Papers

We prove that the maximum number of mutually unbiased orthonormal bases in $\C^6$ is three, resolving Zauner's dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification …

For all positive integers $n$ and $m$, every $A \in M_n(\mathbb{C})$ and every polynomial $P(z) = \sum_k B_k z^k$ with $B_k \in M_m(\mathbb{C})$, the manuscript proves $\|P[A]\| \le 2 \max_{z \in W(A)} \|P(z)\|$, where $P[A] = \sum_k A^k \otimes B_k$ …

We resolve the complete Crouzeix conjecture by proving the sharp constant-two numerical-range inequality for every bounded operator on a complex Hilbert space and every matrix-valued polynomial. No separability assumption is needed. The closure of the numerical range is a complete …

We prove that the exponent of matrix multiplication over the complex numbers is at most $9/4$.

We prove worst-case total-variation cutoff for the zero-field Gaussian Sherrington--Kirkpatrick heat-bath dynamics at every fixed inverse temperature \(0\leq\beta<1\). With rate-one refresh at each spin, the cutoff location is \(\log n/(2\lambda(\beta))\) for a positive deterministic rate \(\lambda(\beta)\). The location for uniformly …

We prove the Sandier--Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background. Combined with Bétermin and Sandier's asymptotic formula, this also proves the Brauchart--Hardin--Saff conjecture for …

We prove that planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment. This resolves the planar no-bigeodesics conjecture under that …

We prove the Euclidean-complete affine Bernstein conjecture in dimensions three through nine: a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete. The same conclusion holds, without an initial graph assumption, …

We determine the unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. It equals the one-shot Holevo capacity and is given by a one-variable maximum. A binary pure-state ensemble attains the one-use optimum, …

For each fixed dimension $d\ge3$ and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on $\mathbb Z^d$ with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open …

We resolve the smooth anisotropic Calderón uniqueness problem with arbitrary same-patch measurements. A smooth Riemannian metric on a compact connected manifold of dimension at least three is determined, up to a diffeomorphism fixing the measured patch, by its zero-frequency Dirichlet-to-Neumann …

We resolve the three-dimensional Ball--Evans approximation problem for $1\le p\le2$. Every $W^{1,p}$ homeomorphism between arbitrary bounded domains in $\R^3$ is a strong $W^{1,p}$ limit of smooth diffeomorphisms onto the same target.

We resolve the three-dimensional Ball--Evans approximation problem for every finite $p>2$. Every $W^{1,p}$ homeomorphism between arbitrary bounded domains in $\R^3$ can be approximated strongly in $W^{1,p}$ by smooth diffeomorphisms onto the same target.

For a smooth strictly convex planar billiard of positive curvature, a continuous foliation of a full grazing annulus by individually invariant essential curves forces an analytic boundary and a jointly analytic collar of smooth strictly convex caustics. The physical-collar rigidity …

We prove the three-dimensional Bochner--Riesz conjecture in its strict-order formulation. The main result is boundedness of the Bochner--Riesz multipliers on $L^3(\RR^3)$ for every positive order. Interpolation and duality then give the full conjectured strict range.

We resolve the generalized Cartan--Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most $\kappa\le0$, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in …

We prove that nearest-neighbor Bernoulli bond and site percolation on $\mathbb Z^3$ have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.

We resolve De Giorgi's conjecture positively in dimension eight: every entire $C^2$ solution $u:\R^8\to(-1,1)$ of $\Delta u=u^3-u$ with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution $v:\R^7\to[-1,1]$ of this equation is a constant …

Yes: a discrete group (countable or not) is amenable if and only if every uniformly bounded representation on a complex Hilbert space is similar to a unitary one; every nonamenable group has a nonunitarizable representation of bound at most 1 …

We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either …

Yes: ||e^{it sqrt(-Delta)} f||_{L^3(R^3 x [1,2])} <= C_eps ||(1-Delta)^{eps/2} f||_3 for every eps > 0, and hence for every 2 < p < infinity the bound with alpha > max{0, 1-3/p} derivatives, which gives AIM's annular estimate for every 3 …

We prove that the ordinary symmetric Fourier partial sums of every complex-valued function in $L\log L$ on the circle converge almost everywhere to the function along the full sequence. This establishes the classical $L\log L$ sufficiency conjecture.

We prove a uniform positive spectral gap for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings, establishing the positive even-periodic formulation of the spin-one Haldane conjecture.

