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.
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.
