数学前沿

We prove that the generic fibre of the Betti moduli space associated to any of the ten Painlev\'e equations coincides with the result of attaching Weinstein handles along the Stokes Legendrian, and provide Weinstein handlebody diagrams for each of them in the process. We moreover extend the procedure for obtaining presentations in Gompf normal form of Weinstein manifolds obtained by attaching handles to Legendrian lifts of cooriented curves in surfaces to their unit cotangent bundle to the case where the surface may have nonempty boundary.

We study the structure and exact certification of nonnegative tensor decompositions supplied through rational factor matrices. Every failure of minimality or essential uniqueness contains an irreducible positive exchange between $p$ supplied terms and $q$ competing terms. A dimension count based on the Lovitz--Petrov splitting theorem bounds the effective order of such an exchange, the sum over the modes of the joint factor-span dimension minus one, by $p+q-2$, and this bound is attained at every supplied length $R\ge2$. After eliminating ambient-row variables by the Farkas lemma, the resulting semialgebraic systems have fewer than $2R^2$ real variables. For input bit length $L$, this gives a deterministic $G(R)L^{O(R^2)}$-time algorithm that computes the nonnegative rank and an exact minimum-length decomposition, decides minimality and essential uniqueness, and returns exact witnesses when either property fails. The exponent is independent of the tensor order and the mode dimensions. For Lebesgue moment tensors, positive quadrature determines the nonnegative rank and the uniqueness behavior, and some rational length-$R$ inputs force a common field of degree at least $\lfloor R/2\rfloor!$ for any exact minimum-length output. As an illustration, an implementation resolves all 114 instances left inconclusive by earlier positive-scattering criteria, of which 95 are minimal and essentially unique and 19 are minimal but nonunique.

We establish modified scattering for the three-dimensional Hartree equation with Coulomb interaction and singular external potentials, including repulsive Coulomb and inverse-square potentials. In the defocusing case, the result holds for arbitrary initial data in $H^{0,1}(\mathbb R^3)\cap H^{1,0}(\mathbb R^3)$. In particular, we obtain large-data modified scattering for the flat defocusing Hartree equation. For the focusing problem with repulsive Coulomb potential $V(x)=\kappa|x|^{-1}$, we prove radial modified scattering under the sharp mass condition $|\lambda|\|u_0\|_2^2\leq 2\kappa$, including the endpoint, and construct standing waves at every larger mass. Our argument obtains strong $L^2$ convergence from rescaled energy and comparison with a frozen-field evolution.

We introduce a general definition of higher-form connections on principal $\infty$-bundles in differential geometry. These connections are built as infinitesimal parallel transports and thus satisfy the fake-curvature condition. We achieve this by developing the formal differentiation and integration of maps from smooth manifolds to derived stacks with sufficient deformation theory. That allows us to introduce the Atiyah $L_\infty$-algebroid of a principal $\infty$-bundle and establish its global sections as the $L_\infty$-algebra of the derived higher symmetry group of the bundle. We define the space of $p$-form connections on an $\infty$-bundle as the space of order $p$ splittings of its Atiyah $L_\infty$-algebroid. This can be cast equivalently as lifting the classifying map of a bundle on a manifold to the order $p$ truncation of the de Rham stack of the manifold. We demonstrate that our new concept of derived geometric $p$-form connections recovers the known notion of connections on higher U(1)-bundles defined via \v{C}ech-Deligne differential cocycles. We further relate the $L_\infty$-algebras of derived higher symmetries of higher U(1)-bundles and higher Courant algebroids. Some applications in higher gauge theory and in supergravity are mentioned.

We consider a set of generators for the space of Eisenstein series of varying even weight $k$ for any fixed congruence group $\Gamma$ and study the set of all of their zeros taken for all $\Gamma(1)$-conjugates of $\Gamma$ in the standard fundamental domain for $\Gamma(1)$. We describe (a) an upper bound $\kappa_\Gamma + O(1/k)$ for their imaginary part; (b) a finite configuration of geodesics segments to which all zeros converge in Hausdorff distance as $k \rightarrow \infty$; (c) a finite set containing all algebraic zeros for all weights. The bound in (a) depends on the (non-)vanishing of a new generalization of Ramanujan sums. The proof of (b) originates in a method used to study phase transitions in statistical physics. The proof of (c) relies on the theory of complex multiplication. The results can be made quantitative for specific groups. For $\Gamma=\Gamma(N)$ with $4 \nmid N$, $\kappa_\Gamma=1$ and the zeros tend to the unit circle, whereas if $4 \mid N$, $\kappa_\Gamma=2$ and the limit configuration includes parts of vertical geodesics and circles of radius $2$. In both cases, for sufficiently large $k$, the only algebraic zeros are at $\mathrm{i}$ and $\exp(2\pi \mathrm{i}/3)$, and, for $4 \mid N$, at $2\mathrm{i}$. For $\Gamma(N)$ with $N$ odd, we use finer estimates to prove a trichotomy for the exact `convergence speed' of the zeros to the unit circle, as well as angular equidistribution of the zeros as $k \rightarrow + \infty$, implying weak convergence of the normalized zero counting measure on $Y(1)$ to the image of the argument measure on the lower arc of the fundamental domain.

