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Paper An infinite finitely presented residually finite 2-group and a finitely presented nil algebra
Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also…
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Paper Amenability, unitarizability, and strong Ulam stability
Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability…
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Paper A group without fixed price
Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away…
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Paper Sharp finite-matrix Lieb–Thirring inequalities and all equality cases
Proves the sharp one-dimensional Lieb–Thirring inequality for 1/2\lt \gamma\lt 3/2 and arbitrary finite-matrix potentials W ≥ 0 with \int\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty: the optimal…
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Paper Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability
Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C∗-algebra, resolving this case of Szabó's…
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Paper Threshold and positive-energy bound states of the BFSS matrix model
Proves that the undeformed relative \mathrm{SU}(N) BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For \mathrm{SU}(2), a…
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Paper Bloch's law, its lattice correction, and the spherical magnetization law
Proves Bloch's T3/2 law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ3.…
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Paper Polynomial-time unitary synthesis from a Boolean oracle
Solves the constant-error Aaronson–Kuperberg unitary synthesis problem: a uniform polynomial-size quantum oracle circuit approximates every n-qubit unitary channel within diamond-norm error 1/2, after a suitable Boolean…
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Paper The optimal quartic separation between randomized and quantum queries
Shows that the universal bound R(f)=O((1+Q(f))^4) for total Boolean functions is sharp in its exponent, ruling out every smaller power and disproving the conjectured cubic relation. Here R and Q are randomized and…
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Paper Strong Kadison–Kastler stability and its spatial boundaries
Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit…
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Paper Failure of rational injectivity for maximal coarse assembly
Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with…
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Paper Gromov’s integral scalar-curvature bound for simplicial volume
Proves \intM(\mathrm{Scal}g^-)^{n/2}\,dVg\ge an\lVert M\rVert for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with an\gt 0 depending only on dimension. Here…
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Paper Spectral scalar curvature, Urysohn width, and macroscopic dimension
Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying -4\Delta+\mathrm{Scal}\ge1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n-2 whose…
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Paper Sharp singular-set bounds for stationary integral varifolds
Proves that every stationary integral m-varifold in a Euclidean open set has singular set of Hausdorff dimension at most m-1, a sharp bound in every positive dimension and codimension. On round spheres, the family…
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Paper Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample
Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in…
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Paper Nonuniqueness with local conservation for the hard-sphere Boltzmann equation
Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and…
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Paper Joint metric and connection recovery from one boundary patch
Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least…
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Paper An L^{3} bound for the trilinear Hilbert transform
Proves that the principal-value trilinear Hilbert transform with shifts x-t, x-2t, x-3t is bounded from L^3(\mathbb R)^3 to L^1(\mathbb R). This resolves the L3 exponent case of the standard…
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Paper Metric Markov cotype of \ell_{1} and Hilbert-space Lipschitz extension
Proves that real ℓ1 has metric Markov cotype two, answering Mendel and Naor's question. Consequently, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ1 extends to the whole space with a…
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Paper Nonnegative-curvature Einstein classification and an L^{2} topological gap
Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S4, Fubini–Study \mathbb{CP}^2,…
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Paper Primitive roots for every admissible integer base
Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least ca x/(\log x)^2 primes in every sufficiently large interval…
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Paper Modularity of elliptic curves over imaginary quadratic fields
Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place. Manuscript…
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Paper Global quantum geometric Langlands at irrational level
Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level $`c\in\mathbb…
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Paper Uniform black-box noncommutative identity testing across characteristics
For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that…
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Paper Subset Sum in O(2^{0.49n}) time
Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary O(2^{0.49n}) word-RAM time on polynomial-bit inputs, where n counts the integers. The time bound holds on every execution and…
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Paper Existential–universal real sentences in the counting hierarchy
Proves that the existential theory of the reals lies in the counting hierarchy. More generally, truth of existential–universal real sentences can be decided at one fixed level of that hierarchy, even when their integer…
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Paper Deterministic polynomial factorization over prime fields
Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽p, including multiplicities, in bit complexity polynomial in (n+1)\log p. The prime is…
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Paper Pointwise multiple ergodic averages for mixing transformations
Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is…
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Paper A torsion-free group algebra that is not directly finite
Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective…
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Paper Sharp singularity rates for symmetric random sign matrices
Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give \Pr(\det An=0)=(1/2+o(1))^n; for fixed bias…
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