数学前沿

A general method of resolution can take part in forming problems once their language represents the method itself. A fixed formation rule gives each sound resolver a presentation it cannot answer. Admitting the diagnosis yields a sound successor with an instance of its own. For standard arithmetical theories $\mathrm{T}$, the instance is $\mathsf{HA}$-provably equivalent to $\mathrm{Con}(\mathrm{T})$. A theory that decides its own instance asserts a false sentence. The $m$-th iterated independence question is equivalent to the instance $m$ stages later. No effective admission policy with consistent union closes the regress. A noncomputable answer function resolves all program indices, while a false assertion of resolution supplies no warrant. At feasible scale, bounded self-consistency is cheap. A clocked G\"odel sentence gives an unconditional gap relative to its short binary description, but no superpolynomial lower bound in the window size. The arithmetic identity motivates studying bounded consistency of the successor theory, as in a conjecture of Pudl\'ak. We state its finitely axiomatized case and prove that it implies $\mathcal{NP}\ne\mathrm{co}\mathcal{NP}$. The conjecture motivates a feasible analogue, but does not establish all clauses of the unbounded barrier.

We prove that every finite symmetric tensor category over an algebraically closed field of positive characteristic admits a symmetric tensor functor to a higher Verlinde category. This was conjectured for arbitrary symmetric tensor categories of moderate growth by Benson, Etingof and Ostrik. The proof uses higher Frobenius functors, introduced by the authors in [CF1]. The main effort is directed towards a proof of a conjecture from [CF1] about thick tensor ideals in the representation category of elementary abelian $p$-groups, which implies that the higher Frobenius functors are symmetric monoidal. In fact, we prove a refinement of the latter conjecture that relates tensoring with restricted Steinberg modules to an iterated generalised cohomology of modules of elementary abelian $p$-groups.

We present a comprehensive generalization of Lusztig's braid group symmetries for quasi-split iquantum groups. Specifically, we give 3 explicit rank one formulas for symmetries acting on integrable modules over a quasi-split iquantum group of arbitrary Kac-Moody type with general parameters. These symmetries are formulated in terms of idivided powers and iweights of the vectors being acted upon. We show that these symmetries are consistent with the relative braid group symmetries on iquantum groups, and they are also related to Lusztig's symmetries via quasi $K$-matrices. Furthermore, through appropriate rescaling, we obtain compatible symmetries for the integral forms of modified iquantum groups and their integrable modules. We verify that these symmetries satisfy the relative braid group relations.

Modeling to Generate Alternatives (MGA) methods explore the near-optimal solutions of energy system optimization models (ESOMs), i.e., alternative planning pathways whose cost lies within a small predefined slack of the optimum but which can differ substantially in technology composition, regional distribution, and environmental impacts. Among MGA methods, ORACLE uniquely guarantees convergence to the near-optimal set within a targeted tolerance by iteratively refining inner and outer polytope approximations. At each iteration, given a trial point, it solves a problem as hard as the ESOM to find the nearest near-optimal solution to it. However, the trial point is obtained by solving a bilevel problem whose solution time grows rapidly with the number of iterations and eventually dominates the computational cost of the method. To eliminate this bottleneck, we present G-ORACLE, which exploits the fact that an optimal solution of the bilevel problem is always attained at a vertex of the outer approximation. G-ORACLE therefore replaces the bilevel problem with incremental vertex enumeration by the double description method and one convex projection per vertex, while retaining the convergence guarantee of ORACLE. A supporting-hyperplane check further avoids unnecessary recomputation of these projections as the inner approximation grows. We evaluate G-ORACLE on an electricity-only PyPSA-Eur model, exploring the near-optimal space of the installed capacities of six generation and two storage technologies. While ORACLE's bilevel problem exceeds 20 minutes at iteration 64, G-ORACLE performs the trial-point search in at most 14 seconds throughout 240 iterations, leaving the ESOM solves as the only significant computational cost.

