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Paper The second Kahn–Kalai conjecture with an edge-count bound
Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in G(n,p) is at most C p{\mathrm E}(n,H)(1+\log2 h), with universal C.…
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Paper Hilbert's sixteenth problem: uniform bounds for limit cycles
Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree.…
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Paper The entropy-rate dimension formula for self-similar measures
For every self-similar measure on the line generated by finitely many contracting similarities, proves \dim{\mathrm H}\mu=\min\{1,h{\mathrm{RW}}/\chi\}, where h{\mathrm{RW}} is the entropy rate of random…
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Paper Shareshian–Wachs elementary positivity
Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in \mathbb N[q]. The…
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Paper Sharp logarithmic exponents for off-diagonal Ramsey numbers
For every fixed integer s ≥ 5, proves r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)} as t\to\infty, determining the logarithmic exponent and matching the classical upper bound at that scale. Here r(s,t) is the least…
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Paper The Erdős–Gallai cycle-decomposition conjecture
Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most Cn simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai…
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Paper Finite lattice representation and undecidability
Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation,…
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Paper Tensor saturation for even spin groups
Proves saturation factor one for \mathop{\mathrm{Spin}}\nolimits (2n), n ≥ 2: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already…
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Paper Saxl’s conjecture and universal tensor squares
Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every Sn with n\notin\{2,4,9\}…
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Paper Critical percolation on every quasi-transitive graph
Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with pc\lt 1: at the critical probability, there is almost surely no…
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Paper The Benjamini–Schramm nonuniqueness conjecture
Proves pc\lt pu for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range…
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Paper Perceptron free energies and microscopic jamming exponents
Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern…
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Paper Continuum phase transitions for radial pair potentials
Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout…
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Paper Rigidity of the Turing degrees
Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability. () Manuscript September 24,…
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Paper Single-fold Diophantine representations and undecidability under an at-most-one-solution promise
Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and…
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Paper Quasi-isometric recognition of virtually polycyclic groups
Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a…
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Paper The spin-one Haldane gap
Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open…
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Paper Area laws and tensor networks for two-dimensional gapped systems
Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On…
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Paper The entropy photon-number inequality
Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output's entropy photon number is at least the…
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Paper QMA-hardness of continuum Coulomb energy
Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only…
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Paper Classical capacity of generalized amplitude damping
Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by…
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Paper A ZFC counterexample to Naimark's problem
Gives an alternative to of a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted…
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Paper Quillen's conjecture in rational homology
Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero…
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Paper The Hovey–Strickland and Chai conjectures
Proves Chai's invariant-ideal conjecture for Lubin–Tate deformation rings over finite residue fields, at every prime and positive height n. Through the implication of Barthel–Heard–Naumann, this proves the…
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Paper The Grothendieck homotopy hypothesis
Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces. () Manuscript…
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Paper Finite generation for the K(n)-local sphere
Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the K(n)-local sphere is a finitely generated ℤp-module, for every prime, positive height and integer degree. The…
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Paper Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds
Proves that every smooth Riemannian metric on Sn, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The same conclusion holds on every closed manifold admitting a finite smooth spherical…
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Paper Unique tangent flows at the first surface singularity
Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ3. At every singular point, all fixed-center backward tangent flows agree as area…
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Paper Scalar curvature and finite-time Ricci-flow singularities
Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded…
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Paper A finite-time singularity of Calabi flow
Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth U(10)-invariant Kähler metric on \mathbb{CP}^{10} whose flow develops a finite-time singularity. The metric lies in…
372 posts