This paper studies the relationship between an analytic compactification of the moduli space of flat SL_2(ℂ) connections on a closed, oriented 3-manifold M defined by Taubes, and the Morgan–Shalen compactification of the SL_2(ℂ) character variety of the fundamental group of M . We exhibit an explicit correspondence between ℤ/2 harmonic 1-forms, measured foliations, and equivariant harmonic maps to ℝ-trees, as initially proposed by Taubes. As an application, we prove that ℤ/2 harmonic 1-forms exist on all reducible or Haken manifolds with respect to all Riemannian metrics. We also prove the existence of manifolds which support singular ℤ/2 harmonic 1-forms but which have compact SL_2(ℂ) character varieties, and this resolves a folklore conjecture.
We show that there exist infinitely many pairwise non-isotopic splitting 3-spheres for two unlinked, unknotted 2-spheres in S^4 . This answers a question posed by Hughes, Kim, and Miller.
The linearized operator ℒ associated with the three-dimensional cubic nonlinear Schrödinger equation is shown to possess no embedded eigenvalues in the essential spectrum. This result rigorously confirms the key spectral hypothesis underpinning the construction of center-stable manifolds by Schlag (Ann. Math. (2) 169(1):139–227, 2009), and thereby provides the final analytical link for an unconditional stability theory near the ground state soliton. Unlike the one-dimensional case, the 3D model lacks integrability; the non-self-adjointness of ℒ and the non-explicit profile of the ground state render classical techniques insufficient to exclude eigenvalues in the continuum. The proof rests on the introduction of a weight-modulated positivity trap to exploit the hyperbolic instability of the fundamental mode, combined with a novel constrained shooting method and delicate comparison arguments for higher angular momenta. Beyond the cubic model, the machinery introduced here provides a robust analytical framework for addressing the spectral coercivity conjectures central to the Merle–Raphaël’s log-log blow-up on mass-critical NLS.
We develop a new theory of maximizing sets in dynamical systems, for the study of ergodic optimization in systems with weak hyperbolicity but where the Mañé cohomology lemma does not hold. This leads to new solutions of the Typical Periodic Optimization problem in the Lipschitz category: existence of an open dense set of Lipschitz functions such that each member has a unique maximizing measure and this measure is periodic (an equi-distribution on a single periodic orbit). The theory yields a structural theorem, that isolates the part of the system responsible for any robust non-periodic optimization. The structural theorem is developed further in the setting of symbolic dynamics: given any shift space, for typical Lipschitz functions the maximizing measure is shown to be either periodic or supported on the Markov boundary of the shift space. It follows that Contreras’ Typical Periodic Optimization theorem (Contreras in Invent. Math. 205:383–412, 2016) for shifts of finite type can be extended to a wide class of shift spaces, including every sofic shift. The structural theorem is used to provide the first known example of a shift space where Typical Periodic Optimization fails despite periodic measures being dense in the set of all invariant measures.
We show that there exist Brauer classes over a field F which are not split by any genus one curve over F. This answers a question of Clark and Saltman.
The complement of the union of a collection of disjoint open disks in the 2-sphere is called a Schottky set. We prove that a subset S of the 2-sphere is quasiconformally equivalent to a Schottky set if and only if every pair of complementary components of S can be mapped to a pair of open disks with a uniformly quasiconformal homeomorphism of the sphere. Our theorem applies to Sierpiński carpets and gaskets, yielding for the first time a general quasiconformal uniformization result for gaskets. Moreover, it contains Bonk's uniformization result for carpets as a special case and does not rely on the condition of uniform relative separation that is used in relevant works.
We prove a new lower bound on the Ramsey number r(ℓ, Cℓ) for any constant C > 1 and sufficiently large ℓ, showing that there exists ε=ε(C)> 0 such that r(ℓ, Cℓ) ≥(p_C^-1/2 + ε)^ℓ, where p_C ∈ (0, 1/2) is the unique solution to C = log p_C/log(1 - p_C). This provides the first exponential improvement over the classical lower bound obtained by Erdős in 1947.
We prove a Witt vector version of the usual Grauert–Riemenschneider vanishing theorem over perfect fields of positive characteristic, solving a question raised by Blickle, Esnault, Chatzistamatiou and Rülling. We then deduce some rationality consequences for F –rational singularities.
We prove that in any dimension n there exists an origin-symmetric ellipsoid ℰ⊂ℝ^n of volume c n^2 that contains no points of ℤ^n other than the origin, where c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in ℝ^n whose density is at least cn^2 · 2^-n. Previously known constructions of sphere packings in ℝ^n yielded densities of at most C n log n · 2^-n. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least c n^2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.