Symmetric polyconvexity provides a sufficient condition for weak lower semicontinuity in small-strain elasticity and a tractable lower bound for the relaxation of non-(quasi)convex energies. Since classical algorithms for computing such functions can be computationally expensive, neural network approximations have recently emerged as an alternative. A central difficulty with establishing such neural network based approaches for symmetric polyconvexity is the additional monotonicity property. As we show, the corresponding weight restrictions on ReLU-based input-convex networks do not yield universal approximation of the lifted representers. Here, we introduce symmetric polyconvex neural networks (SPCNNs) for constitutive modeling and approximation of symmetric polyconvex envelopes. The deep architecture combines convexity with the monotonicity required of a representer in the space of minors. We prove a universal approximation theorem for symmetric polyconvex energies, without requiring isotropy, and quantify the number of affine pieces attainable by the lifted network and derive reduced architectures for isotropic energies. Numerical examples illustrate stress-based regression and approximation of the symmetric polyconvex envelope of a double-well potential. Symmetric polyconvexity is guaranteed by construction, while the accuracy of the fitted energies and envelopes is assessed numerically.

This paper develops a rigorous analytic framework for the hyperbolic Monge-Amp\`ere equation on strip-like domains, which model wrinkled patterns in thin elastic sheets. Our work addresses the rigid side of the classical rigidity-flexibility dichotomy by defining this regime not by high smoothness, but by the more fundamental property of partial convexity. The hodograph transformation is the natural tool for this setting, as its validity is predicated on partial convexity. It converts the nonlinear Monge-Amp\`ere equation into a linear damped wave equation, allowing us to formulate a well-posed Cauchy-Goursat problem. A key challenge is the corner singularity that arises where characteristic and non-characteristic boundary data meet. To resolve this, we develop a parametrix-corrector decomposition that captures the solution's inherent singular behavior. This method recasts the problem as a singular Volterra integral equation, for which we prove the existence and uniqueness of a new class of hodograph weak solutions. Finally, we derive energy estimates to establish the quantitative stability of these rigid solutions under perturbations of the underlying curvature function.

A linear monodimensional lattice Boltzmann scheme with three velocities tackling the advection equation is investigated. We prove that it is second-order accurate and provide necessary and sufficient stability conditions through an explicit formula involving the two relaxation parameters. Relying on existing results for finite difference schemes, we prove convergence for smooth initial data with initialization of the non-conserved moments at equilibrium. Several numerical simulations illustrate these theoretical findings.

This paper develops a piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation. The discrete space imposes continuity of vertex sums on $(d-2)$-faces and of averaged normal derivatives across interior $(d-1)$-faces. A local algebraic characterization of $P_2$ is first established through the unisolvence of an enriched polynomial space and the resulting compatibility conditions. For each interior element, a $3^d$-element patch is then introduced, and the associated homogeneous patch space is shown to be one-dimensional. The corresponding patch functions, extended by zero, form a basis of the global homogeneous discrete space. A local quadratic interpolation operator is further constructed in arbitrary dimensions. Its approximation properties, together with a consistency estimate derived from the weak continuity conditions, lead to the optimal broken $H^2$ error estimate for the biharmonic problem.

We take the first steps towards developing a tropical homotopy theory, which we use to construct explicit finite CW complex structures on smooth complex varieties. Our approach decomposes a variety into a set of radial and angular parts, a construction we call polar decomposition. We derive a combinatorial formula for the $f$-vector of the resulting CW complex in terms of a weighted sum of local angle $f$-vectors, and we determine these $f$-vectors explicitly for toric varieties and complete intersections in terms of Stirling numbers. To construct these decompositions, we introduce the notion of a rational tropical manifold, which we view as a tropical manifold with a non-reduced structure. The local topology of these is dictated by Kummer coverings of affine linear spaces in a complex torus, as opposed to the affine linear spaces in the theory of tropical manifolds. Examples of such manifolds include the Fermat and Dwork hypersurfaces. Using Kato-Nakayama spaces, we associate a diagram of topological spaces to any rational tropical manifold. We then construct a cofibrant replacement of this diagram using sealed Kato-Nakayama spaces to show that the homotopy colimit is homotopy equivalent to a general nearby fiber. To construct CW complex structures, we then show that the spaces in the diagram can be replaced by their angle sets. More precisely, for any Kummer covering of a very affine linear space, we prove that the angle map is a homotopy equivalence, extending earlier work of Salvetti and Bj\"{o}rner--Ziegler. We construct compatible cellulations for these angle spaces and show that these give rise to an explicit CW complex structure on the nearby fibers. For complete intersections, we make these spaces and all the CW complex structures completely explicit using refined Coxeter arrangements.

