Recent progress in mathematics, discussed by the people who make it.
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Recent activity
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0 votes 0 replies 9 views
Paper Milne’s rationality conjecture and algebraic specialization
Proves Milne's rationality conjecture for abelian varieties over \overline{\mathbb Q} with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology…
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Paper Donaldson's hypersymplectic deformation conjecture
Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by…
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Paper The full BSD formula from low Selmer corank
Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group.…
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Paper Hilbert’s tenth problem over ℚ
Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively. Manuscripts (2) October 6, 2026 ·…
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Paper Termination of projective and Kähler fourfold minimal model programs
Proves termination of every existing generalized log canonical flip sequence on globally Weil ℚ-factorial compact Kähler fourfolds, with rational boundary, fixed rational analytically nef b-data, and projective small…
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Paper Goldfeld’s conjecture: densities and mean analytic rank
Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist…
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Paper The local p-adic section conjecture and global consequences
Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚp correspond bijectively to conjugacy classes of sections of its full arithmetic étale…
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Paper A counterexample to Wall's finite D(2) problem
Constructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite…
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Paper Unrestricted pro-modularity at the prime two
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of G{\mathbb Q} unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves…
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Paper Universal computation in forced Navier–Stokes flows
Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing. A terminating compiler turns a Turing machine and…
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Paper Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity
Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef ℚ-Cartier adjoint is semiample. It also…
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Paper The rational Hodge conjecture for CM abelian varieties
Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields…
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Paper Numerical semiampleness and generalized minimal models
Proves numerical semiampleness for nef adjoints KX+B+M with KX+B pseudo-effective and M nef rational, for projective klt rational pairs over algebraically closed characteristic-zero fields and smooth compact…
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Paper Gaussian fields and interfaces for triangular-lattice Lipschitz heights
Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments 0,\pm2 and two-arc boundary values \pm1, and zero-boundary…
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Paper The joint critical Ashkin–Teller current limit
Identifies the joint scaling limit of Ashkin–Teller heights and both complete current-cluster collections throughout the critical line, including the four-state Potts endpoint. In bounded Jordan domains with admissible…
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Paper Donaldson's tamed-to-compatible conjecture
Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure…
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Paper Annular variation and dyadic absolute bounds for the triangular Hilbert transform
Proves maximal and annular r-variation bounds, for every r > 2, from complex L^3(\mathbb R^2)\times L^3(\mathbb R^2) to L^{3/2}(\mathbb R^2). The maximal estimate controls both hard truncation endpoints and…
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Paper Correspondence coloring with a fixed forbidden subgraph
Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need OF(\Delta/\log\Delta) colors when their maximum degree Δ is sufficiently large. Also…
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Paper Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan
Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over \overline{\mathbb F}q under the four stated Lie-theoretic characteristic hypotheses.…
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Paper The ℓ¹-Bass conjecture for all discrete groups
Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over \ell^1(G) are supported on finitely many finite-order conjugacy classes. The algebraic companion proves…
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Paper Virasoro constraints for complete intersections and projective-bundle towers
Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also…
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Paper Quasipolynomial algorithms for mean-payoff, stochastic and parity games
Gives deterministic algorithms using 2^{O((\log(L+2))^2)} bit operations, for complete binary input length L, for ordinary mean-payoff games and two separate extensions. They compute exact values and optimal…
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Paper Conformal universality for weakly interacting and random-bond Ising models
Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE3…
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Paper Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions
Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area,…
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Paper Positive-temperature Bose–Einstein condensation and exact quantum depletion
Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature,…
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Paper Uniform Laughlin gap and stability under bounded scalar disorder
Proves the fermionic Laughlin spectral-gap conjecture for the full V1 interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body…
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Paper Counterexamples to stable-Morse and strong Arnold fixed-point bounds
Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate…
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Paper The quasi-Riemann hypothesis
Proves that every Dirichlet L-function, including \zeta(s), is zero-free in \Re s\gt 7/8, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over…
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Paper Function-field reconstruction from Milnor K-theory and Galois data
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from K^{\mathrm M}1/\ell, K^{\mathrm M}2/\ell, and their product. These data recover the perfect closure and…
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Paper The Margulis–Platonov conjecture over global fields
Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract…
372 posts