数学前沿

Let $(X, \Omega)$ be an affine Calabi--Yau variety, by which we mean a klt affine variety $X$ with a non-vanishing algebraic volume form $\Omega$. In connection with complete Calabi--Yau metrics of Euclidean volume growth on $(X, \Omega)$, we study algebraic aspects of polarizations of $X$ given by a negative valuation. We show that such a valuation is finitely generated with a klt degeneration if and only if $X$ admits a dlt log Fano type compactification, generalizing a compactification considered by Tian--Yau. We further give a valuative criterion detecting K-semistability of the degeneration by a negative valuation.

We compute the smallest non-abelian quotients of spherical and affine Artin groups. Usually, these are isomorphic to the smallest non-abelian quotients of the corresponding Coxeter groups. As applications, we deduce that two affine Artin groups are isomorphic if and only if their Coxeter diagrams are isomorphic and that irreducible spherical Artin groups are distinguished from each other by their finite quotient groups.

A smooth projective variety equipped with a suitable torus action admits a cell decomposition, called the Bia{\l}ynicki-Birula decomposition. The singularities of the closures of these cells are not well understood in general. Schubert varieties in a flag variety~$G/B$ are classical examples of such closures, and several criteria for their smoothness are known. In this paper, we study the closures of Bia{\l}ynicki-Birula cells in regular semisimple Hessenberg varieties~$\mathrm{Hess}(s,h)$, called Hessenberg Schubert varieties. We first investigate the smoothness of intersections of Schubert varieties with $\mathrm{Hess}(s,h)$, obtaining a sufficient condition for a Hessenberg Schubert variety to be smooth. We also establish several combinatorial sufficient conditions for the smoothness of Hessenberg Schubert varieties in type $A$.

We construct piecewise smooth solutions of the full compressible Euler equations that evolve from entropy waves into vortex sheets while retaining a strictly positive Taylor jump. For every integer $\kappa\geq1$, we obtain solutions satisfying initial material compatibility for pressure and velocity through order $\kappa$, whose tangential velocity jump has a nonzero leading term of order $t^{\kappa+1}$. These solutions are periodic in the tangential variables and admit a smooth realization in a slab with impermeable walls. We also construct solutions on a local nonperiodic tangential patch that satisfy initial material compatibility at every order, yet have a tangential velocity jump that is nonzero for every sufficiently small positive time and flat at $t=0$. The finite-order construction uses a normal Cauchy-Kowalevski problem for velocity potentials, whereas the all-order construction uses a homogeneous reduction to a smooth hyperbolic Cauchy problem.

We consider the small-time asymptotic expansions for the Dirichlet and Open heat contents of convex planar domains. We prove that the small-time asymptotic expansion for the Open heat content can determine a triangle among triangles, a parallelogram among parallelograms, and an acute trapezium among acute trapezia of the same height. In addition, for a sequence of convex polygons converging to a convex set with smooth boundary in the sense of Hausdorff, we show that the coefficients of order $t$ in the small-time asymptotic expansions of the Dirichlet and the Open heat contents converge to those of the limiting set. While, for a sequence of convex sets with smooth boundary converging to a convex polygon in the sense of Hausdorff, we show that the coefficients of order $t$ do not converge to that of the limiting polygon.

The space of weak-$L^{\infty}$ on measure space was introduced by Bennett et al. (Ann. of Math., 113(1981), 601-611) and proved to contain the space of $L^\infty$ and possess the appropriate interpolation property. In this paper, we are aimed to continue this line of research on the Euclidean space $\mathbb R^n$ endowed with Hausdorff content, which is a low-dimensional outer measure on $\mathbb{R}^n$, and fails to be additive or absolutely continuous with respect to Lebesgue measures. But low-dimensional subsets, such as fractals, of $\mathbb{R}^n$ are Hausdorff capacitable. Precisely, in the setting of Hausdorff content, an analogue of weak-$L^\infty$ is defined and proved to possess an improved Marcinkiewicz interpolation theorem. Moreover, we show that the space of functions with bounded mean oscillation associated to the Hausdorff content is embedded into this new weak-$L^\infty$ space by establishing a key estimate for the distribution via a packing condition. As a by-product, the boundedness of the Hardy-Littlewood operator on the space of weak-$L^{\infty}$ is established.

