Papers

We prove that, for real Banach spaces of dimension greater than one, each of the identities $\|Id+T^2\|=1+\|T^2\|$ and $\|Id-T^2\|=1+\|T^2\|$, required for every rank-one operator $T$, characterizes the Daugavet property, thereby answering a question posed by Kadets, Martín, and Merí (2007). Consequently, every extremely non-complex Banach space of dimension greater than one has the Daugavet property, answering a question of Martín and Merí (2011). We also compare pointwise versions of the square Daugavet properties with Daugavet points, $\Delta$-points, and $\nabla$-points.

We establish a Hamiltonian-to-pancyclic analogue of Bondy's theorem for graph systems under an aggregate degree condition. Let $\G=(G_1,\ldots,G_n)$ be a graph system on a common $n$-vertex set $V$, and write $\delta(v)=\min_{i\in[n]}d_{G_i}(v)$. If $\G$ contains a rainbow Hamilton cycle and \[ \sum_{v\in V}\delta(v)\ge \left\lceil\frac{n^2}{2}\right\rceil-1, \] then $\G$ is rainbow pancyclic, unless $n$ is even and every member is the same balanced complete bipartite graph. For even $n$ the threshold is exact at the integer level. Unlike the usual transversal Dirac- or Ore-type hypotheses, our condition is not layerwise: the member attaining $\delta(v)$ may depend on $v$, and some vertices may have $\delta(v)<n/2$. Relative to a fixed rainbow Hamilton cycle, we count shortcuts whose colors are released by the Hamilton arcs they replace. A missing cycle length forces complementary shortcut supports to cross-intersect. A counting gap settles even shortening, while equality or near equality in odd shortening yields a distance-two exchange whose orbits force the balanced bipartite obstruction. At the lower integer threshold an exact defect identity shows that only one or two units of slack are available. \noindentKeywords: graph system; rainbow cycle; pancyclicity; Hamilton cycle; extremal graph theory.

We study the convergence rate of the heat-bath Glauber dynamics for the Discrete Gaussian Free Field on arbitrary connected finite graphs. We show that, when starting from super-harmonic initial conditions, the evolution enjoys a strong form of monotonicity. This allows us to get a sharp mixing profile as the size of the graphs diverges. More precisely, we show that mixing occurs at time $\frac{1}{2\lambda}\log(\mathcal{E})$ with window $\mathcal{O}(1/\lambda)$, where $\lambda$ is the spectral gap of the graph Laplacian and $\mathcal{E}$ is the energy of the super-harmonic initial condition. This result holds for arbitrary graphs that do not exhibit extreme connectivity properties (one way or the other). In particular, it holds for finite boxes of the grid $\mathbb{Z}^d$, in dimension $d\geq 3$.

We consider the central Hardy–Littlewood maximal operator in $\mathbb{R}^n$. For every integer $n\geq1$, we construct two bounded, nonnegative functions $f,g\in L^1(\mathbb{R}^n)$ that differ on a set of positive measure and satisfy $M_cf(x)=M_cg(x)$ at every point $x\in\mathbb{R}^n$. Thus the central maximal operator is not injective on the nonnegative cone of $L^1(\mathbb{R}^n)$. The functions can have compact support; when $n\geq2$, they can also be chosen radial, either with compact support or strictly positive throughout the space. The construction makes a small redistribution of mass invisible to the maximal operator by excluding averaging spheres that cross its support. We also give an explicit formula, in terms of the incomplete beta function, for the $n$-dimensional volume of the intersection of two balls with arbitrary radii and centers, and an elementary formula for the central maximal function of a ball in $\mathbb{R}^3$.

