-
作者：Sharifi, Romyar
摘要：For primes p >= 5, we propose a conjecture that relates the values of cup products in the Galois cohomology of the maximal unramified outside p extension of a cyclotomic field on cyclotomic p-units to the values of p-adic L-functions of cuspidal eigenforms that satisfy mod p congruences with Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an isomorphism of certain spaces that allows us to compare the value of a reciprocity map on a particular norm compatible system o...
-
作者：Gross, Mark; Siebert, Bernd
摘要：We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect that our deformation parameter...
-
作者：Deift, Percy; Its, Alexander; Krasovsky, Icor
摘要：We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general nondegenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.
-
作者：Bernig, Andreas; Fu, Joseph H. G.
摘要：We give in explicit form the principal kinematic formula for the action of the affine unitary group on C-n, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce also several other canonical bases for the algebra of unitary-invariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone ...
-
作者：Avni, Nir
摘要：Let Gamma be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if Gamma has the congruence subgroup property, then the number of n-dimensional irreducible representations of Gamma grows like n(alpha), where alpha is a rational number.
-
作者：Guionnet, Alice; Krishnapur, Manjunath; Zeitouni, Ofer
摘要：We study the empirical measure LA of the eigenvalues of nonnormal square matrices of the form A(n) = UnTnVn, with U-n, V-n independent Haar distributed on the unitary group and T-n diagonal. We show that when the empirical measure of the eigenyalues of T-n converges, and T-n satisfies some technical conditions, L-An converges towards a rotationally invariant measure mu on the complex plane whose support is a single ring. In particular, we provide a complete proof of the Feinberg-Zee single rin...
-
作者：Hiary, Ghaith Ayesh
摘要：The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Schonhage's method, or Heath-Brown's method. The complexities of these methods have exponents 1/2, 3/8, and 1/3 respectively. In this article, three new fast and potentially practical methods to compute zeta are presented. One method is very simple. Its complexity has exponent 2/5. A second method relies on this author's algorithm to compute quadratic exponential su...
-
作者：Kalinin, Boris; Katok, Anatole; Rodriguez Hertz, Federico
摘要：We consider an ergodic invariant measure mu for a smooth action alpha of Z(k), k >= 2, on a (k + 1)-dimensional manifold or for a locally free smooth action of R-k, k >= 2, on a (2k +1)-dimensional manifold. We prove that if mu is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Z(k) has positive entropy, then mu is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general cl...
-
作者：Hedenmalm, Haakan; Montes-Rodriguez, Alfonso
摘要：A Heisenberg uniqueness pair (HUP) is a pair (Gamma, Lambda), where Gamma is a curve in the plane and Lambda is a set in the plane, with the following property: any finite Borel measure mu in the plane supported on Gamma, which is absolutely continuous with respect to arc length, and whose Fourier transform (mu) over cap vanishes on Lambda, must automatically be the zero measure. We prove that when is the hyperbola x(1) x(2) = 1 and Lambda is the lattice-cross Lambda = (alpha Z x {0}) boolean ...
-
作者：Buergisser, Peter; Cucker, Felipe
摘要：The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltran and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a...
