We consider $K3$ surfaces collapsing to a three-dimensional affine base. We show that certain affine lines on the base lift to degenerating sequences of special Lagrangian two-spheres and tori in the collapsing $K3$ surface. In particular, we construct special Lagrangian two-spheres connecting pairs of Taub-NUT bubbles. These examples fit into the broader program of reconstructing special submanifolds from graphs and combinatorial data on a collapsed affine limit.

Pál's isominwidth inequality (1921) answered the Kakeya needle problem (1917) for convex sets. It states that among convex bodies of fixed minimum width $w$ in the Euclidean plane, the regular triangle has minimal area. The isominwidth inequality was generalized to the $2$-dimensional sphere by Bezdek and Blekherman and Freyer and Sagmeister (arXiv:2411.11462). Interestingly, in hyperbolic space, no minimizer exists, as shown by Böröczky, Freyer and Sagmeister (arXiv:2502.04427). The stability of the Euclidean Pál inequality with respect to the Hausdorff metric and the symmetric difference metric was proved by Lucardesi and Zucco (arXiv:2405.18294). Fodor, Robock and Sagmeister (arXiv:2602.19300) proved $r$-ball convex analogs of the isominwidth inequality in all three constant curvature planes connecting Pál's theorem with the Blaschke--Lebesgue inequality. In this paper, we prove optimal stability versions of this statement with respect to the Hausdorff distance and the symmetric difference metric in all three constant curvature planes.

We prove a sparse transference theorem for induced Ramsey graphs. The theorem transfers the weighted random-host proof of Aragão, Campos, Dahia, Filipe, and Marciano to the sparse random setting. It follows that, for every fixed graph $H$ with no isolated vertices and at least two edges, and every $η>0$, there is $C>0$ such that, whenever $N\ge r^{Cr}$ and $N^{-1/m_2(H)+η}\le p\le \frac12$, with high probability every $r$-colouring of the edges of $G(N,p)$ contains a monochromatic induced copy of $H$. Here, $m_2(H)$ denotes the usual maximum 2-density of $H$.

The maximal entries of Dynkin friezes over the positive integers have recently been determined for all finite Dynkin types except $B_n$ and $D_n$. In this note, we explicitly construct large positive integral points on affine cluster varieties of type $B_n$ (resp. $D_n$), giving rise to friezes of types $B_n$ (resp. $D_n$) over the positive integers with largest entries $F_{n+1} F_{n+2} - 1$ (resp. $F_n F_{n+1} - 1$) where $F_k$ is the $k$-th Fibonacci number. We conjecture that these are the maximal possible entries for their respective Dynkin types.

Identifying ideal Perron similarities is a problem of central interest in the longstanding nonnegative inverse eigenvalue problem (NIEP). A normalized ideal Perron similarity is called totally extremal if every entry has modulus one. Recently, Gershnik et al. [J. Algebra 694 (2026), 782--800] proved that the character table of a finite Abelian group is totally extremal and conjectured the converse. In this paper, we settle this conjecture in the affirmative by first showing that the rows of a totally extremal normalized ideal Perron similarity form a group under the Hadamard product. Then, it is shown that the rows of a nonsingular matrix form a group under the Hadamard product if and only if it is the character table of a finite Abelian group, and we further show that this group is isomorphic to the underlying group. These results extend the classical theorem due to Romanovsky and Karpelevič on the unimodular eigenvalues of stochastic matrices in the complex unit disk to the setting of spectratopes in the unit ball of complex Euclidean space.

We prove that stationary random metric measure spaces have 0,1,2 or a Cantor space of ends. This notion includes stationary random graphs, manifolds and discrete subgroups. In the case of surfaces, we classify all possible homeomorphism types, in analogy with the work of Biringer and Raimbault on unimodular Riemannian manifolds. Our approach relies on a general "no geometric core" principle and an analysis of finite versus infinite expected return times.

We present the first rigorous analysis of decentralized geometric routing in Geometric Inhomogeneous Random Graphs (GIRGs), a weight-agnostic variant of the greedy routing protocol. While greedy routing in GIRGs is known to explain the algorithmic small-world phenomenon by finding ultra-short paths of length $Θ(\log \log n)$, it assumes additional knowledge of vertex weights beyond geometry, an assumption that is often restrictive or unavailable. We investigate whether the underlying geometry alone is sufficient for efficient navigation. We prove that for power-law weight exponent $τ\in (2,3)$ and geometric decay parameter $α> τ- 1$, geometric routing succeeds with constant probability and finds ultra-short paths of length $Θ(\log \log n)$, matching the optimal asymptotic guarantees for greedy routing. Our analysis further reveals that, upon success, both protocols follow a similar two-phase trajectory, consisting of a rapid ascent to the heavy vertices, followed by efficient navigation to the target. These results demonstrate that, in the appropriate regime, the network's geometry alone implicitly guides the path to the target through its high-weight core.