We consider the global dynamics of finite-energy solutions to the $2$-equivariant harmonic map heat flow from $\bbR^2$ to $\bbS^2$ and $\dot H^1$-bounded radial solutions to the energy-critical nonlinear heat equation in dimension $6$. Building upon soliton resolution, we obtain a classification of their multi-bubble dynamics. First, we prove that all such solutions are global. For the harmonic map heat flow, the number and sign of the bubbles are determined by the energy class of the initial data, while for the nonlinear heat equation the bubble signs necessarily alternate. In both models, the largest scale is determined, up to a positive multiplicative constant, by the spatial tail of the initial datum. The remaining scales concentrate according to exponential and iterated-exponential laws determined by the largest scale. Finally, global bubble trees with an arbitrary number of bubbles exist for both models. For the harmonic map heat flow, this follows directly from the classification; for the nonlinear heat equation, we construct alternating bubble trees with any prescribed radial $\dot H^1$ tail, thereby realizing the scale laws arising in the classification.

We prove that there exists $A\subseteq \mathbb{N}$ such that \[ R_A(N)=\frac{\pi}{4}N+ O\!\left(N^{1/4}\sqrt{\log N}\right), \] where $R_A(N)=\#\{(a,b)\in A^2:a+b\le N\}$. This improves the record $O(N^{1/4}\log N)$ obtained by Ruzsa in 1997.

We prove Ebenfelt's SOS conjecture, which implies the Huang–Ji–Yin gap conjecture.

Let $M^n\subset\mathbb{S}^{n+1}(1)$ be a closed minimal hypersurface with constant $S=|h|^2$ and constant $f_3=\operatorname{tr} h^3$, where $h$ is the shape operator. We prove that $S>n$ implies $S\geq2n$, and that the equality images are precisely the minimal Cartan isoparametric hypersurfaces with three principal curvatures, which occur only in dimensions $n=3, 6, 12, 24$.

Under suitable group actions, it is known that $A_\infty$-structures descend to orbit categories. We study the converse problem of lifting an $A_\infty$-structure to a prescribed covering. We show that a compatible group grading determines a canonical lift, uniquely characterized by the strictness of the covering projection. As an application, we construct $A_\infty$-categories associated with admissible fractional Brauer graph algebras by lifting Brauer graph $A_\infty$-categories. Their geometric data augment the surface models of Brauer graph algebras with a Nakayama character encoding the covering. We prove that admissible fractional Brauer graph algebras with equivalent geometric data are derived equivalent. Furthermore, we establish a set of combinatorial derived invariants and prove their completeness in reduced genus zero and in reduced genus at least two when the defining ribbon graph is non-bipartite.

We study finite-horizon distributionally robust linear-quadratic control with disturbances that can be arbitrarily correlated in time. The ambiguity set is modeled as a single 2-Wasserstein ball centered at a nominal elliptically contoured distribution of the disturbances. Despite the infinite-dimensionality of both the policy space and ambiguity set, we prove that the optimal policy is affine and that the worst-case distribution is an affine push-forward of the nominal distribution. These affine maps can be computed efficiently via a best-response algorithm based on the Frank–Wolfe algorithm. Experiments demonstrate improved out-of-sample performance over LQG and its distributionally robust extensions when the true disturbances are correlated, incurring only a small loss of performance under uncorrelated disturbances.

The spectral limited-memory preconditioner (sLMP) clusters leading eigenvalues of symmetric positive definite matrices to accelerate conjugate gradient (CG) convergence. In practice, the cluster point is often chosen to be unity. In some cases, however, this choice can fail to accelerate convergence relative to unpreconditioned CG, even when highly accurate spectral information is available. Alternative cluster points have been proposed based on exact-arithmetic convergence analysis, but such analysis does not explain this finite-precision behaviour. We study how the cluster point influences two sources of numerical error in sLMP-preconditioned CG. First, we analyse the propagation of floating-point rounding errors during application of the preconditioner and derive computable relative-error bounds. For the dominant subspace (spanned by the eigenvectors associated with the leading eigenvalues of the unpreconditioned system) and its orthogonal complement (spanned by the remaining eigenvectors), these bounds are minimized by a weighted median and a weighted arithmetic mean of the leading eigenvalues, respectively. Our analysis explains why small cluster points can strongly amplify errors in the dominant subspace. Second, we investigate sensitivity to perturbations in the dominant spectral information when constructing the preconditioner. The resulting perturbation bound is minimized by a weighted median of the perturbed dominant eigenvalues, with weights determined by the eigenvector perturbation magnitudes. Numerical experiments on synthetic problems illustrate the predicted rounding-error and sensitivity behaviour. Together, these results show that cluster-point selection in finite precision should account for exact-arithmetic convergence, rounding errors, and inaccuracies in the available spectral information.