We prove global well-posedness for the cubic nonlinear Schrödinger equation for periodic initial data in the mass-critical dimension d=2 for initial data of arbitrary size in the defocusing case and data below the ground state threshold in the focusing case. The result is based on a new inverse Strichartz inequality, which is proved by using incidence geometry and additive combinatorics, in particular, the inverse theorems for Gowers uniformity norms by Green-Tao-Ziegler. This allows to transfer the analogous results of Dodson for the non-periodic mass-critical NLS to the periodic setting. In addition, we construct an approximate periodic solution that implies the sharpness of the results.
We prove the K(n)-local analogue of the Hahn-Wilson conjecture on fp-spectra, which states that the truncated Brown-Peterson spectra generate the category of fp-spectra as a thick subcategory. As a corollary, we deduce the original conjecture at height 1. Along the way, we prove the existence of K(n)-local finite complexes with particularly regular rings of homotopy groups.
We extend Falconer's 1988 landmark result on the dimensions of self-affine fractals to encompass the dimensions of their projections, showing furthermore that their families of exceptional projections contain algebraic varieties which are preserved by the underlying linear algebraic group. The techniques which we develop allow us to construct examples of additional new phenomena: firstly, we give general examples of equilibrium measures on self-affine fractals which admit non-exact-dimensional projections. Secondly, we construct strongly irreducible self-affine sets which have small sumsets without any arithmetic resonance in their construction.
In 1969, Fulton introduced classical Hurwitz spaces parametrizing simple d-sheeted coverings of the projective line in the algebro-geometric setting. He established the irreducibility of these spaces under the assumption that the characteristic of the ground field is greater than d, but the irreducibility problem in smaller characteristics remained open. We resolve this problem in the current paper and prove that the classical Hurwitz spaces are irreducible over any algebraically closed field. On the way, we establish the irreducibility of Severi varieties in arbitrary characteristic for a rich class of toric surfaces, including all classical toric surfaces. Our approach to the irreducibility problems comes from tropical geometry, and the paper contains two more results of independent interest - a lifting result for parametrized tropical curves and a strong connectedness property of the moduli spaces of parametrized tropical curves.
We resolve the long-standing problem of elucidating the cutoff phenomenon for a vast class of Markov processes, namely Markov diffusions with non-negative Bakry-Émery curvature. More precisely, we prove that any sequence of non-negatively curved diffusions exhibits cutoff in total variation as soon as the product condition is satisfied. Our result holds in Euclidean spaces as well as on Riemannian manifolds, and for arbitrary non-random initial conditions. Reversibility is not required. This simplifies, unifies and generalizes a number of works that have established cutoff through a delicate and model-dependent analysis of mixing times. The proof is elementary: we exploit a new differential relation between varentropy and entropy to produce a universal bound on the width of the mixing window.
A decorated surface 𝕊 is an oriented topological surface with marked points on the boundary considered modulo the isotopy. We consider the moduli space ℳ_𝕊 of hyperbolic structures on 𝕊 with geodesic boundary, such that the hyperbolic structure near each marked point is a cusp, equipped with a horocycle. The space ℳ_𝕊 carries a volume form Ω . Let us fix the set K of the distances between the horocycles at the adjacent cusps, and the set L of the geodesic lengths of boundary circles without cusps. We get a subspace ℳ_𝕊(K, L) with the induced volume form Ω _𝕊(K, L) . However, if the cusps are present, the volume of the space ℳ_𝕊(K, L) , or its variant without horocycles, is infinite. We introduce the exponential volume form e^-WΩ , where W is the potential - a positive function on ℳ_𝕊 , given by the sum over the cusps of the hyperbolic areas under the horocycles. We show that the following exponential volume is finite: 1 ∫ _ℳ_𝕊(K, L)e^-WΩ _𝕊(K, L). We suggest that moduli spaces ℳ_𝕊(K, L) with the exponential volume forms are the true analogs of moduli spaces ℳ_g,n with the Weil–Petersson volume form, e.g. relevant to the open string theory. We prove unfolding formulas, expressing integrals ∫ _ℳ_𝕊(K, L)f e^-WΩ _𝕊(K, L) , where f is an integrable function, as finite sums of similar integrals for the three elementary decorated surfaces. They generalise Mirzakhani’s recursions for the volumes of moduli spaces of hyperbolic surfaces. We show that exponential volumes for elementary decorated surfaces give rise to a commutative algebra ℰ, which we call the positive Hecke-Whittaker algebra for PGL_2(ℝ) . Exponential volumes for all decorated surfaces and unfolding formulas extend the algebra ℰ to all decorated surfaces.
We develop the theory of geometric Eisenstein series and constant term functors for ℓ -adic sheaves on stacks of bundles on the Fargues-Fontaine curve. In particular, we prove essentially optimal finiteness theorems for these functors, analogous to the usual finiteness properties of parabolic inductions and Jacquet modules. We also prove a geometric form of Bernstein’s second adjointness theorem, generalizing the classical result and its recent extension to more general coefficient rings proved in [6]. As applications, we decompose the category of sheaves on Bun_G into cuspidal and Eisenstein parts, and show that the gluing functors between strata of Bun_G are continuous in a very strong sense.
