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Paper Torus-packet equidistribution in prime, quartic, and sextic degrees
Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order…
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Paper Uchida’s conjecture for open homomorphisms of Galois groups
Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the…
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Paper Log-canonical threefold abundance in numerical dimension one
Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p > 3 when the effective boundary is rational and the ℚ-Cartier adjoint is nef of numerical…
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Paper Nagata’s conjecture and maximal Seshadri constants
Proves Nagata's strict inequality \sumi mi\lt d\sqrt r for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal…
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Paper Shafarevich counterexamples in dimension two and with large fundamental group
Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate…
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Paper A stable-coordinate counterexample in four variables
Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine…
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Paper Fundamental groups of special complex varieties and root orbifolds
Proves Campana's abelianity conjecture: special compact Kähler manifolds have virtually abelian fundamental groups. Using this theorem, establishes the same conclusion for order-two root orbifolds of smooth projective…
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Paper Real ultraflat Littlewood polynomials and unbounded binary merit factors
Constructs polynomials with N consecutive coefficients in \{-1,1\} whose modulus is (1+o(1))\sqrt N uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood…
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Paper The geometric case of the Erdős similarity conjecture
For every fixed q\in(0,1), constructs compact subsets of [0,1] with measure arbitrarily close to one containing no translated and nontrivially dilated copy of \{q^n:n\ge1\}, with dilations of either sign.…
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Paper Hyperbolicity cones without semidefinite lifts
Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real…
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Paper The sharp simplex conjecture for isotropic constants
Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave…
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Paper Approximate counting of common integer polymatroid bases
Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each…
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Paper Quantitative trace-reconstruction bounds with a uniform decoder
At every fixed deletion probability in (0,1), reconstructing an arbitrary length-n binary string requires n^{\Omega(\log\log n)} independent traces, ruling out polynomial-sample reconstruction. A uniform decoder…
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Paper Exponential semidefinite complexity of perfect matching
Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of…
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Paper Classwise permanence for weakly reversible mass-action systems
Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact…
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Paper Weak mixing of triangular billiards with an irrational angle
Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the…
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Paper Talagrand’s expectation thresholds, discrete convexity, and graph decompositions
Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand…
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Paper Power savings for intersective polynomial differences and prime arguments
For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of \{1,\ldots,N\} avoiding nonzero values h(1),h(2),\ldots as differences has size…
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Paper Uniform influence and sharp thresholds for graph and hypergraph properties
Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed 0\lt \varepsilon\lt 1/2, every nontrivial increasing relabeling-invariant property crosses from…
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Paper Nonsingular systems of equations over arbitrary groups
Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group…
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Paper The geometric phase diagram, diffusion, and spectra of random planar maps
Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for 0\lt q\le4 and to the Brownian continuum random tree for q > 4, establishing the surface-to-tree geometric transition. For…
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Paper Canonical O(3) continuum limit and exact O(4) mass asymptotics
Constructs the canonical continuum limit of the two-dimensional nearest-neighbor O(3) model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice O(4)…
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Paper Critical and near-critical XY scaling and BKT universality
For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations C{\betac}(r)\sim Ar^{-1/4}(\log r)^{1/8} and the Berezinskii–Kosterlitz–Thouless essential singularity…
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Paper GOE bulk universality for regular graphs with weak Anderson disorder
For every fixed degree d ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random d-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy…
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Paper Directional zero–one laws beyond iid environments and iid ballisticity
On ℤd, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid…
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Paper Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling
Proves chordal SLEκ limits for critical square-lattice random-cluster interfaces for 0\lt q\le4, with \kappa=4\pi/\arccos(-\sqrt q/2): bounded Jordan domains are allowed for q ≥ 1, and smooth Jordan domains for…
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Paper Critical and quenched near-critical universality for Poisson–Voronoi percolation
Proves Cardy's formula for annealed critical Poisson–Voronoi crossing probabilities in every bounded Jordan quadrilateral. With each model normalized by its own expected unit-square pivotal count, the conditional joint…
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Paper Critical SK autocorrelation processes and dynamics across the temperature transition
For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the \log n scale for fixed 0\le\beta\lt 1, mixing time n^{2/3+o(1)} at β = 1, and…
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Paper Exact three- and four-state reconstruction thresholds and four-state tree capacity
Proves the exact reconstruction threshold d\lambda^2\gt 1, with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (d ≥ 2) and observed Poisson trees…
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Paper Limiting random SAT thresholds, sharp variance and computability
For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance \Thetak(n) for every fixed k ≥ 3, and computability of…
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