Let $\Omega\subset\mathbb R^3$ be an unbounded domain with smooth boundary and let $\Sigma$ be a complete, noncompact, orientable, immersed free boundary minimal surface in $\Omega$, with compact boundary $\partial \Sigma\subset\partial \Omega$ and finite Morse index. We prove that if the mean curvature of $\partial \Omega$ satisfies $H_{\partial \Omega}\ge 0$ along $\partial \Sigma$ and $H_{\partial\Omega}>0$ at some point of $\partial\Sigma$, then \[ \textrm{Ind}(\Sigma)\ \ge\ \frac13\Bigl(2g+k+2\sum_{j=1}^r(d_j+1)-2\Bigr), \] where $g$ is the genus of $\Sigma$, $k$ is the number of connected components of $\partial\Sigma$, $r$ is the number of ends of $\Sigma$ and $d_1,\dots,d_r$ are their respective multiplicities. When $H_{\partial\Omega}$ is only assumed to be nonnegative we obtain $\textrm{Ind}(\Sigma)\ge(2g+k-2)/3$, with the sharp bound $(2g+k-1)/3$ under a mild condition on the ends. The proofs use the harmonic one-form method of Ros and Chodosh–Máximo, with weighted $L^2$ spaces, adapted to the free boundary setting in the spirit of Ambrozio–Carlotto–Sharp. The main new ingredient is the computation of the dimension of the space of harmonic one-forms on a punctured compact Riemann surface with boundary which are tangential along the boundary and square integrable with respect to a weight. For a class of admissible weights $\rho$, we prove that this dimension is $2g+k-1+2\sum_jN_j-\varepsilon$, where $N_j$ is the maximal order of pole allowed by $\rho$ at the $j$-th puncture and $\varepsilon\in\{0,1\}$. For the weight of Chodosh–Máximo one has $N_j=d_j+1$.

We prove an upper bound on the homogenization length scale which is a fixed power of the coarse ellipticity ratio for coefficient fields that satisfy a finite range of dependence assumption.

We study graph hypersurfaces associated with higher-dimensional ($n>2$) Cobb–Douglas production functions with variable returns to scale. We prove a rigidity theorem showing that every flat model in this class is necessarily homogeneous. The proof relies on an affine dimension-reduction principle that transfers the higher-dimensional flatness condition to suitable two-dimensional restrictions. We then obtain a complete classification in the non-degree-one case: every such flat model is a power of a positive linear form. For degree one, we give a local structural description in terms of envelopes on the rank-one set, with linearity on open rank-zero regions. These results extend the two-input analysis recently developed in [8] to the higher-dimensional setting, where the degree-one case exhibits a genuinely different behavior.

We study a discrete-time auction model in which multiple sellers update their bids according to their performance in the preceding round. Bidders aim to maximize their profits and adjust their bids solely based on their most recent outcome (myopic behavior). In each round, the auctioneer purchases the lowest-priced $p$-fraction of the total quantity offered by the bidders. We find a system of differential equations governing the macroscopic dynamics and derive it as a scaling limit of the microscopic model. We find an explicit solution for the max-price evolution $q_t$ and show that, in the long run, bidders coordinate, i.e., their bids converge to a common value depending only on their initial distribution and the fraction $p$. For Poisson-distributed initial bids, we establish hydrodynamic limits for the empirical bid distribution and the max-price trajectory and conjecture the corresponding Gaussian fluctuations for $q_t$. Finally, we generalize the model to allow for heterogeneous bid-update velocities: in this case, the max-price velocity becomes proportional to the harmonic mean of the update velocities of bidders at the max-price.

We study a family of coupled geometric evolution equations describing the deformation of a Riemannian metric and a non-abelian gauge field on closed manifolds, which generalizes the Ricci–Yang–Mills flow. We derive interior curvature estimates, find a preserved integral curvature condition, and discover a scaling invariant monotone functional analogue to Perelman's entropy. In particular, we prove strong dominance of the Riemannian curvature over the gauge curvature at finite-time singularities in all dimensions. We also provide a non-trivial explicit example of a shrinking self-similar solution on a $SU(2)$ bundle over $S^4$.