The generalized Turán number $ex(n, K_{t, t}, K_{2, t+1})$ is the maximum number of copies of $K_{t, t}$ that a $K_{2, t+1}$-free graph on $n$ vertices can contain. Recently, Pohoata, Tidor, and Yu established that $ex(n, K_{t, t}, K_{2, t+1}) = Θ_t(n^2)$ for all integers $t \geq 3$. In this short note, we use an explicit construction to establish that when $t$ is a prime power and $n = t^{2e - 1}$, then $$ ex(n, K_{t, t}, K_{2, t+1}) = (1 + o(1))\frac{n^2}{2t(t-1)}. $$

Peskine and Szpiro's Intersection Theorem for finitely generated modules was generalized by Foxby to all modules and bounded complexes of such. We strengthen Foxby's result and prove a dual result, which for complexes with finitely generated homology recovers Iversen's Amplitude Inequality but also applies to derived complete complexes.

We study Cheeger-type inequalities for persistent Laplacians associated with inclusions of simplicial complexes $\mathcal{K}\hookrightarrow \mathcal{L}$. We introduce a persistent up $p$-Laplacian $Δ_{q,p,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$ for $p\geq 1$. For $p=2$, this recovers the usual persistent up Laplacian, while for $p=1$ it yields a nonzero persistent Cheeger constant $φ_q^{\mathcal{K},\mathcal{L}}$. We prove a Cheeger-type inequality relating $φ_q^{\mathcal{K},\mathcal{L}}$ to the smallest nonzero eigenvalue of $Δ_{q,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$. This gives a persistent extension of recent work by Jost and Zhang (Ann. Sc. Norm. Super. Pisa Cl. Sci., 2024; arXiv:2302.01069). We then study two more structured settings. Under a locally complete $q$-skeleton assumption on $\mathcal{K}$, we extend the complete-skeleton isoperimetric inequality of Parzanchevski--Rosenthal--Tessler (Combinatorica, 2016; arXiv:1207.0638) to the persistent setting. For orientable $(q+1)$-dimensional pseudomanifolds, we prove a Kron-type reduction of the persistent up Laplacian to a vertex- and edge-weighted graph Laplacian, possibly with Dirichlet boundary terms, and obtain two-sided Cheeger inequalities; this is related to the dual-graph perspective in the work of Steenbergen--Klivans--Mukherjee (Adv. Appl. Math., 2014; arXiv:1209.5091). We also describe the nonzero persistent Cheeger constant $φ_q^{\mathcal{K},\mathcal{L}}$ explicitly in terms of the dual graph in the non-branching pseudomanifold case. Finally, for graph inclusions $H\hookrightarrow G$, we compare the persistent Cheeger constants introduced here with the Kron-reduction Cheeger constants of Mémoli et al. (SIAM J. Math. Data Sci., 2022; arXiv:2012.02808).

The classical no-$k$-in-line problem asks for the largest number of points that can be placed on an $n \times n$ grid without having $k$ of them collinear. A natural extension, motivated by the analogous question by Erde for $k\in \mathbb{Z}$, is the \emph{extensible no-$(k(n)+1)$-in-line problem}, which seeks a subset of points in $\mathbb{Z}^2$ with maximal possible density such that at most $k(n)$ points are collinear within the subgrid $[1,n]^2$. We construct optimal sets for linear functions and positive-density sets for power functions. We prove that any configuration achieving $\liminf\frac{S_n}{n k(n)} \ge 0.897$ must satisfy $k(n) = Ω( n^c)$ for some $c>0$ constant; therefore, the extensible no-$k$-in-line problem has no configuration with this property when $k$ is a constant. Finally, we reduce the problem to the extensible no-$k$-in-line problem, showing that if a positive-density point-set exists for a constant limiter function, then one also exists for any sufficiently regular function $k(n)$.