The inviscid generalized surface quasi-geostrophic (gSQG) equations are a family of active scalar equations parametrized by a scalar $\alpha$ that includes the two-dimensional Euler equations ($\alpha = 0)$ and the SQG equation $(\alpha = 1)$. We prove that, for every $\alpha \in (\frac{7}{4}, 2)$ and every $T > 0$, there exists a self-similar patch solution whose boundary is a smooth, unbounded graph for $0 \leq t < T$ and develops a corner singularity at time $T$.

Let $W(r,k)$ be the least $N$ such that every $r$-colouring of $\{1,\dots,N\}$ contains a monochromatic $k$-term arithmetic progression. Many of the best lower bounds for $W(r,k)$ with $r\ge3$ come from blow-up recurrences of Blankenship, Cummings and Taranchuk and of Xu, in which a known colouring is lengthened by a periodic block. Call an $h$-colouring of $Z_q$ $k$-polychromatic if every $k$-term progression with nonzero common difference meets each of the $h$ colours. The argument behind these recurrences shows that such a colouring gives $W(r,k)>q(W(r-\lceil r/h\rceil,k)-1)$ for all $r\ge h$. We show that cubic, quartic, quintic and sextic residue colourings give $k$-polychromatic blocks with $q$ much larger than $k$ (e.g. $q=1,376,317$ for $h=3$ and $k=28$), and use them to get explicit lower bounds. For three colours and $9\le k\le28$ we improve on every published bound we know of, among them $W(3,9)>4,002,705$ and $W(3,28)>159,594,201,561,100,726$. For $3\le r\le6$ and $9\le k\le28$, $77$ of the $80$ cases improve on what the published bounds give under these recurrences and the other three tie. For $r=5$ and $12\le k\le15$ the best values come from palettes that are cyclic intervals of colours indexed by quintic residues. We checked every block by exhaustive enumeration and every base colouring by computer. We also prove that the block modulus is at most $(W(2,k)-1)/(k-1)$, so the method does not improve the asymptotics. For large $k$ the largest cubic block modulus lies between $k^{4-\varepsilon}$ and $k^2 4^k$.

This paper presents a model order reduction framework for linear systems where the $\mathcal{H}_2$ optimization is combined with the Petrov-Galerkin projection to preserve a wide range of system properties, including stability, bounded realness, sector boundedness, and dissipativity. The model reduction problem is formulated as a nonconvex optimization problem on a generalized Stiefel manifold, aiming to minimize the $\mathcal{H}_2$ norm of the pproximation error between the full-order and reduced-order models. An explicit expression for the Euclidean gradient of the objective function is derived, and a Riemannian gradient procedure with an efficient generalized Cayley update is applied to seek a (locally) optimal projection, followed by a theoretical analysis of the algorithm. Finally, the performance of the proposed method is demonstrated by a numerical example.

We study spectral sequences as an obstruction theory to deforming cohomology classes via the Rees construction. A filtered complex is interpreted as an equivariant family over the affine line, in which higher spectral sequence differentials arise as obstructions to lifting cohomology classes through infinitesimal neighborhoods of the special fibre. This leads to various geometric and module-theoretic degeneration criteria. We give various applications to $\mathrm{Ext}$ spectral sequences for complex Lagrangians and the formality results of Melani-Rubi\'o amongst others. For double complexes, we revisit the Khovanov-Qi decomposition on the Rees side and introduce a two-parameter Rees construction over the affine plane. The higher differentials of the two associated spectral sequences correspond to torsion modules supported on the thickened coordinate axes. We obtain degeneration criteria in terms of these modules and an intrinsic characterisation of the $\partial\bar\partial$-lemma.

Let $k$ be a perfect field of characteristic $p>0$. We prove that if a $p$-torsionfree ring $R$, defined by homogeneous equations over $W(k)$, admits a perfectoid tower arising from $(R,(p))$, then its special fiber is $F$-liftable. We also give explicit examples admitting no such tower, including examples with $F$-liftable special fibers.

The Chern-Schwartz-MacPherson class $c_{SM}(X)$ is defined in the Chow group $A_*(X)$ of a scheme $X$, and generalizes $c(T_X)\cap [X]$ if $X$ is smooth. In this paper we independently compute $c_{SM}(S^nX)\in A_*(S^nX)$ modulo $n!$ torsion; we use this result to compute the pushforward to $S^nX$ of $c_{SM}(Quot^n_X(E))$ (and in particular $c_{SM}(Hilb^n X)$) when $X$ is smooth and $E$ is a vector bundle on $X$. Finally, when $X$ is smooth and complete of dimension $3$ we propose a refined version of the the MNOP conjecture that computes the image of the pushforward of $[Hilb^n X]^{vir}$ to $A_0(S^nX)$ and prove it modulo $n!$ torsion when $X$ is Calabi-Yau.