We establish the extension of Milnor's inequality, relating total curvature with the bridge index, to the $C^1$-closure of a knot class, without any restriction on the self-intersections of the limit curve. Together with recent work of Reiter–von der Mosel, this resolves the circular elastic knot conjecture, first predicted by Gallotti–Pierre-Louis in 2007 and then formulated as a mathematical conjecture by Gerlach–Reiter–von der Mosel in 2017. More precisely, if the bridge and braid indices of a tame knot class coincide, then the multiply covered circle is the unique elastic knot.

We establish a sufficient condition for an additively idempotent semiring to be inherently nonfinitely based: if its generated variety is locally finite and every Zimin word is minimal in the additive order, then it is contained in no finitely based locally finite variety. We characterize Zimin minimality by the membership of a countable flat factor semiring $\Finf$ in the generated variety and prove that $\V(\Finf)$ is a minimal inherently nonfinitely based variety. We also obtain general restrictions on the additive order of finite inherently nonfinitely based ai-semirings. As applications, the six-element ai-semirings $\A$ and $\B$ are shown to be inherently nonfinitely based. Finally, we prove that the six-element ai semiring $\Abar$ is not inherently nonfinitely based and is finitely based. Its multiplicative reduct is inherently nonfinitely based; thus, to the best of our knowledge, $\Abar$ is the first finitely based ai-semiring with an inherently nonfinitely based multiplicative reduct.

We resolve the longstanding geometric ideal triangulation conjecture arising from Thurston's theory of hyperbolic 3-manifolds: every finite-volume cusped hyperbolic 3-manifold admits a finite geometric ideal triangulation realizing its complete hyperbolic structure.

Blest's rank correlation $\nu$ is a variant of Spearman's rho $\rho$ that weights the leading ranks of one variable more heavily, at the price that $\nu$ is not symmetric in its arguments. We quantify both features by determining the exact region of $(\rho,\nu)$ over all bivariate copulas, as well as that of $(\eta,\nu)$, where $\eta$ is the symmetrized Blest coefficient of Genest and Plante. The latter region is a linear image of the set of all pairs $(\nu(C),\nu(C^\top))$ formed by a copula $C$ and its transpose. Consequently, Blest's coefficient differs from Spearman's rho by at most $1/4$, and interchanging the two variables changes it by at most $27/64$, improving on the bound $1/2$ implied by the first inequality. For every given value of $\rho$ or $\eta$, each corresponding extreme value of $\nu$ is attained by exactly one copula, given in closed form and supported on finitely many line segments. Near countermonotonicity, the upper extremizers of the $(\eta,\nu)$-region are supported on the graph of a function of the second coordinate, yet their conditional laws given the first coordinate carry two atoms. The proofs rest on a rearrangement inequality with equality case and on explicit Kantorovich potentials.

Duke's interpolation theorem states that the orientable genera of a connected graph form an integer interval, and Stahl established the corresponding result for nonorientable embeddings. In 1991, Širáň showed that this interpolation property fails for signed graph embeddings: the Euler-genus spectrum of a signed graph may contain gaps. He subsequently asked whether all such gaps must occur at the lower end of the spectrum. In this paper, we establish a characterization of the Euler-genus spectrum of a connected signed graph. We prove that, for each parity class, the Euler genera form a step-two interval. Moreover, whenever both parity classes are nonempty, their maximum elements differ by one. As an consequence, if two consecutive integers $k$ and $k+1$ belong to the Euler-genus spectrum, then every integer from $k$ to the maximum Euler-genus also belongs to the spectrum, thereby answering Širáň's question affirmatively. Our proof uses the pre-signed graph representation of signed embeddings together with ordered adjacent-exchange operations and a matching interpretation of face numbers.

