Let μ and ν be compactly supported probability measures on ℝ, and let μ⊞ν denote their additive free convolution. We show that for all sufficiently large real z , 0.1 ∫ _-∞^∞log (z - x) (μ⊞ν )( dx) = sup _Π{𝐄_Π[log (z - (X+Y))] - H(Π|μ⊗ν ) } , where the supremum is taken over all couplings Π of μ and ν . Analogous formulas hold for multiplicative free convolution μ⊠ν and free compression [μ ]_τ . In this way, integrals of a log-potential against free convolutions can be expressed as entropic optimal transport problems. The corresponding optimal couplings admit explicit formulas, from which the standard R - and S -transform descriptions of additive and multiplicative free convolution are recovered. In particular, the optimizer Π _z in (0.1) encodes the subordination equations via couplings of random variables. Our approach is based on a large deviation principle on the symmetric group, combined with the quadrature method of Marcus–Spielman–Srivastava.
We establish the scalar curvature and distance bounds, extending Perelman’s work on the Fano Kähler–Ricci flow to general finite time solutions of the Kähler–Ricci flow. These bounds are achieved by our Li–Yau type and Harnack estimates for weighted Ricci potential functions of the Kähler–Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov–Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is obtained for a fibre over every relatively compact open subset of the regular base of the fibration. We also apply our estimates to show that every solution of the Kähler–Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.
Take a closed monotone symplectic manifold containing a smooth anticanonical divisor. The quantum connection on its cohomology has singularities at zero and infinity (in the quantum parameter). At zero it has a regular singular point, by definition. We show that the singularity at infinity is of unramified exponential type. The argument involves: realizing cohomology as a deformation of the symplectic cohomology of the divisor complement; the corresponding deformation of the wrapped Fukaya category; a new categorical interpretation of the Fourier-Laplace transform of D -modules; and the regularity theorem of Petrov-Vaintrob-Vologodsky in noncommutative geometry.
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.