We prove that every surjective topological dynamical system on a compact metric space can be approximated in the $C^0$-Gromov--Hausdorff topology by homeomorphisms with any prescribed topological entropy in $[0,+\infty]$, even when simultaneous approximation of its sets of invariant and ergodic measures is required. We also establish a rigidity result under uniform geometric bounds in a fixed dimension: for a homeomorphism of a closed, connected Riemannian manifold, topological stability in the sense of Gromov--Hausdorff guarantees its persistence as a factor of sufficiently close perturbations within the same geometric class. This persistence prevents topological entropy from decreasing and ensures that every invariant or ergodic measure of the original system admits a lift of the same type to the perturbed system.

We determine exactly in which dimensions the sharp Hardy coefficient on the positive integer lattice agrees with that on the positive Euclidean cone, with Dirichlet conditions on every coordinate face. Equality holds in dimensions one and two, whereas the discrete coefficient is strictly smaller in every dimension at least three. An exact transformation identifies the lattice problem with a weighted Dirichlet problem on a cube. A tangential integration identity evaluates the defect of an explicit critical profile and proves the dimensional transition. The continuous coefficient remains the lower edge of the essential spectrum of the transformed operator; below that edge, the sharp discrete coefficient is a simple eigenvalue with a positive eigenfunction. In dimension three, an explicit supersolution places the coefficient strictly between twelve and forty-nine quarters. A representation on a product of three-dimensional spheres relates the spectral threshold to the local geometry of the transformed problem.

This paper develops a unified theory of joint $q$-numerical ranges $\Fq(F)$ for continuous functions $F:\Om \to \B(\cH)$, where $\Om$ is a compact Hausdorff space and $\B(\cH)$ denotes the $C^*$-algebra of bounded linear operators on a complex Hilbert space $\cH$. We prove that $\Fq(F)$ is totally bounded in $\cC(\Om)$, the space of continuous complex-valued functions on $\Om$, and establish the upper semicontinuity of the set-valued map $F\mapsto \Fq(F)$, and show that $\Fq(F)$ is compact when $\cH$ is finite-dimensional. The main result of the paper is a realization theorem. We prove that, for every totally bounded subset $K$ of $\cC(\Om)$ and every $q\in \C$ with $|q|=1$, there exist a Hilbert space $\cH$ and a continuous function $F:\Om\to\B(\cH)$ such that $qK\subset \Fq(F)\subset q\cok$. Consequently, if $K$ is closed and convex, then $\Fq(F)=qK$. As applications, we derive analogous realization results for bounded subsets in $\C^n$ and for totally bounded subsets in arbitrary Banach spaces.

We consider the one-dimensional stirring process on the segment $\{-N,\ldots,N\}$, coupled to boundary dynamics that inject particles from the right reservoir and remove particles from the left reservoir, each acting on a window of size $K$. We investigate the non-equilibrium fluctuations of the system, starting from a product measure associated with a smooth initial profile. The limit Ornstein-Uhlenbeck process is given in terms of the Laplacian and gradient operators acting on test functions satisfying time-dependent Robin boundary conditions that are obtained by linearizing the boundary conditions in the hydrodynamic equation. A central ingredient in our analysis is the derivation of sharp bounds on the space and space-time $v$-functions of arbitrary degree for the centered occupation variables; these bounds were proved in the companion article \cite{birmpa2025nonequilibriumfluctuationsstirringprocess}.

Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with measurements. Consequently, these problems are naturally framed as stochastic inverse problems. In this paper, we explore three aspects of this problem: direct inversion, variational formulation with regularization, and optimization via gradient flows, drawing parallels with deterministic inverse problems. A key difference from the deterministic case is the space in which we operate. Here, we work within a probability space rather than Euclidean or Sobolev spaces, making tools from measure transport theory necessary for the study. Our findings reveal that the choice of metric---both in the design of the loss function and in the optimization process---significantly impacts the stability and properties of the optimizer.