-
作者：Colding, Tobias H.; Minicozzi, William P., II
摘要：It has long been conjectured that starting at a generic smooth closed embedded surface in R-3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key to showing this conjecture is to show that shrinkin...
-
作者：Sun, Binyong; Zhu, Chen-Bo
摘要：Let G be one of the classical Lie groups GL(n+1)(R), GL(n+1)(C), U(p, q + 1), O(p, q + 1), On+1(C), SO(p, q + 1), SOn+1(C), and let G' be respectively the subgroup GL(n)(R), GL(n)(C), U(p, q), O(p, q), O-n(C), SO(p, q), SO (C), embedded in G in the standard way. We show that every irreducible Casselman-Wallach representation of G' occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of G. Similar results are proved for the Jacobi groups GL(n)(R) (sic) H2n+...
-
作者：Borgs, C.; Chayes, J. T.; Lovasz, L.; Sos, V. T.; Vesztergombi, K.
摘要：We consider sequences of graphs (G(n)) and define various notions of convergence related to these sequences including left-convergence, defined in terms of the densities of homomorphisms from small graphs into G(n), and right-convergence, defined in terms of the densities of homomorphisms from G(n) into small graphs. We show that right-convergence is equivalent to left-convergence, both for simple graphs G(n), and for graphs G(n), with nontrivial nodeweights and edgeweights. Other equivalent c...
-
作者：Ding, Jian; Lee, James R.; Peres, Yuval
摘要：We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph G is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on G, scaled by the number of edges in G. This allows us to resolve a number of open questions. We give a deterministic polynomial-time algorithm that computes the cover time to within an O(1) factor for ...
-
作者：Duminil-Copin, Hugo; Smirnov, Stanislav
摘要：We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to root 2 + root 2 This value has been derived nonrigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self-avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence...
-
作者：Epstein, Charles L.
摘要：We prove the relative index conjecture, which in turn implies that the set of embeddable deformations of a strictly pseudoconvex CR-structure on a compact 3-manifold is closed in the C-infinity-topology.
-
作者：Frank, Rupert L.; Lieb, Elliott H.
摘要：We derive the sharp constants for the inequalities on the Heisenberg group H-n whose analogues on Euclidean space R-n are the well known Hardy-Littlewood-Sobolev inequalities. Only one special case had been known previously, due to Jerison-Lee more than twenty years ago. From these inequalities we obtain the sharp constants for their duals, which are the Sobolev inequalities for the Laplacian and conformally invariant fractional Laplacians. By considering limiting cases of these inequalities s...
-
作者：Zagier, D.
摘要：A formula is given for the special multiple zeta values occurring in the title as rational linear combinations of products zeta(m)pi(2n) with m odd. The existence of such a formula had been proved using motivic arguments by Francis Brown, but the explicit formula (more precisely, certain 2-adic properties of its coefficients) were needed for his proof in [2] of the conjecture that all periods of mixed Tate motives over Z are Q[(2 pi i)(+/- 1)]-linear combinations of multiple zeta values. The f...
-
作者：Agler, Jim; McCarthy, John E.; Young, N. J.
摘要：We prove generalizations of Lowner's results on matrix monotone functions to several variables. We give a characterization of when a function of d variables is locally monotone on d-tuples of commuting self-adjoint n-by-n matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.
-
作者：Howard, Benjamin
摘要：We study the intersections of special cycles on a unitary Shimura variety of signature (n - 1, 1) and show that the intersection multiplicities of these cycles agree with Fourier coefficients of Eisenstein series. The results are new cases of conjectures of Kudla and suggest a Gross-Zagier theorem for unitary Shimura varieties.