We develop a general discrete approximation scheme for time-changed Brownian motions on $\mathbb{R}^d$. Our approximation scheme works for any smooth measure with full quasi-support on $\mathbb{R}^d$ with suitable initial distributions. Under some mild conditions on the smooth measure, the discrete approximation scheme works for every starting point. Our results in particular give a discrete approximation scheme for Liouville Brownian motions.

We consider the quasilinear Hardy-H\'enon equation $$u_t=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T),$$ in the range of exponents and dimension $$N\geq3, \quad \sigma\in(-2,\infty), \quad m>1, \quad p>p_S(\sigma)=\frac{m(N+2\sigma+2)}{N-2}.$$ We establish sharp ranges of exponents $(m,p,\sigma)$ guaranteeing existence of \emph{peaking solutions} in backward self-similar form $$u(x,t)=(T-t)^{-\alpha}f(|x|(T-t)^{-\beta}).$$ By peaking solutions we understand solutions blowing up at a finite time $T\in(0,\infty)$ but allowing for a continuation after the blow-up time. As a remarkable outcome, a number of ranges of exponents $\sigma$ and $m$ for which peaking solutions exist for any $p>p_S(\sigma)$ are deduced, contrasting to the previously considered exponent $\sigma=0$, where a Lepin-type critical exponent $p_L$ provides an upper bound for $p$.

It is well known that the sum of two bent functions on $\mathbb{F}^n$ has algebraic degree at most $n/2$ for even $n\ge 4$. Tokareva conjectured that the converse holds: every Boolean function with degree at most $n/2$ can be written as a sum of two bent functions. The conjecture has been verified for even $n\leq 6$ and for several special classes of Boolean functions. We disprove this conjecture by showing that $x_1x_2(x_3x_4+x_5x_6+x_7x_8)$ is not a sum of two bent functions on $\mathbb{F}^8$. We extend this construction to obtain counterexamples in every even dimension $n\ge 8$. Using Shaporenko's reduction, we also obtain explicit counterexamples to Tokareva's derivative conjecture in every even dimension $n\ge 10$.

We undertake a systematic comparison of cluster algebras associated to Legendrian knots of the same smooth knot type. We describe a cluster structure on the m-graded augmentation variety of Legendrian representatives of a family of double twist knots that includes all orientably Lagrangian fillable twist knots. For different Legendrian representatives of the same smooth knot type, we show that these cluster structures are equivalent. Along the way, we characterize the representatives that admit Maslov-m exact Lagrangian fillings, construct a Catalan number of such fillings for each representative, and show that any Maslov-0 fillable representative is Legendrian isotopic to the (-1)-framed closure of a positive braid. Finally, we give two Legendrian representatives of a cabled twist knot that yield equivalent cluster structures of infinite type.

Fuzzy logics with truth values in the unit interval vary in their choice of propositional connectives and their semantics, ranging from just minimum and maximum as in Zadeh logics to truncated addition and subtraction as in {\L}ukasiewicz logics or their extensions with truth constants, known as Pavelka logics. In the more specific world of fuzzy modal logics, non-expansive fuzzy modal logic has recently been proposed as a sweet spot between {\L}ukasiewicz (more precisely, Pavelka) and Zadeh fuzzy modal logic; it restricts {\L}ukasiewicz propositional connectives to have one argument constant, equivalently extends the Zadeh propositional base with constant shift operators. It retains reasonable expressiveness both in practical terms and abstractly; in particular, it characterizes the standard notion of behavioural distance on fuzzy Kripke frames. At the same time, it allows for satisfiability checking in the same complexity as for its two-valued correspondent (the modal logic K, or the description logic ALC), in contrast to the notorious computational hardness of {\L}ukasiewicz modal logic. In the present work, we show that non-expansive fuzzy modal logic admits uniform interpolation, a key property in modularization, information hiding, summarization, and other applications that fails for both {\L}ukasiewicz (or Pavelka) and Zadeh modal logic.