We prove a uniform interior $C^{1,\alpha_d}$ estimate for bounded infinity-harmonic functions in every dimension $d\ge3$. For some $\alpha_d\in(0,1/3]$ depending only on the dimension, the gradient's supremum norm and $\alpha_d$-Hölder seminorm on $B_{1/2}$ are bounded by a dimension-dependent constant times the …

We prove that sufficiently close unital von Neumann algebras are conjugate by a unitary arbitrarily close to the identity. The tolerance depends only on the prescribed distance of that unitary from the identity, uniformly over all algebras, representations, and Hilbert …

We prove that every bounded complex-linear unital algebra homomorphism from a unital complex $C^*$-algebra into the bounded operators on an arbitrary Hilbert space is similar to a $*$-homomorphism. This resolves Kadison's similarity conjecture positively. We also obtain one universal hyperreflexivity …

We prove the subcritical Hénon--Lane--Emden conjecture: for every dimension $n\ge2$, positive powers $p,q$, and real weights $A,B$, the system has no strictly positive entire solution when $(n+A)/(p+1)+(n+B)/(q+1)>n-2$. Solutions need only be continuous at the origin and classical elsewhere, without a …

We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in $\R^n$ has volume product at least $4^n/n!$, with equality exactly for invertible linear images of Hanner polytopes.

We disprove the Modified Integer Round-Up Conjecture for bin packing by constructing instances with arbitrarily large additive gaps between the configuration-LP value and the integral optimum. We also prove that, for every fixed nonnegative integer $c$, distinguishing instances that fit …

We prove that the weak Ma--Trudinger--Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities …

We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's …

We prove that seven is the critical dimension for the one-phase Bernoulli problem: every nonzero one-homogeneous global minimizer in dimensions at most six is flat, while a nonflat one-homogeneous global minimizer exists in dimension seven. It follows that the interior …

We prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on …

Exact worst-case reconstruction of a binary word from independent deletion traces requires $n^{\Omega(\log\log n)}$ samples for every fixed deletion probability $q\in(0,1)$, even with unrestricted computation and any fixed positive success probability. This gives a negative answer to the polynomial-sample question …

We construct an entangled state on $\C^{10}\otimes\C^{10}$ with zero distillable secret key for the local-instrument protocols specified here. They allow joint processing of all copies and unlimited two-way public communication, with the input as the only shared private resource and …

For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part …

We prove the boundedness and persistence conjectures for finite weakly reversible mass-action systems with positive constant reaction rates. For every positive initial condition, the solution exists for all forward time, and every concentration remains bounded above and bounded away from …

We prove nonuniqueness for the three-dimensional periodic hard-sphere Boltzmann equation among global entropy solutions satisfying exact local conservation of mass, momentum, and kinetic energy. We construct one nonnegative initial density with bounded velocity support and finite mass, energy, and absolute …

We construct two distinct uniformly positive bounded measurable scalar conductivities on a ball in $\R^3$ with the same full Dirichlet-to-Neumann operator. Both conductivities equal one near the boundary. This gives nonuniqueness in the scalar Calderón problem at bounded measurable regularity.

We prove the directional zero--one conjecture for nearest-neighbor random walks in independent and identically distributed strictly elliptic environments on $\ZZ^d$, $d\ge3$: the probability of escape in each fixed nonzero real direction is zero or one. Only strict positivity of the …

We prove that almost-sure transience in a fixed direction implies a deterministic limiting velocity with positive projection in that direction for nearest-neighbor random walks in independent and identically distributed uniformly elliptic environments on $\mathbb Z^d$, $d\ge2$. This resolves the ballisticity …

We disprove Sidorenko's conjecture with a bipartite graph on \(35\) vertices and \(66\) edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one …

We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies. We also prove the sharp entropy bound $h(f)\ge m+\tfrac12\log\det\Cov(f)$ for every log-concave probability density on $\R^m$, with equality …

Yes: for every p > 3 there is C_p, depending only on p, with ||Eg||_{L^p(R^3)} <= C_p ||g||_{L^infinity(S^2, sigma)} for every bounded measurable complex g on S^2..

For every fixed $1\le q<4$, critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal $\SLE_{\kappa(q)}$, where $\kappa(q)=4\pi/\arccos(-\sqrt q/2)$, confirming the Rohde--Schramm prediction in this parameter range. This holds in every bounded Jordan domain under uniform marked boundary approximation. …

We prove that the standard sine map of the two-dimensional torus has positive metric entropy with respect to normalized area for every sufficiently large positive parameter. This gives a full parameter tail, and hence answers Sinai's positive-parameter-measure conjecture affirmatively.

We prove that the full static displacement-to-traction map uniquely determines both real smooth Lamé moduli on every bounded connected smooth domain in $\R^3$, provided $\mu>0$ and $3\lambda+2\mu>0$ on the closure. This resolves the smooth three-dimensional isotropic elastic Calderón uniqueness problem …

We prove a uniform \(L^2\) bound for the Hilbert transform along a Lipschitz unit vector field in the plane, with integration restricted to a fixed absolute multiple of the reciprocal Lipschitz seminorm. The bound is uniform in the inner truncation …

We prove that the limit shape of undirected nearest-neighbor first-passage percolation on $\Z^2$ with independent exponential edge weights is strictly convex and has a $C^1$ boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More …

We prove that every finite loopless \(k\)-edge-connected multigraph on at least two vertices, with \(k\ge1\), has a spanning tree meeting each cut in at most \(C/k\) times the size of the cut, where \(C\) is a universal constant. This resolves …