We study symmetric differentials on weighted hypersurfaces. Let $k$ be an algebraically closed field and let $$ X_d\subset \mathbb{P}_k(a_0,\ldots,a_N),\qquad N\ge3, $$ be a weighted hypersurface, with $a_{\min}:=\min_i a_i$. In characteristic zero we prove twisted vanishing in the range $t<m a_{\min}$ for quasi-smooth hypersurfaces of degree $d\ge2a_{\min}$, with the corresponding reflexive statement on the coarse space. We also prove a tame positive-characteristic analogue; in particular, reflexive symmetric differentials of every positive order vanish. In the second part we specialize to $k=\mathbb{C}$ and to surfaces in weighted projective three space. Using the quotient-singularity criterion of Asega–De Oliveira–Weiss together with weighted Kummer and Segre constructions, we produce singular weighted hypersurfaces whose minimal resolutions have big cotangent bundle. Combining the vanishing and bigness results with simultaneous resolution yields deformation-equivalent smooth projective surfaces for which the symmetric plurigenera jump from zero in every positive order to cubic asymptotic growth.

We study asymmetric simple binary hypothesis testing between $H_0:P_0^{n}$ and $H_1:P_1^{n}$, based on $n$ independent and identically distributed observations. Leveraging a variational representation of Rényi divergence of order $\alpha$, we derive our main result: a finite-sample converse with $\alpha>1$. The bound uses both directions of the divergence $D_\alpha(P_1\|P_0)$ and $D_\alpha(P_0\|P_1)$, tensorises under product measures, and contains familiar data-processing converses as boundary cases. For comparison, we apply the same variational approach to general $f$-divergences and specialise it to total variation, $E_\gamma$, Hellinger, and Kullback Leibler divergences, thereby recovering familiar converses within a unified framework. Together with an achievability bound involving Rényi divergence with $\alpha\in (0,1)$, the main converse recovers the phase transition of the optimal Type II error under the exponentially decaying Type I error constraint $\varepsilon_n=e^{-nr}$. Under regularity conditions, the optimal Type II error vanishes exponentially when $r<D(P_1\|P_0)$ and converges exponentially fast to one when $r>D(P_1\|P_0)$. We also derive sample-complexity bounds and extend both the converse and achievability analyses to locally differentially private observations, quantifying the cost of privacy and recovering the non-private achievability bound as the privacy constraint vanishes.

We determine, up to isomorphism of the covering graphs, the connected arc-transitive regular covers of cubic edge-primitive graphs whose covering transformation group is cyclic or elementary abelian of order $p^2$, where $p$ is a prime. Combining the known classifications for the base graphs ${\rm K_{3,3}}$ and ${\rm DC_{14}}$ with new arguments for ${\rm F30A}$ and ${\rm F102A}$ gives the full list in these two classes of covering groups. In the cyclic case, the covers of ${\rm F30A}$ and ${\rm F102A}$ are ${\rm F90A}$ and ${\rm F204A}$, respectively. In the elementary abelian case, neither ${\rm F30A}$ nor ${\rm F102A}$ admits an arc-transitive regular $\mathbb{Z}_p^2$-cover, so the base graph is ${\rm K_{3,3}}$ or ${\rm DC_{14}}$.

The purposes of this paper are two-folded. First, we define a class of local psh functions on Berkovich spaces, and study their properties. In particular, we show that this class of psh functions is equivalent on the one hand to the one defined by Chambert-Loir–Ducros, and is equivalent on the other hand to the class of global psh functions of Boucksom–Favre–Jonsson on projective varieties satisfying the envelope property. Second, making use of our local theory of non-archimedean psh functions, we prove the envelope conjecture for projective normal varieties over a discretely or trivially valued non-archimedean field of residue characteristic zero.

We construct an adelic counterpart of Scheithauer's lifting map from scalar-valued modular forms to vector-valued modular forms transforming with the Weil representation. More precisely, we introduce suitable spaces of vector-valued adelic automorphic forms and define an adelic lifting map between the corresponding scalar-valued and vector-valued automorphic forms. We then prove that this adelic construction is compatible with Scheithauer's classical lifting map under adelization. In this way, the relation between scalar-valued and vector-valued modular forms is transferred naturally to the adelic setting.

