Weekly Recap

A single adversarial direction can force a specific intersection size between two point sets in finite affine planes.

The combinatorial landscape of pattern-avoiding involutions is significantly clarified, revealing exact enumerative formulas for families avoiding 4321 and a specific length-four pattern.

Random graph pebbling on paths exhibits a sharp phase transition in stackability probability governed by a fixed logarithmic offset, revealing a critical threshold for configuration density.

Hyperspaces of compact subsets of connected, locally finite, regular graphs are proven to be plastic, revealing a surprising structural rigidity in these complex spaces.

Maximum weight matching in random graphs exhibits Gaussian fluctuations, even when graph structure dominates edge weight randomness.

A single adversarial direction can force a specific intersection size between two point sets in finite affine planes.

The combinatorial landscape of pattern-avoiding involutions is significantly clarified, revealing exact enumerative formulas for families avoiding 4321 and a specific length-four pattern.

Random graph pebbling on paths exhibits a sharp phase transition in stackability probability governed by a fixed logarithmic offset, revealing a critical threshold for configuration density.

Hyperspaces of compact subsets of connected, locally finite, regular graphs are proven to be plastic, revealing a surprising structural rigidity in these complex spaces.

Maximum weight matching in random graphs exhibits Gaussian fluctuations, even when graph structure dominates edge weight randomness.

The Krohn-Rhodes complexity of graph monoids reveals a sharp threshold for cycles and paths, with MR(G) becoming complexity 2 for cycles $\ge 4$ and paths $\ge 13$.

Exact Ramsey numbers for cycles and eleven-vertex wheels are finally settled for $n \ge 14$, revealing a simple $2n-1$ pattern.

Hash tables can now achieve arbitrary load factors up to 1.0, delivering 2x throughput and 18% memory reduction over hashbrown.

A novel multi-view LP framework tightens bounds for locally recoverable codes, revealing information loss in simpler models and achieving new exact constructions for availability $a=2$.

The chromatic number of the associahedron is proven to grow only double-logarithmically with its size.

A novel incidence bound for hyperplanes in $\R^d$ unlocks improved bounds for Falconer's distance set problem in $\R^3$ and $\R^4$.

A canonical boundary crossing for mock theta functions is achieved by continuing modular functional equations and applying Écalle--Borel summation to resurgent asymptotic series.

A pseudo-null module quotient $M/TM$ is shown to be pseudo-null if and only if its associated primes are contained within the primes associated with $T$, unifying criteria across commutative and noncommutative rings.

Quantum state tomography protocols exhibit a strict performance hierarchy based on their tomographic rate functions, even when sample complexity suggests equivalence.

Pluricanonical forms extend across smooth Kähler families with nef canonical bundles, and can be extended from a bigness-propagating central fiber in one-parameter degenerations.

Algebraic integral points on arithmetic surfaces exhibit predictable limit distributions, revealing a deep connection between Galois orbits and local probability measures.

A Poincaré-Hopf formula is established for anisotropic quasilinear elliptic systems, revealing topological invariants for critical points.

The space of vacuum initial data sets is revealed to possess rich, non-trivial global topology, challenging prior assumptions about its structure.

Minimal submanifolds in locally symmetric spaces reveal a surprising connection to higher-dimensional expander families, with their volume directly reflecting the manifold's scale.

A single adversarial token can bypass most safety filters.

Higher-genus Gromov--Witten theory of a Calabi--Yau threefold is shown to be governed by topological recursion on its mirror curve, proving long-standing conjectures.

Rational surfaces with infinite automorphism groups are now constructible, challenging prior assumptions about their finiteness.

Proper covers trivializing isocrystals can be replaced by finite étale covers, revealing a fundamental connection between geometric triviality and finite monodromy.

De Jonquières Cremona maps of degree 4 can require up to five quadratic maps for decomposition, a significant increase from their non-de Jonquières counterparts.

A new class of non-commutative minifolds reveals geometric helices whose Euler matrices are irreducible to commutative Fano threefolds, challenging existing classification paradigms.

Genus formulae for Atkin-Lehner quotients of modular curves are derived by constructing semistable models over Witt rings.

Bridgeland stability on K3 surfaces directly implies the existence of special Lagrangian spheres, confirming a key prediction of the Thomas-Yau-Joyce conjectures.

Twisted loop symmetric varieties reveal fundamental purity and parity properties in their orbit closure singularities, unlocking new insights into affine representation theory and Langlands duality.

Chirality is now explicitly encoded in 3D shape descriptors, enabling precise discrimination of mirror-image pairs where prior methods failed.

A new (infinity, 2)-categorification of Lusztig's map unlocks a fresh proof for the classification of unipotent representations in split finite groups of Lie type.

Finite-time singularities in Kahler-Ricci flow exhibit controlled scalar curvature growth, paving the way for analytic minimal model program applications.