We prove a sharp Clifford-threshold focal-radius estimate and rigidity for immersed hypersurfaces. Under a $p$-form curvature condition, formulated by the Weitzenböck curvature term together with $\mathrm{Ric}_p\ge p$, any closed two-sided immersion $F:Σ^m\to M^{m+1}$ with $b_p(Σ;\mathbb R)\neq0$ and $1\le p\le m/2$ satisfies \[ r_f(F,M)\le\fracπ{4}. \] The equality case is rigid: if the ambient manifold is complete, equality forces the hypersurface to be locally the Clifford hypersurface $S^p(1/\sqrt2)\times S^{m-p}(1/\sqrt2)\subset S^{m+1}(1)$; if the ambient manifold is compact and connected, it is a spherical space form. The curvature condition follows from $\sec\ge1$ for $p=1$, from normalized $\mathrm{PIC1}\ge1$ for $p=2$, and from curvature operator bounded below by one in all degrees. By quotient lifting and the Hopf fibrations, we also obtain focal-radius estimates in $\mathbb{CP}^n$ and $\mathbb{HP}^n$, with projective Clifford rigidity, without any Betti-number assumption.

We study the semi-algebraicity of the neuromanifold $\mathcal{M}_\mathbf{d}$ of a feedforward ReLU neural network and its symmetries. We prove that $\mathcal{M}_\mathbf{d}$ is not a semi-algebraic quotient of the space of weights of the network. We introduce and study the notion of \emph{honest} open subset of the space of weights, where the network does not show any hidden symmetries. Finally, we conjecture that the maximal honest open is always semi-algebraic and prove that in the shallow case it is even Zariski.

Given an ordered set $B$ of a finite field, a combinatorial simplex over $B$ is defined as the set of vectors such that the positions of the entries, with respect to $B$, sum up to a fixed integer. CAP codes are Reed-Muller type codes defined over a combinatorial simplex. They were recently introduced by Kopparty et al. as a high-rate alternative to classical Reed-Muller codes, capable of achieving arbitrarily high rates close to one for any fixed minimum distance. In this paper, we use tools from commutative algebra to analyze a combinatorial simplex and its associated CAP code. We give a universal Gröbner basis for the vanishing ideal of a combinatorial simplex. We describe the generalized Hamming weights of a CAP code in terms of the footprint of the vanishing ideal. For the minimum distance case, we proved a closed formula. We give a set of polynomials whose evaluations on the combinatorial simplex generate the dual of the CAP code. We describe the affine permutations that leave invariant a combinatorial simplex and use this information to prove that, in some cases, the permutation group of a CAP code is a symmetric group.

We show that the Gromov--Hausdorff convergence of non-collapsed polarized constant scalar curvature Kähler (cscK) surfaces can be realized as convergence in a Hilbert scheme. We also derive uniform estimates of Bergman kernels on the effective regular set. As an application, we establish the Zariski openness of cscK metrics for certain smooth polarized families, following the approach of Donaldson.

The nullcone membership problem, deciding whether an orbit closure contains the origin, is fundamental in computational invariant theory. For self-adjoint groups, Bürgisser, Franks, Garg, Oliveira, Walter and Wigderson gave a geodesic optimization algorithm whose complexity is controlled by the gap, a condition number of the representation. We study the gap for quiver representations under the action of the special linear group. We prove that the inverse gap is polynomially bounded in the number of vertices and the maximum dimension for type A and $\hat{A}$, as well as tree quivers with uniform dimension vectors. Consequently, the algorithm of Bürgisser et al. solves the nullcone membership problem in polynomial time for these families. In contrast, we construct families of quivers and dimension vectors where the gap is exponentially small in the number of leaves, furthermore, for every connected quiver we exhibit dimension vectors such that the weight margin (a related condition number) is exponentially small in the number of vertices. We also extend our results to $σ$-semistability, thereby giving a new proof of a recent result of Iwamasa, Oki, and Soma.

We prove a folklore conjecture identifying two categorical enhancements of the automorphic side of the local Langlands correspondence. Concretely, we construct an equivalence for torsion coefficients between the category considered by Zhu and the one considered by Fargues-Scholze. To achieve this, we revisit Scholze's analytification functor and apply the first author's theory of kimberlites. We discuss unconditional applications to the splitting of the semi-orthogonal decomposition on BunG, and the compatibility with Eisenstein functors. Finally, we formulate a linearity conjecture for our functor with which we can show new vanishing statements for the cohomology of local Shimura varieties, and perverse exactness statements for Hecke operators.

Let $S_κ$ be the space of cusp forms of weight $κ$ and level one, and let $S_κ^{\ast}$ denote its dual space. In this paper, we find explicit spanning subsets of $S_κ$ consisting of Rankin-Cohen brackets of Eisenstein series and explicit subsets of periods that span $S_κ^{\ast}$.

