Authors:Logan C. Hoehn; Lex G. Oversteegen Pages: 177 - 216 Abstract: We show that the only compact and connected subsets (i.e. continua) X of the plane \({\mathbb{R}^2}\) which contain more than one point and are homogeneous, in the sense that the group of homeomorphisms of X acts transitively on X, are, up to homeomorphism, the circle \({\mathbb{S}^1}\) , the pseudo-arc, and the circle of pseudo-arcs. These latter two spaces are fractal-like objects which do not contain any arcs. It follows that any compact and homogeneous space in the plane has the form X × Z, where X is either a point or one of the three homogeneous continua above, and Z is either a finite set or the Cantor set. The main technical result in this paper is a new characterization of the pseudo-arc. Following Lelek, we say that a continuum X has span zero provided for every continuum C and every pair of maps \({f,g\colon C \to X}\) such that \({f(C) \subset g(C)}\) there exists \({c_0 \in C}\) so that f(c 0) = g(c 0). We show that a continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable (i.e., every subcontinuum is indecomposable) and has span zero. PubDate: 2016-06-01 DOI: 10.1007/s11511-016-0138-0 Issue No:Vol. 216, No. 2 (2016)

Authors:David Hoffman; Martin Traizet; Brian White Pages: 217 - 323 Abstract: For every genus g, we prove that \({\mathbf{S}^2\times\mathbf{R}}\) contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the \({\mathbf{S}^2}\) tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in \({\mathbf{R}^3}\) that are helicoidal at infinity. We prove that helicoidal surfaces in \({\mathbf{R}^3}\) of every prescribed genus occur as such limits of examples in \({\mathbf{S}^2\times\mathbf{R}}\) . PubDate: 2016-06-01 DOI: 10.1007/s11511-016-0139-z Issue No:Vol. 216, No. 2 (2016)

Authors:Yong Huang; Erwin Lutwak; Deane Yang; Gaoyong Zhang Pages: 325 - 388 Abstract: A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems. PubDate: 2016-06-01 DOI: 10.1007/s11511-016-0140-6 Issue No:Vol. 216, No. 2 (2016)

Authors:Nicolas Bergeron; John Millson; Colette Moeglin Pages: 1 - 125 Abstract: Let S be a closed Shimura variety uniformized by the complex n-ball associated with a standard unitary group. The Hodge conjecture predicts that every Hodge class in \({H^{2k} (S, \mathbb{Q})}\) , \({k=0,\dots, n}\) , is algebraic. We show that this holds for all degrees k away from the neighborhood \({\bigl]\tfrac13n,\tfrac23n\bigr[}\) of the middle degree. We also prove the Tate conjecture for the same degrees as the Hodge conjecture and the generalized form of the Hodge conjecture in degrees away from an interval (depending on the codimension c of the subvariety) centered at the middle dimension of S. These results are derived from a general theorem that applies to all Shimura varieties associated with standard unitary groups of any signature. The proofs make use of Arthur’s endoscopic classification of automorphic representations of classical groups. As such our results rely on the stabilization of the trace formula for the (disconnected) groups \({GL (N) \rtimes \langle \theta \rangle}\) associated with base change. PubDate: 2016-03-01 DOI: 10.1007/s11511-016-0136-2 Issue No:Vol. 216, No. 1 (2016)

Authors:Gang Tian; Zhenlei Zhang Pages: 127 - 176 Abstract: In this paper, we will establish a regularity theory for the Kähler–Ricci flow on Fano n-manifolds with Ricci curvature bounded in L p -norm for some \({p > n}\) . Using this regularity theory, we will also solve a long-standing conjecture for dimension 3. As an application, we give a new proof of the Yau–Tian–Donaldson conjecture for Fano 3-manifolds. The results have been announced in [45]. PubDate: 2016-03-01 DOI: 10.1007/s11511-016-0137-1 Issue No:Vol. 216, No. 1 (2016)

Abstract: We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction. PubDate: 2016-12-01 DOI: 10.1007/s11511-017-0146-8

Abstract: We determine the asymptotics of the independence number of the random d-regular graph for all \({d\geq d_0}\) . It is highly concentrated, with constant-order fluctuations around \({n\alpha_*-c_*\log n}\) for explicit constants \({\alpha_*(d)}\) and \({c_*(d)}\) . Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs. PubDate: 2016-12-01 DOI: 10.1007/s11511-017-0145-9

Abstract: We construct families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed, in particular establishing the existence of waves of large amplitude. A Riemann–Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudodifferential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler–Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation. PubDate: 2016-12-01 DOI: 10.1007/s11511-017-0144-x

Abstract: We give lower bounds for the numbers of real solutions in problems appearing in Schubert calculus in the Grassmannian \({\mathop{\rm Gr}(n,d)}\) related to osculating flags. It is known that such solutions are related to Bethe vectors in the Gaudin model associated to \({\mathop{\rm gl}_n}\) . The Gaudin Hamiltonians are self-adjoint with respect to a non-degenerate indefinite Hermitian form. Our bound comes from the computation of the signature of that form. PubDate: 2016-09-01 DOI: 10.1007/s11511-016-0143-3

Abstract: We construct approximate transport maps for perturbative several-matrix models. As a consequence, we deduce that local statistics have the same asymptotic as in the case of independent GUE or GOE matrices (i.e., they are given by the sine-kernel in the bulk and the Tracy–Widom distribution at the edge), and we show averaged energy universality (i.e., universality for averages of m-points correlation functions around some energy level E in the bulk). As a corollary, these results yield universality for self-adjoint polynomials in several independent GUE or GOE matrices which are close to the identity. PubDate: 2016-09-01 DOI: 10.1007/s11511-016-0142-4

Abstract: We prove a form of Arnold diffusion in the a-priori stable case. Let $$H_{0}(p)+\epsilon H_{1}(\theta,p,t),\quad \theta \in {\mathbb{T}^{n}},\,p \in B^{n},\,t \in \mathbb{T}= \mathbb{R}/\mathbb{T},$$ be a nearly integrable system of arbitrary degrees of freedom \({n \geqslant 2}\) with a strictly convex H 0. We show that for a “generic” \({\epsilon H_1}\) , there exists an orbit \({(\theta,p)}\) satisfying $$\ p(t)-p(0)\ > l(H_{1}) > 0,$$ where \({l(H_1)}\) is independent of \({\epsilon}\) . The diffusion orbit travels along a codimension-1 resonance, and the only obstruction to our construction is a finite set of additional resonances. For the proof we use a combination of geometric and variational methods, and manage to adapt tools which have recently been developed in the a-priori unstable case. PubDate: 2016-09-01 DOI: 10.1007/s11511-016-0141-5