discriminants, are logarithms of absolute values of algebraic numbers. The conjecture was proved by Li, Bruinier, and Yang. Mellit proposed a complex refinement of this conjecture, predicting that the normalized CM values of a complex lift of the Green function are logarithms of algebraic numbers modulo periods. For $\Gamma=\mathrm{SL}_2(\mathbb Z)$ and $k\in{4,6,8,10,14}$, we follow Zagier's construction of CM invariants from Eichler integrals of the Eisenstein series $E_4$. By applying Maass raising operators in both variables to a normalized iterated Eichler integral of a Bergman-type kernel, we obtain a complex-valued extension $\mathcal G_{k/2}^\bullet(z,w)$ whose real part is a constant multiple of $G_{k/2}(z,w)$. We study the normalized CM values of $\mathcal G_{k/2}^\bullet$ and describe their dependence on the choice of representatives of quadratic form classes. In particular, we show that replacing the quadratic form corresponding to the first variable by an equivalent one changes the exponential of the normalized value only by a root of unity. We also prove the algebraicity of these exponentials in the special case where the second CM point is $w_*=(1+i\sqrt d)/2$, with $d>3$ and $d\equiv3\pmod4$, and the class of the quadratic form corresponding to the first variable is equal to its inverse.

In this paper, we study the following magnetic nonlinear Schrödinger equation \[ \left(\frac{\varepsilon}{i}\nabla-A(x)\right)^{2}u+V(x)u=f(|u|^{2})u\,\ \text{in}\,\ \mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $A\in C(\mathbb{R}^{2}, \mathbb{R}^{2})$ is a magnetic potential, $V\in C(\mathbb{R}^{2}, \mathbb{R})$ is a multi-well potential, and $f\in C^1(\mathbb{R}, \mathbb{R})$ exhibits exponential critical growth. By variational methods and a penalization technique, we prove the existence and multiplicity of nontrivial solutions for sufficiently small $\varepsilon>0$. If, in addition, $A\in C^1(\mathbb{R}^2,\mathbb{R}^2)$, we show that these solutions have multiple peaks and we establish their concentration behavior and exponential decay.

This paper develops a Willis-type thermal homogenization framework for two-dimensional particulate composites under harmonic heat transfer and establishes a unique effective constitutive relation without introducing residual fields. The local thermal fields are obtained using the dual-reciprocity inclusion-based boundary element method, in which two eigen-fields are introduced to simulate the mismatches in thermal conductivity and heat capacity. Instead of retrieving effective kernels from several independent macroscopic solutions generated by residual fields, the complete constitutive operator is constructed upon relations between eigen-fields and generalized macroscopic responses. The effective constitutive relations are validated against a one-dimensional analytical solution and an extended Fourier-series homogenization scheme. The Fourier formulae further establish the opposite-wavenumber reciprocity relations between two coupling tensors and reflection-symmetry parity relations, which explain when transverse coupling components vanish and the sign reversal of off-diagonal conductivity components under reflection. Numerical results further demonstrate that the de-phased average is required to extract physically consistent effective thermal properties under the Bloch conditions, and reveal their nonlocal frequency/wavenumber-dependent behavior. The framework provides a general computational method for Willis-type homogenization of particulate composites and is extendable to three-dimensional, multiphysical problems.

We introduce a systematic framework for constructing holonomic recurrences from meromorphic modular forms. Its central ingredient is a modular analogue of creative telescoping, which turns finite-dimensional quotients of spaces of meromorphic modular forms into explicit recurrence operators. This provides a common explanation for several previously isolated examples: it recovers Apéry's recurrence, Beukers' modular constructions and Zagier's seven sporadic recurrences. The framework also produces a large collection of new examples across different weights and groups. The modular origin of these sequences also confers them with non-trivial arithmetic properties, such as denominator bounds, irrationality statements, and $p$-adic congruences.