Liftings and parallel redrawings are classical topics in applied discrete geometry, concerned respectively with vertically lifting a $(d-1)$-picture – a realisation of a vertex-hyperplane incidence geometry in $\mathbb{R}^{d-1}$ – to a $d$-dimensional polyhedral scene, and with redrawing a $d$-picture (or hyperplane arrangement) within $\mathbb{R}^d$ while preserving prescribed hyperplane normals. In this paper, we develop a unified framework for the analysis of forced-symmetric liftings and forced-symmetric parallel redrawings. In particular, we show that the classical duality for these theories also appears in the forced-symmetric setting. We establish the orbit lifting matrix and the orbit concurrence geometry matrix, and show that they are the appropriate symmetry-adapted analogues of the standard lifting matrix and concurrence geometry (or parallel redrawing) matrix. We also make explicit the relationship with the orbit version of Whiteley's parallel design matrix for graphs. Using these tools, we derive necessary conditions for symmetric pictures to be forced-symmetric flat, and for symmetric hyperplane arrangements to be forced-symmetric robust, expressed as sparsity counts on the group-labelled quotient graphs associated with the symmetric incidence geometries. Finally, we discuss conjectures regarding the sufficiency of these conditions for generic configurations.

The design of profile extrusion dies remains a challenging task due to the complex rheological behavior of polymer melts and the geometric intricacies of flow channels. Traditional manual optimization approaches, which rely heavily on human experience, are inefficient and often employ unvalidated heuristics. To address these challenges, we present a deterministic and explainable framework for automatic die design based on adjoint-based shape optimization. This approach enables the computation of sensitivities that directly indicate beneficial modifications to the flow channel geometry. A major difficulty in such optimization processes lies in generating boundary-conforming meshes that evolve consistently with changing geometries. To overcome this issue, we employ non-boundary conforming geometry representation methods that eliminate the need for an explicit surface representation along physical boundaries. A dedicated reconstruction technique is developed to recover accurate sensitivity information at the virtual interface between fluid and solid regions. The proposed algorithm is demonstrated on 3D geometries with varying complexity, including realistic extrusion die flow channels. Several objective functionals relevant to industrial applications, such as flow balance at the outflow, are considered. The results highlight significant improvements in performance metrics while maintaining numerical robustness. This work showcases the potential of adjoint-based techniques for automated die design in a domain still largely governed by manual trial-and-error procedures, establishing a foundation for data-efficient, sustainable manufacturing workflows using computational rheology.

We study the parabolic vectorial Allen–Cahn equation in two space dimensions for a fixed smooth potential with finitely many non-degenerate wells. Our result provides a parabolic extension of Bethuel's elliptic compactness theorem: at almost every positive time, the diffuse energy, potential energy, and gradient tensor converge to measures concentrated on a countably \(1\)-rectifiable interface and satisfy all the Bethuel-type relations. In the scalar case, the tangential defect vanishes and the usual mean-curvature-flow structure is recovered. For general vectorial potentials, the limiting stress and dissipation yield a two-mobility system: the normal energy flux balances weighted curvature, while a tangential internal mobility transports residual tangential energy. Thus the result extends the Allen–Cahn-to-Brakke framework of Ilmanen from scalar to vector-valued systems, with an additional non-negative dissipation defect in the localized energy inequality.

We study joint oscillation conditions for commutators of bilinear rough singular integrals with bounded mean-zero angular kernels. For second-order commutators with two possibly distinct complex-valued symbols, we prove lower bounds under a geometric non-degeneracy assumption. These bounds distinguish the commutation positions: mixed commutators control the product of the separate quadratic oscillations, while repeated-position commutators of the partial transposes additionally control the quadratic average of the product of the centered symbols. We also establish sparse domination for iterated commutators, retaining the symbol oscillations inside the local averages. For each fixed commutator order, the sparse constant is $O(s/(s-1))$ as the averaging exponent $s$ decreases to $1$. The resulting joint oscillation conditions yield strong bounds in the Banach range and weak bounds in the quasi-Banach range. At the natural exponent triples, the necessary and sufficient conditions differ by an arbitrarily small increase in the oscillation exponents.