We propose Adaptive Accelerated Mirror Descent (AAMD), a flow-based method that combines nonlinear preconditioning, acceleration, and adaptivity in mirror geometry. The key ingredient is an accumulated Lyapunov perturbation budget: local descent failures are allowed as long as the total budget remains nonpositive, so line search is used only when stability is at risk. We prove accelerated convergence under dual relative smoothness/convexity and a mirror-geometry compatibility condition, and obtain an $O(1/k^2)$ rate for convex objectives by homotopy under a bounded-sublevel-set assumption. Experiments on relative-smoothness problems show that combining preconditioning, acceleration, and adaptivity gives substantial gains over methods using only part of this structure.

The paper is concerned with $α$-stable SDEs with singular time-inhomogeneous general drift. Our drifts satisfy a condition that is close to the minimal possible scaling-invariance and can introduce local blow-ups of the type arising in some particle systems with strong attracting interactions.

All stabilizing PID controllers form a set in three-dimensional space. A novel viewpoint to its boundary as a ruled surface (or surfaces) being cut with 3D planes is presented. The characterization, being not too new, contributes to an understanding of the stability set as the whole, instead of the classical view as a stack of 2D slices, say, on the P-coefficient. The viewpoint gives clear insight on the structure of the PID stability region, and, in particular, splits its boundary into continuous parts. It is followed by natural 2D unwrapping of the stability set boundary. It also correctly handles pure imaginary zeros in transfer function. A wireframe 3D visualization reveals the structure of the stability set. The presentation is valid both for ideal and filtered PID controllers, as well as for time-delay systems and other linear systems. Finally, based on the viewpoint, a simple formula for stability (fragility) radius is provided.

Devnani and Eyyunni recently studied the maps pre$_k$ on integer partitions, which arise from applying elementary symmetric polynomials to the parts of a partition. They asked whether there exists $n \ge 1$ such that exactly one partition of $n$ lies in the image of pre$_2$. We show that this occurs only for $n$ in {1, 2, 4}, and that for all $n \ge 5$, at least two partitions of n are in the image of pre$_2$.

We investigate the equilibrium (trivial solution) stability, also known as Lyapunov stability, of a certain first-order linear complex constant coefficient dynamic equation on the time scale $\p_{α,β}$, which has continuous intervals of length $α>0$ followed by discrete gaps of length $β>0$. In particular, we establish results in the case of this specific time scale, for coefficient values in the complex plane, including where the exponential function alternates in sign. In our analysis, we employ the Lambert $W$ function. For increasing gap size $β$ relative to $α$, we prove that the complex constant coefficient undergoes a bifurcation in its parameter space. We establish interesting results for both the delta dynamic equation and the nabla dynamic equation. Lastly, we connect these results to those related to Hyers--Ulam stability of the same nabla equations.

Let $r_n^{k}(N)$ be the number of representations of $N$ as the sum of $n$ generalized $k$-gonal numbers and $r_n^{\square}(N)$ be the number of representations of $N$ as the sum of $n$ squares. By modifying the Heath-Brown circle method, we prove a closed-form asymptotic relation between $r_n^{k}(N)$ and $r_n^{\square}(8(k-2)N+n(k-4)^2)$ for any $k\geq3$ and any $n\geq4$. Consequently, we determine the asymptotics of $\sum_{N\leq x}r_4^{k}(N)^2$ and, via a result of Bringmann, Jang, Kane, and Tse, prove a similar closed-form asymptotic relation between the number $r_{4,+}^{k}(N)$ of representations of $N$ as the sum of four ordinary $k$-gonal numbers and $r_4^{\square}(8(k-2)N+n(k-4)^2)$. We also show that if $4\mid k$, any strictly increasing infinite subsequence on which $r_{4,+}^{k}$ is bounded converges $2$-adically to $(k-4)^2/(4-2k)\in\mathbb{Z}_2$, supplementing a result of Meng and Sun, and if $4\nmid k$, there is no strictly increasing infinite subsequence on which $r_{4,+}^{k}$ is bounded.