Every 3-regular graph of girth at least $g$ has a cut containing at least a fraction $M_g$ of its edges, and the best rigorous bounds at large girth are $0.912\le \lim_{g\to\infty}M_g \le 0.92386$. Lower bounds on $M_g$ follow from the performance of local algorithms: on a graph of girth $g \ge 2p+2$ a depth-$p$ local algorithm sees the same tree around every edge, so its expected cut fraction on that tree bounds $M_g$ from below. We apply this strategy to the Variational Iterative Rotation Algorithm (VIRAL), a classical local algorithm, and evaluate it on the depth-$p$ tree for every even depth up to $p=32$. The resulting bounds exceed those recently obtained from the Quantum Approximate Optimization Algorithm (QAOA) for every $p\ge 4$, by $0.005$ to $0.011$, and they extend the range of girths covered from $g=36$ to $g=66$. Notably, at $g=66$ we obtain $M_{66} \ge 0.9127$ at $99.99999\%$ confidence level, which raises the asymptotic lower bound from $0.912$ to $0.9127$.

A teardrop geodesic on a hyperbolic surface is a closed curve that is geodesic except at exactly one point, called its vertex. One imagines a closed curve with one point held down as it deforms to a geodesic. For each teardrop geodesic, we give an explicit geometric construction of the closed geodesic in its free homotopy class. This construction also yields a relation between the length of the teardrop geodesic, the angle at its vertex, and the length of its geodesic representative. We then apply the construction to a self-intersecting closed geodesic at a chosen self-intersection point. Since, at that point, the geodesic splits into two directed teardrop geodesics, our construction gives a geometric description of Turaev's cobracket in terms of geodesic representatives. Using this description we prove that if two self-intersection points yield a pair of canceling terms in the cobracket, then the angle at these points and the lengths of the small loops determined are the same.

Let $(M^n,g_0)$ be a complete Riemannian manifold with $\Ric_{g_0}\ge-(n-1)g_0$ whose local universal covers are uniformly $\delta$-Reifenberg-close to $\R^n$ below a fixed radius $\rho$. We show that for every $\alpha\in(0,1)$ there are $\delta(n,\alpha)>0$ and $T(n,\rho,\alpha)>0$ such that, if $\delta\le\delta(n,\alpha)$, a complete Ricci flow $g(t)$ with $g(0)=g_0$ and $|\Rm_{g(t)}|\le\alpha/t$ exists on $(0,T]$. On closed manifolds this is a uniform lower bound for the lifetime of the Ricci flow; on noncompact manifolds, where the initial curvature may be unbounded, it is an existence theorem, and the flow is unique among complete Ricci flows from $g_0$ with a curvature bound of the form $A/t$. The key step is a finite-buffer comparison of two Ricci flows through the Ricci–harmonic map heat flow, following Lee, which transfers a Type-I curvature bound from a complete selected flow to the incomplete flow lifted to a local cover. Consequently, below a dimensional Reifenberg threshold, collapsing closed manifolds admit singular nilpotent fibrations over their Gromov–Hausdorff limits. The Reifenberg hypothesis cannot be weakened to noncollapsing of the local covers: products of shrinking circles with Nifa's four-manifolds have $\Ric\ge0$ and uniformly noncollapsed local covers, but admit no continuous Gromov–Hausdorff approximations onto their limit.

We study the complexity of high-dimensional integration of periodic functions from Besov spaces $S^r_{p,q}B(\mathbb{T}^d)$ with mixed smoothness in the quantum model of computation. We construct new quantum algorithms based on the multivariate Faber-Schauder decomposition of such functions and analyze their convergence rates. We show a gain of one convergence order compared to the well-known deterministic rates of convergence. We additionally obtain lower bounds showing the optimality of our results up to logarithms.

For every countable discrete group, we construct a character for finite-dimensional unitary quasi-representations, with values in the inverse limit of the equivariant $K$-theory of its Rips complexes. The character is a complete invariant for quasi-representations up to stable equivalence for amenable groups and, more generally, for groups satisfying the Haagerup property and QR-quasidiagonality. For Haagerup groups satisfying quantitative QR-quasidiagonality, we also obtain a quantiative version of the theorems. These conclusions apply when the maximal group $C^*$-algebra is quasidiagonal and has the lifting property, including amenable groups, free groups, and closed orientable surface groups. For surface groups, rank and winding number give complete invariants for the classification. Finally, composing with the equivariant Chern character gives a delocalized cohomological obstruction to stable equivalence of a quasi-representation with genuine representations.