A sampled robust control problem averages future values over disturbance laws, while its continuous-time counterpart evolves with an averaged drift. We establish a fluid limit connecting these two models for Wasserstein ambiguity sets. Under a uniform disturbance moment bound and subcritical growth, sampled Bellman values converge locally uniformly to the upper and lower values of a discounted differential game. The argument accommodates unbounded disturbances without requiring square-integrable increments. We also derive an explicit bound on the change in either game value when both the radius and center of the ambiguity set vary. For Euclidean disturbance space, uniform Lipschitz continuity in the disturbance, common state and cost Lipschitz bounds, and a discount rate exceeding the drift's state Lipschitz constant, this estimate converts distributional confidence bounds into uniform value approximation bounds. Static Wasserstein duality reformulates pointwise distributional optimization; affine disturbance dependence permits an exact reduction to a ball of means. Experiments illustrate the common limit and decreasing value error with sample size. For a nonlinear disturbance model, uniform shifts overestimate worst-case recovery speed by about 36\%. In a two-product inventory model, increasing ambiguity raises production near balance while reducing the region of full production in deeper backlogs.

We present a rigorous construction of novel time-periodic solutions to the radially symmetric cubic conformal wave equation on the Einstein cylinder $\mathbb{R}\times\mathbb{S}^{3}$. Their existence, hinted by the earlier numerical studies (arXiv:2504.10513), complements prior results on Cantor sets of infinitesimally small solutions. Our proof, based on the contraction principle, requires extensive computations that are performed on a computer by the means of rational arithmetic. We are able to significantly reduce their amount by a thorough analysis of the inverse of the associated linear operator, leading to sharper analytical bounds, as well as introduction of simplified versions of certain relevant operators. We provide examples of three solutions with specific frequency; however, our techniques and tools can be directly adapted to prove the existence of other solutions conjectured in (arXiv:2504.10513).

For p, q >= 3 with gcd(p,q) = 1 we show that the symmetry group of the Turk's head knot Th(p,q), the closure of the braid (sigma_1 sigma_2^{-1} sigma_3 sigma_4^{-1} ... sigma_{p-1}^{+-1})^q, is the dihedral group of order 4q. This proves Conjecture R of Di Prisa and \c{S}avk for q >= 3; the case p = 3 is due to Sakuma and Weeks. The argument is diagrammatic: by the flyping theorem of Menasco and Thistlethwaite, the symmetries of a knot with a prime, reduced alternating diagram without non-trivial flypes are the symmetries of the diagram (we write out a proof for the diagrams at hand), and we show that the standard diagram of Th(p,q) has no non-trivial flypes and exactly 4q symmetries. The chirality of Th(p,q) for even p, which also follows from a theorem of Murasugi, is visible in the group. We compare the argument with the geometric one of Sakuma and Weeks for p = 3.

Let $I = I(G)$ be the edge ideal of a finite simple graph $G$. We investigate the relationship between the associated primes and $v$-numbers of powers $I^t$ and their integral closures $\overline{I^t}$. For the third power, we establish the associated prime containment $\operatorname{Ass}(S/\overline{I^3}) \subseteq \operatorname{Ass}(S/I^3)$, which yields $v(I^3) \le v(\overline{I^3})$. In contrast, for $t \ge 4$, we show that $\operatorname{Ass}(S/\overline{I^t})$ and $\operatorname{Ass}(S/I^t)$ are in general incomparable. Furthermore, for every fixed $t \ge 3$ and $r\ge 4$, we construct families of graphs for which the discrepancies $v(\overline{I^t}) - v(I^t)$ and $v(I^r)-v(\overline{I^r})$ are arbitrarily large.

We study the Cauchy problem for the semilinear Schr\"{o}dinger equation with a harmonic potential and combined power-type nonlinearities. The goal is to classify initial data that lead to either global existence or blowup solutions by using an enhanced cross-constrained variational theory. First, we review and analyze this variational approach, pinpointing the main issue for our specific problem. We then introduce new variational structures by carefully selecting terms in the energy functional and adjusting their weights, enabling us to create eight cross-constrained variational combinations and related manifolds to distinguish initial data that result in global solutions or finite-time blowup. Using these structures, we examine three initial energy levels: negative, non-negative low, and high energy. Notably, we prove that for any initial data with negative energy, solutions blow up in finite time regardless of their size. Throughout the study, we utilize these new variational frameworks and functionals to systematically classify initial data for global existence or finite-time blowup. We highlight the comprehensiveness of our approach as we analyze numerous combinations of the energy and mass terms, despite the complexity involved. Interestingly, we find that when initial energy is minimized, classifications based on different functionals coincide. For high initial energy, an unresolved issue even for the standard Schr\"{o}dinger equation, we develop a strategy controlling bounds of energy to classify the solution dynamics, including global existence or blowup, for our model. We also provide a complete proof for the standard case addressing this open problem.