We propose a new geometric interpretation of Eguchi's construction, which associates to a statistical divergence, an essential ingredient of Information Geometry, a statistical structure on a differentiable manifold. Our approach is based on bi-Lagrangian geometry. Building on the work of O.~E.~Barndorff-Nielsen and P.~E.~Jupp~\cite{Barndorff}, we show that a statistical divergence naturally induces a bi-Lagrangian structure, hence a para-K\"ahler structure. Within this framework, the Riemannian metric $g^D$ and the pair of dual connections $(\nabla^D,{}^*\nabla^D)$ of Eguchi's construction are induced by the para-K\"ahler structure: $g^D$ is the restriction to the diagonal of the para-K\"ahler metric $G$ of signature $(d,d)$, while $\nabla^D$ and ${}^*\nabla^D$ are obtained, through the two natural projections, from the Bott connections of the two transverse Lagrangian foliations, which are themselves the restrictions of the Levi-Civita connection of $G$ (the Hess, K\"unneth or bi-Lagrangian connection). This perspective shows that the statistical structure associated with a divergence is not merely the formal outcome of successive differentiations, but the manifestation of an underlying bi-Lagrangian geometry. Our approach thus provides an intrinsic and unified understanding of Eguchi's construction, highlights the connections between information geometry, symplectic geometry and para-K\"ahler geometry, and opens new research perspectives at the interface of these fields.

In multi-objective model predictive control (MOMPC), a single Pareto-optimal solution has to be selected at each sampling instant by an automated a priori decision-making (DM) mechanism, since real-time constraints do not permit a human decision-maker to be involved. This paper considers MOMPC for fixed point stabilization, where the DM translates a high-level preference into the solution to be implemented, while closed-loop stability has to be preserved. To this end, an approach called individual minima-informed DM is introduced. These methods leverage the individual minima, which are characteristic Pareto-optimal points, to facilitate the translation of a high-level preference to a point on the Pareto front. They can be implemented through two sequential optimizations, regardless of the number of objectives, thereby facilitating a real-time implementation of MOMPC. Two novel methods are proposed and compared with four established ones. These methods are embedded into a stabilizing MOMPC scheme that builds on a quasi-infinite horizon framework from the literature. Closed-loop stability is proven under a descent condition that is less restrictive than the one in the original framework and retains the largest DM space among all descent conditions that prevent any objective from increasing. The required terminal ingredients can be constructed with a practical procedure. In a numerical case study, the resulting scheme stabilizes the fixed point and illustrates the potential benefit of adapting the preference online.

We establish necessary and sufficient conditions for global H\"{o}lder solvability of weighted Poisson equations and related fractional problems. On domains satisfying a capacity density condition, a nonnegative measure with bounded support depth admits a bounded globally H\"{o}lder continuous zero-boundary solution if and only if it satisfies an interior Morrey-type growth condition. Conversely, uniform boundary decay estimates for solutions with localized nonnegative sources characterize the capacity density condition. Analogous characterizations are established for the restricted and spectral fractional Laplacians. Our approach is based on weighted strong barriers and also yields two-weight Hardy inequalities, compact trace embeddings, and global H\"{o}lder solvability for nonlinear Dirichlet-to-Neumann operators.

In this work, we correct results in the published paper [Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. 118 (2024), article 22]. We determine the non-hit kernel of the first reduced power in polynomial degree $p+1$, identify the published Frobenius construction as zero on indecomposables, and replace the multiplier assertion by a theorem with explicit degree hypotheses. We also classify the monomial multipliers of Frobenius that preserve hits in one variable. For the third algebraic transfer, we correct the target dimensions and establish surjectivity throughout the four principal degree families. Explicit invariant calculations and Minami's degree reduction give isomorphisms in seven infinite families at $p=3$ and $p=5$, including a family with an initial change from dimension one to dimension two. For every odd prime, compatibility with the Frobenius operation on $\operatorname{Ext}$ gives an exact sequence comparing successive transfer kernels and forces a nonzero invariant extraction kernel at every second-family dimension jump. We compute the first cohomology of a nonzero extraction kernel and determine the exact obstruction to injectivity from rank-two products in the first and fourth families. Coefficient extraction and changes of coordinates give an additional infinite isomorphism family and a spanning result for annihilated polynomial homology. At each extraction isomorphism, the quotient by all rank-two products is preserved as a general linear group module. We distinguish these results from the unresolved invariant dimensions at arbitrary normalized index differences, and give the precise limitations of composition-factor counts and Dickson multiplication. An appendix supplies explicit representatives for all the nonzero rank-two invariant lines in Crossley's dimension classification.

In this paper, we classify connected amply regular graphs with diameter $d \geq 4$ and parameters $(v, k, \lambda, \mu)$ satisfying $\mu = \frac{k-1}{2}$, where $k\geq 5$ is odd. We prove that such a graph must be exactly one of the following: the $5$-cube, the graph $\K_2 \square \Lambda$, where $\Lambda$ is the unique bipartite $(0,2)$-graph on $14$ vertices, or the point--block incidence graph of a group divisible design with the dual property, namely a $GDDDP\left(2, k+1;\, k;\, 0, \frac{k-1}{2}\right)$. For the last family, we give equivalent characterizations in terms of bipartite $Q$-regular graphs and relation graphs of symmetric association schemes with five classes. Furthermore, we present constructions of such amply regular graphs, yielding infinite families of examples derived from Paley graphs, Peisert graphs, and Paley digraphs.