Unbalanced normal bundles in rational curve families are characterized by explicit numerical criteria derived from a novel two-term resolution of the relative normal bundle.

Diagonal cohomology representatives are proven to be interval-independent, revealing a fundamental structural property of Ekedahl's equivalence.

Orthogonal quivers unexpectedly yield the full spectrum of Kleinian surface singularities, mirroring classical results for simply-laced quivers.

Analytic torsion for projective spaces surprisingly lacks logarithmic terms in its asymptotic expansion, simplifying its structure.

Finite local measurements can stably reconstruct functions from their geodesic ray transform, even in complex geometries.

Symmetry of optimal transport solutions is proven for overdetermined Monge-Ampère problems, even with complex boundary conditions and target domains.

Smooth solutions to generalized SQG equations persist for nearby singularity parameters, bridging the gap between standard SQG and more singular active scalar models.

Smooth complex quasi-projective varieties have fundamental groups that are precisely finite direct products of free groups and cyclic groups of order $m$.

Critical immersions of even-dimensional manifolds into $\mathbb R^m$ are proven to be real-analytic, extending fundamental regularity results for geometric variational problems.

High-codimension mean curvature flows in dimensions 5 and 6 now have their singularities understood across the full pinching range, thanks to a new Kato-type inequality.

A critical ellipticity moment unlocks quantitative homogenization and large-scale regularity for nondivergence-form equations previously lacking a positive lower bound.

The classical limit of Magnus expansion in QFT is precisely mapped to worldline QFT via a Hopf algebra, yielding a new formula for Murua coefficients.

Simple singularities and Fermat hypersurfaces reveal a functional equation linking phi-functions to their reflections, generalizing F-invariant characterizations of regular local rings to characteristic zero.

Universal Coxeter groups unlock new insights into the Zariski-dense orbits of birational automorphisms on Calabi-Yau manifolds.

Supersingular Tate conjecture is proven for $K3^{[n]}$, OG6, OG10, and $\mathrm{Kum}^n$ varieties, unlocking new insights into their arithmetic geometry.

The full proof for Teukolsky wave-transport systems and energy-Morawetz estimates on subextremal Kerr perturbations is finally delivered, completing a decade-long quest for black hole stability.

A rigorous mathematical framework for compressible free-surface fluid dynamics is established, proving local well-posedness for the first time without surface tension.

Non-locally-conformally-flat metrics on the ball can lead to infinitely many unbounded solutions for the boundary Yamabe equation, even with fixed background and prescribed curvatures.

A direct proof of complex Hessian gradient estimates on Hermitian manifolds bypasses established complex analytic tools, revealing a new path via stability and comparison arguments.

Polynomial roots achieve $W^{1,q}$ regularity for $q$ up to $d/(d-1)$, and a simple Lipschitz reparameterization renders them Lipschitz continuous for all $q$.

KPZ universality is now proven for a broad class of nonlinear SPDEs in half-space, extending beyond integrable systems.

Soliton resolution is now proven for the derivative NLS in the optimal critical $L^2$ space, revealing quantized $R$-bubbles in finite-time blow-up.

Even-odd perturbations of Boussinesq solitary waves exhibit a strict dichotomy: they either scatter globally or blow up in finite time.

Singular sets in parabolic homogenization exhibit codimension-two Minkowski bounds and scale-invariant doubling properties, even below the homogenization scale.

A single magnetic field can prevent buckling in slender ferromagnetic rods, enabling precise control over their stability and leading to predictable, hysteretic switching behaviors.

A direct geometric interpretation of divisorial $d$-gonality emerges for pseudocompact metric graphs via harmonic morphisms to trees.

Neumann boundary conditions introduce a fundamental 1/4 decay loss in wave equation Green functions, even for smooth strictly convex domains.

The optimal placement of many small traps in heterogeneous media switches from an arithmetic to a geometric average of diffusivity and particle distribution as trapping strength increases, with the latter dominating most practical scenarios.

Recovering unique impedance pairs from just two resonant frequencies on annular and spherical shells is now possible via explicit Green's functions and resonance spectrum bilinearity.

Periodic advection can accelerate or decelerate reaction-diffusion wave propagation by an amount proportional to the spatial period, with an explicit formula for this leading correction.

A novel three-level regularization scheme guarantees the existence of weak solutions for consensus-based kinetic image segmentation models with non-smooth kernels and degenerate diffusion on bounded domains.

A novel degenerate dissipation mechanism allows viscosity in one phase to drive the dynamics of an inviscid phase in the low Mach number limit of two-phase flow.

Optimal polynomial stability is achieved for coupled second-order evolution equations with asymmetric damping and arbitrary memory kernels, even when wave speeds differ.

Log-concavity of generating polynomials unlocks efficient approximate counting for a broad class of combinatorial problems, resolving long-standing conjectures.

Dense subsets of $\mathbb F_q^{2m}$ are nearly determined by quadratic form values, revealing a surprising geometric rigidity.