We study the free energy fluctuations of spherical and Ising perceptron models with general even activations. These include the symmetric binary perceptron, the $U$-perceptron, and the bipartite Sherrington--Kirkpatrick model. Our main result is a Gaussian CLT for the free energy that holds both in the ``high-temperature'' interior of a second moment RS regime and on its ``critical'' boundary. At criticality, the CLT has scale $(\frac{\log N}{6})^{1/2}$, which matches the free energy fluctuations of the critical Sherrington--Kirkpatrick model and suggests this scaling is universal. The proof shows that the perceptron model's free energy fluctuations are explained, up to an additive $o(1)$ error with high probability, by those of a corresponding spherical Hopfield model, and that this relation persists even at criticality. This proof strategy builds on a critical reweighted moment method developed by the first and third authors \cite{du2026fluctuations} to identify the critical fluctuations of the SK free energy, which links these fluctuations to the spectral edge spacing of the disorder.

The scaling limits of interfaces in the critical two-dimensional Ising model are described by the Schramm--Loewner evolution $\mathrm{SLE}_3$. While, in the scaling limit, the law of the whole near-critical FK-Ising configuration is known to be singular with respect to the critical law in the thermal direction, and the same is expected for the spin configuration, we prove in this paper that the law of a single interface is not. More precisely, we consider the Ising model with mesh $\delta$ in the Dobrushin square at $\beta=\beta_c-m\delta$, where $m\ne0$ is fixed, and we prove that the Radon-Nikodym derivative of the law of the interface with respect to its critical law has moments of every positive and negative order, bounded uniformly in $\delta$, so that the law of every subsequential scaling limit of the interface is mutually absolutely continuous with respect to that of $\mathrm{SLE}_3$. Our analysis explains the mechanism behind this result: the logarithmically growing specific heat (that is, the variance of the Hamiltonian, normalized by the number of edges) cancels between the full domain and the domain slit along the interface.

This paper is devoted to the study of source reconstruction algorithms for coupled systems of heat equations, with either constant or spatially dependent coupling terms, where internal measurements are available from a reduced number of observed states. Two classes of systems are considered. The first comprises parabolic equations with constant zero-order coupling terms (through a matrix potential term or via the diffusion matrix). The second type considers parabolic equations coupled by a matrix potential that depends on spatial variables, which leads to the analysis of a non-self-adjoint operator. In all configurations, the source is assumed to be of separate variables, the temporal part is a known scalar function, and the spatial dependence is an unknown vector field. Several numerical examples using the finite element method in 1D and 2D are presented to show the reconstruction of space-dependent sources.

We prove sharp bounds for the upper and lower intermediate dimensions of orthogonal and radial projections of compactly supported measures. Both dimension functions are preserved almost everywhere when the quasi-Assouad dimension does not exceed the target dimension. We obtain exceptional-set estimates using the entire source dimension function, with sharpness for explicit families. We also construct a fixed measure whose projected dimension functions vary on open direction sets, even when all source dimensions lie below the target dimension.

We give a complete characterization of the kernels $\Omega\in \Lup^1(\Sp)$ for which the rough singular integral $\T_\Omega$ and the associated maximal operator $\M_\Omega$ satisfy pointwise bounds by $\Int_1(|\nabla \cdot|)$ and $\M_1(|\nabla \cdot|)$, respectively, where $\Int_1$ is the Riesz potential and $\M_1$ is the fractional maximal operator. These bounds hold if and only if $\Omega$ belongs to the homogeneous dual Sobolev space $\dot \W^{-1,\infty}(\Sp)$. Equivalently, $\Omega=\text{div}_\Sp \mathbf{V}$ for some bounded vector field $\mathbf{V}$ on the sphere, or $\big|\int_E\Omega\big|\leq C\,\mathcal H^{d-2}(\partial E)$ for every open set $E\subseteq\Sp$ with smooth boundary. On the Lebesgue scale, we show that for $d\geq3$ the condition $\Omega\in\Lup^{d-1,\infty}(\Sp)$, together with mean zero, suffices, and that the exponent $d-1$ is sharp. Finally, in dimension $d=2$, we use complex-analytic methods to prove a new pointwise bound in terms of the complex derivative $\bar\partial f$ that is valid for every mean-zero $\Omega\in\Lup^1(\mathbb S^1)$. As a consequence, we obtain new Sobolev bounds for rough singular integrals in the plane.