Let $\mathcal{A}$ be the mod $2$ Steenrod algebra, and $P(n) = \mathbb{F}_2[x_1, \dots, x_n]$ be the polynomial algebra viewed as an unstable module over $\mathcal{A}$. The symmetric hit conjecture asks whether the symmetrization of a hit monomial in $P(n)$ is always hit in the symmetric invariant subalgebra $B(n) = P(n)^{Σ_n}$. While resolved for $n \leq 3$, the case $n=4$ presents significant obstructions due to combinatorial complexity, orbit cancellations intrinsically tied to $Σ_4$-stabilizers, and the emergence of strongly spike-free survivor modules. This paper introduces a conditional structural reduction to overcome these obstructions in the domain where the numerical weight satisfies $μ(d) \leq 4$. By integrating Walker-Wood duality with a new $Σ_4$-stabilizer parity analysis, we reduce the global conjecture to localized algebraic conditions: a symmetric lower-spike reduction and a strengthened four-row digital-engineering hypothesis. Assuming these inputs, the conjecture follows by lexicographic induction on the column-sum and row-sum sequences of the binary exponent matrices. Our approach isolates the four-variable repeated-row anomaly into exact local identities, utilizing global Steenrod-kernel functionals lifted from local spike-free quotients to detect potential survivor elements. Finally, we provide explicit monomial-level computations in degrees $8$, $12$, and $14$, explicitly illustrating the stabilizer mechanism in practice and framing the precise algebraic identities required for a future unconditional proof.

In this note we explore rational approximations to irrational numbers whose denominators are restricted to a subset of the natural numbers. We introduce a specific notion of density for such subsets and use it to establish a restricted analog of a theorem of Dirichlet's.

In this paper, we study the Bohr phenomenon for a generalized subclass of harmonic mappings defined by a second-order differential inequality in the unit disk. Specifically, we consider the class $\mathcal{BH}_0(γ, δ)$, which extends several known subclasses of harmonic and analytic functions. By employing sharp coefficient estimates and growth results, we establish improved versions of Bohr-type inequalities, including refined Bohr radii and Bohr--Rogosinski radii for this class. Furthermore, we derive generalized inequalities involving higher-order coefficient sums and area terms, thereby extending classical Bohr inequalities in a harmonic setting. The sharpness of the obtained results is verified through extremal functions. In addition, we obtain Landau-type theorems for the class $\mathcal{BH}_0(γ, δ)$, providing explicit bounds for the radius of univalence and the size of schlicht disks contained in the image domain. Our results not only unify and extend several earlier works but also provide new insights into the geometric behavior of harmonic mappings under differential constraints.

This work investigates a nonlinear two-point boundary value problem arising in diffusion-reaction processes in catalyst slabs with power-law kinetics and fractional reaction order. The model exhibits a free-boundary structure, where an unknown interface separates a dead-core region with vanishing concentration from an active region with positive concentration. We propose a Physics-Informed Neural Network (PINN) framework that incorporates a structured, hard-constrained trial solution embedding the asymptotic behavior near the interface. The dead-core location is treated as a trainable parameter, enabling the simultaneous approximation of the concentration profile and identification of the free boundary without explicit interface tracking. The method is validated against analytical solutions and high-precision numerical shooting. Numerical experiments demonstrate that the approach accurately captures both the solution profile and the free-boundary location while maintaining a computationally manageable training cost.

The paper deals with dead-core solutions to an isothermal reaction-diffusion problem with power-law kinetics for a single reaction that takes place in a chemical reactor represented by a bounded domain in two dimensions. The model boundary value problem for the stationary non-linear diffusion-reaction equation is solved numerically using an appropriate time-marching method. The spatial discretization is based on the lumped finite element method for piecewise linear functions. The effects of the reaction order and Thiele modulus on the concentration profiles and the size of dead zones are investigated numerically. The paper demonstrates that the formation of multiple dead zones is possible for particular reactor geometries.

We investigate operator spreading in random quantum circuits with gates drawn from orthogonal-invariant or symplectic-invariant ensembles, revealing several key distinctions from the well-studied unitary-invariant case. We find that the ensemble-averaged Pauli-string weights relax to a ternary-valued structure, instead of the binary structure of unitary-invariant circuits. For orthogonal- or symplectic-invariant circuits, the domain wall separating trivial and scrambled regions has a finite width even for Haar-random gates, whereas domain walls are sharp for Haar-distributed random unitary circuits. We further find a fundamental dichotomy between random circuits with two-qubit gates from the two disconnected components of the orthogonal group: While the butterfly velocity for the special orthogonal ensemble lies between zero and the Haar value, the negative-determinant sector exhibits a non-zero lower bound for any gate distribution. Moreover, for qudit size $q=2$, the butterfly velocity can exceed that of the Haar-random ensemble.