Real and complex analysis

We prove that a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components. This establishes the removability-to-rigidity direction of the He–Schramm Conjecture.

We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe's circle-domain conjecture positively.

We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then \(|\varphi'|^s\) is area-integrable for every \(4/3\lt s\lt 4\). We also prove the sharp inverse-square integral-means exponent \(B_{\mathcal S}(-2)=1\) for the normalized schlicht class \(\mathcal S\).

We prove a uniform upper bound for inverse-first-power integral means of normalized univalent disk maps with exponent strictly below 1/4. Consequently, the bounded universal integral-means spectrum satisfies \(B_b(-1)\lt 1/4\), disproving Kraetzer's conjectured spectrum at p = −1.

We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝ^d with Hausdorff dimension greater than \(d/2\) determines a set of Euclidean distances of positive Lebesgue measure.

We prove the four-dimensional Hausdorff-dimension Kakeya conjecture: every subset of ℝ⁴ containing a unit line segment in every direction has Hausdorff dimension four. No compactness or regularity assumption is imposed on the set or its witnessing line family.

We prove the Kakeya maximal conjecture in three dimensions. For every ε > 0, the maximal average over unit tubes of radius δ maps \(L^3(\mathbb R^3)\) to \(L^3(S^2)\) with norm at most \(C_\varepsilon\delta^{-\varepsilon}\).

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.

For every \(\varepsilon\in(0,1)\) and every sufficiently large integer N, there is a polynomial of length N with coefficients in \(\{-1,1\}\) whose modulus lies between \((1-\varepsilon)\sqrt N\) and \((1+\varepsilon)\sqrt N\) everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be chosen separately at each length.

For every η > 0 and every sufficiently large integer N, there is a polynomial with N consecutive coefficients in \(\{-1,1\}\) whose modulus lies between \(\sqrt N/16\) and \((1+\eta)\sqrt N\) everywhere on the unit circle.

We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in \(\{-1,1\}\) is \((1+o(1))\sqrt N\), as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor at length N tends to infinity through all integer lengths, disproving Turyn's conjecture.

We prove the bounded-data Fourier restriction conjecture for the sphere in three dimensions: the Fourier extension operator maps \(L^\infty(S^2)\) boundedly into \(L^p(\mathbb R^3)\) for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp.

We prove the diagonal Fourier extension conjecture for compact smooth positively curved surfaces \(\Sigma\subset\mathbb R^3\), including surfaces with smooth boundary. For every \(3\lt p\lt \infty\), the extension operator maps \(L^p(\Sigma)\) boundedly into \(L^p(\mathbb R^3)\). The compact paraboloid case also yields free Schrödinger local smoothing in two spatial dimensions for every \(3\lt p\lt \infty\) and Sobolev order \(s\gt 2-6/p\).

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

We prove the critical L³ local smoothing estimate for the wave equation in three spatial dimensions, with every positive Sobolev loss. This resolves Sogge's Euclidean local smoothing conjecture in dimension three.

We resolve the Sobolev endpoint of Carleson's pointwise convergence problem in every dimension n ≥ 3. For initial data in \(H^{n/(2(n+1))}(\mathbb R^n)\), the free Schrödinger evolution converges almost everywhere to the initial data as \(t\downarrow0\). The evolution is defined by removing Gaussian regularization on one full-measure spatial set, uniformly over the interval \(0\lt t\lt 1\).

We resolve the planar Sobolev endpoint of Carleson's convergence problem: for initial data in \(H^{1/3}(\mathbb R^2)\), the Schrödinger evolution converges almost everywhere to the initial data as \(t\downarrow0\). The evolution is defined by taking the Gaussian regularization limit first; on one full-measure spatial set, this limit exists for every \(0\lt t\lt 1\), and the resulting evolution is continuous at t = 0.

We prove that an n-Ahlfors–David regular Radon measure on ℝ^d is uniformly n-rectifiable if its n-dimensional Riesz transform is uniformly bounded from scalar \(L^2(\mu)\) to vector-valued \(L^2(\mu)\) over all positive hard truncations, for integers d ≥ 4 and \(2\le n\le d-2\). This gives a positive answer to the David–Semmes Riesz-transform question in its remaining higher-codimension range. The conclusion gives uniform big pieces of Lipschitz images of Euclidean balls.

We prove the annular r-variation estimate for the triangular Hilbert transform from complex \(L^3\times L^3\) to L^3/2 for every r > 2. The partitions may depend on the output point and range over all positive scales. The estimate yields the two-endpoint maximal bound and joint almost-everywhere and norm principal values, and resolves the symmetric scalar triangular Hilbert transform problem.

We prove a uniform \(L^3\times L^3\times L^3\) estimate for the dyadic triangular Hilbert form with unrestricted real inputs. The sum of the absolute local contributions over any finite set of scales is bounded by forty times the product of the input norms. In particular, the bound allows coefficients of modulus at most one to vary independently among admissible interval triples.

For arbitrary complex inputs in \(L^3(\mathbb R^2)\), we prove the pointwise maximal \(L^3\times L^3\to L^{3/2}\) estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and L^3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity. This also proves the conjectured scalar estimate at the symmetric point.

We prove a uniform L² 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 and holds for fields depending on both coordinates. This gives an affirmative answer to Stein's weak-\((2,2)\) conjecture.

We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed \(q\in(0,1)\) and every \(\eta\in(0,1)\), we construct a compact set \(E_{q,\eta}\subseteq[0,1]\) of measure greater than \(1-\eta\) containing no nontrivial affine copy of \(\{q^n:n\ge1\}\), for any translation and either sign of nonzero dilation. The set may depend on q; the result makes no simultaneous assertion for different ratios.

We construct a compact subset of the unit interval, of measure arbitrarily close to one, that contains no affine copy of the dyadic sequence \(\{2^{-n}:n\ge1\}\). The conclusion holds for every translation and every nonzero real dilation, of either sign, proving the dyadic case of the Erdős similarity conjecture.

We prove the endpoint gradient bound \(\|\nabla Mf\|_1\le C\|\nabla f\|_1\) for every real-valued \(f\in W^{1,1}(\mathbb R^2)\), where \(Mf(x)\) is the supremum of the averages of \(|f|\) over disks centered at x, and C is an absolute constant. The maximal function belongs to \(W^{1,1}_{\mathrm{loc}}(\mathbb R^2)\) and has a globally integrable weak gradient. This gives a positive resolution of the planar centered-disk case of the endpoint question of Hajłasz and Onninen.

We prove that the trilinear Hilbert transform with fixed slopes 1, 2, 3 maps \(L^3(\mathbb R)\times L^3(\mathbb R)\times L^3(\mathbb R)\) to \(L^1(\mathbb R)\), resolving this case of the trilinear Hilbert transform conjecture.