Linearized perturbations of all subextremal Kerr black holes are globally bounded, finally proving the Kerr stability conjecture.

Boussinesq systems exhibit significantly longer well-posedness times than previously known, with lifespan scaling precisely with the interplay of nonlinearity, dispersion, and cavitation proximity.

A single product flux measurement can fully reconstruct the forward collision kernel for a reversible reaction, even with imperfect experimental conditions.

Curvature estimates for graphical solutions to curvature quotient equations are proven without convexity or $Γ_{k+1}$-admissibility, unlocking analysis for a broader class of geometric problems.

Quantum convolutions exhibit polymatroidal geometry, unlocking a suite of noncommutative entropic inequalities and optimal bounds on entropy growth.

A complete classification of solutions to the planar isotropic $L_p$ dual Minkowski problem is now available, revealing precise counts and existence conditions for nonconstant positive $C^2$ solutions.

Permanents of a specific class of matrices obey a multiplicative inequality analogous to Koteljanskii's, proving essential for understanding their structural properties.

The complexity of continuous graph homomorphisms from higher-dimensional Schreier graphs is fully characterized, revealing a $\Sigma^0_1$-complete set of receiving graphs.

Finite spaces with simultaneously strict weak, stable, and ordinary Lusternik-Schnirelmann categories are explicitly constructed, resolving a long-standing open problem.

Spatial Fisher information bounds unlock rigorous convergence of a viscous BGK model to Euler equations, providing a novel analytical pathway beyond standard kinetic theory.

Nonlinear Schrödinger equations for open quantum systems are proven well-posed even in less regular energy spaces, relaxing standard monotonicity assumptions.

A graph's global structure is predictable from a single local degree, provided it avoids specific induced subgraphs.

Arbitrarily large chromatic number graphs can be triangle-free, yet their 4-cycle-free subgraphs have a chromatic number of only 3.

A single adversarial token can bypass most safety filters.

A unified framework for global well-posedness of the stochastic Landau--Lifshitz--Baryakhtar equation is established for initial data down to $\mathbb{L}^2$ in $\mathbb{R}^d$, $d \leq 3$.

A tight $O(2^t)$ upper bound for the 2-color bipartite Ramsey number $b_2(t)$ is independently proven, refining previous results.

Khovanov homology for nullhomologous links in $\mathbb{R}P^3$ is fully determined by their Jones polynomial and signatures, a result derived from a novel spanning tree model.

Deterministic perfect quantum state transfer is achieved on signed graphs by eliminating back-scattering through topological routing, even under environmental noise.

A single gain set condition unlocks supersolvability and nice partitions in graphic arrangement deformations, revealing a deep connection to perfect elimination orderings.

Heat kernel regularization unlocks quantitative $L^2$ stability for Kantorovich potentials on Heisenberg groups and RCD spaces, revealing a $W_1^{1/2}$ rate and dimension-independent exponent.

Kazhdan--Lusztig polynomials for sparse paving matroids are unimodal, with only three exceptions to strict unimodality for rank $\ge 5$.

Rank 2 affine MV polytopes of $\widehat{SL_2}$ are now precisely characterized by tropical points of reduced double Bruhat cells, mirroring finite-type structures.

Hypergraphic polymatroids reveal that an additive code's critical exponent is dictated by the weak chromatic number of its associated hypergraph.

A single 32-vertex graph proves that five distinct notions of graph width are indeed separable, challenging assumptions of their inherent equivalence.

The sub-quorum coloring number is precisely characterized for infinite families of caterpillars, revealing structural properties beyond standard graph coloring.

The chromatic number of pseudohemisphere hypergraphs is provably at most four, a significant finding for this newly defined combinatorial structure.

A single vertex can be essential for maintaining connectivity, yet not guarantee an eternal connected vertex cover, revealing a subtle gap between static and dynamic graph defenses.

Hypercube silver colourings are halved by a pair-defensive requirement, limiting colours to $\lfloor(d+3)/2\rfloor$ instead of $d+1$.

The diameter of the additive-multiplicative graph is proven to be 6, and its radius 4, contingent on a specific number-theoretic conjecture.

Support-two monomial ideals with linear resolutions are precisely those whose underlying graphs are co-chordal, a rare property for general monomial ideals.

Kromatic Symmetric Functions can now be recursively computed in a $K$-theoretic monomial basis, revealing graph distinctions missed by their classical counterparts.

Free ultrafilters constructed from partitions of the doubly exponential sequence precisely populate the gaps in the depth filtration of $β\mathbb{N}$.

A closed-form formula for unequal-parameter Kostka--Foulkes polynomials of type $C_n$ at fundamental weights unlocks new explicit calculations for Hecke algebras of non-split orthogonal groups.

Ghost $r$-SICs, previously only conjectured to exist, are now explicitly constructed and linked to Hermitian $r$-SICs via modular dilogarithm identities.