The present work focuses on the study of the renowned Collatz conjecture, also known as the $3x+1$ problem. The analysis rests on the dynamics of an iterative map in binary form, defined on the dyadic rationals of $[1/2,\,1)$. We estimate how the length of the binary expansion of the iterated numbers changes, and we regard the iterative map as a perturbation of a circle map without periodic points, with an explicit bound for the error between both maps. A condition on the periods of periodic orbits is given; it is the classical argument of Crandall, and it recovers the lower bound of Eliahou for the length of nontrivial cycles. This version corrects several errors of the first one; they are listed in the Addendum.

Solutions of second-order elliptic PDE problems on nonconvex or cracked domains with mixed Dirichlet-Neumann conditions typically lie outside $H^2(\Omega)$. Near re-entrant corners, crack tips and points where the boundary condition changes type, the gradient grows like $r^{\lambda-1}$ with $\lambda<1$. We study residual-based adaptive refinement in hybrid variational-collocation PINNs. A multiscale Fourier-feature network is trained on a composite loss. The loss combines a variational residual tested against Legendre bubble functions, a strong-form collocation term and penalties for the Dirichlet and Neumann data. The variational term needs only first derivatives of the network, whereas the collocation term requires second derivatives and provides a residual that can be evaluated at any point. After each training stage, candidate points selected by this residual are added to the collocation set. The method is tested on five two dimensional benchmarks, namely a Mode~III crack, a cross-shaped domain with four re-entrant corners with unequal amplitude, a reaction-diffusion problem with four localized peaks, a monotone semilinear problem with a boundary layer and the Motz problem. Compared with uniform point insertion under the same collocation and iteration budget, residual-based refinement reduced the relative $L^2$ error by up to a factor of $32$, the relative $L^\infty$ error by up to a factor of $38$ and the relative $H^1$ semi-norm error by up to a factor of $15$.

Although classical PID control is widely used in engineering practice, identifying all PID gains that guarantee global regulation of nonlinear uncertain dynamics remains challenging. For a class of second-order nonaffine systems under one-sided derivative bounds, we establish necessary and sufficient conditions for global PID regulation: a fixed PID controller globally regulates the entire infinite-dimensional uncertainty class if and only if it regulates a single boundary linear system. Applying the Routh--Hurwitz criterion to this boundary linear system yields the exact admissible PID gain region. To establish this equivalence, we prove a structured Markus--Yamabe theorem for third-order integrator chains under uniform derivative-ratio conditions. Its proof combines a two-dimensional storage function constructed via viscosity solutions of a Hamilton--Jacobi equation with a scalar comparison argument. We further show that the same PID gain region remains exact for multidimensional affine systems under symmetry assumptions on the state partial Jacobians and the input matrix.

The stable derived category $D_{sg}(R)$ of a Gorenstein ring $R$ is the Verdier quotient of the bounded derived category $D^b(\mathsf{mod}\, R)$ by the thick subcategory of perfect complexes, and was introduced by Buchweitz as a homological measure of the singularities of $R$. This thesis contributes to its study, centered around representation-theoretic and Koszul duality aspects. In Part I, we complete (over $\mathbb{C}$) the classification of homogeneous complete intersection isolated singularities $R$ for which $D^{\mathbb{Z}}_{sg}(R)$ (respectively, $D^b(\mathsf{coh}\, X)$ for $X = \mathsf{proj}\, R$) contains a tilting object. We then use results of Buchweitz-Iyama-Yamaura to describe the indecomposable objects in $D_{sg}^{\mathbb{Z}}(R_Y)$ for $R_Y$ the coordinate rings of $4$ points on $\mathbb{P}^1$ and $4$ points on $\mathbb{P}^2$ in general position, and classify their Betti tables. In Part II, for $R$ a Koszul Gorenstein algebra, we study a natural pair of subcategories whose intersection $\mathcal{H}^{\mathsf{lin}}(R) \subseteq D_{sg}^{\mathbb{Z}}(R)$ consists of modules with eventually linear projective resolutions. We prove that such a pair forms a bounded t-structure if and only if $R$ is absolutely Koszul in the sense of Herzog-Iyengar, in which case $D^b(\mathcal{H}^{\mathsf{lin}}(R)) \cong D_{sg}^{\mathbb{Z}}(R)$. We then relate the heart to modules over the Koszul dual $R^!$. This extends the BGG correspondence to any pair of Koszul dual algebras $(R, R^!)$ with $R$ absolutely Koszul Gorenstein, applying e.g. to elliptic normal curves and del Pezzo surfaces of degree $\geq 4$. Finally, we relate our results to conjectures of Bondal and Minamoto on the graded coherence of AS-regular algebras and higher preprojective algebras; we characterise when these conjectures hold in a restricted setting, and give counterexamples to both in all dimensions $\geq 4$.

More than twenty years ago, Croke and Dairbekov showed that a negatively curved metric $g_0$ on a closed surface has minimal area among all negatively curved Riemannian metrics $g$ on the same surface whose marked length spectrum is greater or equal to that of $g_0$. We show that the curvature assumption on the second metric can be removed and extend this result to reversible Finsler metrics. More precisely, we prove that a reversible Finsler metric $F_0$ on a closed surface whose geodesic flow is Anosov has minimal area among all reversible Finsler metrics $F$ on the same surface whose marked length spectrum is greater or equal to that of $F_0$.

For a semigroup $S$ and a right $\mathbb{Z}[S]$-submodule $J\leq \mathbb{Z}[S]^n$, we study expansivity of the algebraic action of $S$ induced on the Pontryagin dual of $\mathbb{Z}[S]^n/J$. We completely determine the class of semigroups for which expansivity of this action is characterized by the triviality of $J^\perp$, in terms of the existence of a finite subset $K\subseteq S$ such that $S=KS$. This condition is satisfied in particular by every monoid, and more generally by every semigroup with a left unital convolution Banach algebra, in which case we are able to extend the characterization of expansivity in terms of an invertibility condition when $J$ is finitely generated. We also exhibit examples of non-unital semigroups satisfying the hypotheses of our results.

In this article, we apply the derived Morita theory of dg-categories to show how to extend the domain of validity of many identities relating Morita invariants from associative dg-algebras toward non-commutative scheme. Doing so, we obtain that the dg-category of associative algebras can be used to test the exactness of any sequence and the commutativity of any diagram involving Morita invariants depending multilinearly on their arguments, under a mild condition of stability by cofibrant replacement. This gives a simple picture of how the underlying algebra of a non-commutative scheme captures its Morita invariant properties. As an application, we use a K\"unneth formula on non-commutative schemes to factorise the Hochschild cohomology of a product of quasi-phantom categories such as built by Orlov and Gorshinsky in [GO13]. This is an occasion to explore how the derived Morita theory interacts with the \v{C}ech dg-enhancement of a projective scheme and to shed an unifying light upon the landscape of dg-modules over dg-categories.

The heart of the a priori and a posteriori error control in convex minimization problems is the sharp control of the differences of discrete and exact minimal energy. Conforming finite element discretizations for p-Laplace type minimization problems provide upper bounds of the energy difference with optimal convergence rates. Even for smooth solutions, known convergence rates for higher-order non-conforming finite element discretizations for the same problem class with $2 < p < \infty$, however, are exclusively suboptimal. Thus the popular a posteriori error control within the two-energy principle, that generalize hyper-circle identities, appears unbalanced. The innovative point of departure in a refined analysis of two discontinuous Galerkin (dG) schemes exploits duality relations between a discrete primal and a semi-discrete dual problem. The infinite-dimensional dual problem leads to a tiny duality gap that even vanishes for polynomial low-order terms. For a class of degenerated convex minimization problems with two-sided $p$ growth, the novel duality provides improved a priori convergence rates for the error in the minimal energies. This closes the misfit of convergence rates for the conforming and nonconforming schemes at least for the local discontinuous Galerkin schemes at hand. The motivating two-energy principle and some post-processing for a Raviart-Thomas dual variable provides an a posteriori error control, that also may drive adaptive mesh-refining. Computational benchmarks provide numerical evidence for improved convergence rates of the adaptive beyond uniform mesh-refining.

We introduce the Space-Time Markov Chain Approximation (STMCA) for a general diffusion process on a finite metric graph $\Gamma$. The STMCA is a doubly asymmetric (in both time and space) random walk defined on subdivisions of $\Gamma$, with transition probabilities and conditional transition times that match, in expectation, those of the target diffusion. We derive bounds on the $p$-Wasserstein distances between a non-explosive diffusion and its STMCA in terms of a thinness quantifier of the subdivision. An extension to absorbed diffusions uses a boundary-adapted Green-kernel thinness that remains finite even at certain singular exit boundaries. Under the stated regularity and subdivision hypotheses, this gives convergence at any rate strictly less than $\frac14\wedge\frac1p$ in the appropriate thinness quantifier. For suitably adapted subdivisions satisfying a quadratic thinness bound, we show that the rate is doubled: we achieve any rate strictly less than $\frac12\wedge\frac2p$ in terms of the maximum cell size. Additionally, we provide explicit analytical formulas for transition probabilities and times, enabling practical implementation of the STMCA, as well as easily verifiable sufficient non-explosion conditions for the bounds to hold. Numerical experiments illustrate our results.