102,654 research outputs found
Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond
We present recent results on counting and distribution of circles in a given
circle packing invariant under a geometrically finite Kleinian group and
discuss how the dynamics of flows on geometrically finite hyperbolic
manifolds are related. Our results apply to Apollonian circle packings,
Sierpinski curves, Schottky dances, etc.Comment: To appear in the Proceedings of ICM, 201
A Digital Signature Scheme for Long-Term Security
In this paper we propose a signature scheme based on two intractable
problems, namely the integer factorization problem and the discrete logarithm
problem for elliptic curves. It is suitable for applications requiring
long-term security and provides a more efficient solution than the existing
ones
Electrical networks and Stephenson's conjecture
In this paper, we consider a planar annulus, i.e., a bounded, two-connected,
Jordan domain, endowed with a sequence of triangulations exhausting it. We then
construct a corresponding sequence of maps which converge uniformly on compact
subsets of the domain, to a conformal homeomorphism onto the interior of a
Euclidean annulus bounded by two concentric circles. As an application, we will
affirm a conjecture raised by Ken Stephenson in the 90's which predicts that
the Riemann mapping can be approximated by a sequence of electrical networks.Comment: Comments are welcome
Optimal Inverse Littlewood-Offord theorems
Let eta_i be iid Bernoulli random variables, taking values -1,1 with
probability 1/2. Given a multiset V of n integers v_1,..., v_n, we define the
concentration probability as rho(V) := sup_{x} Pr(v_1 eta_1+...+ v_n eta_n=x).
A classical result of Littlewood-Offord and Erdos from the 1940s asserts that
if the v_i are non-zero, then rho(V) is O(n^{-1/2}). Since then, many
researchers obtained improved bounds by assuming various extra restrictions on
V. About 5 years ago, motivated by problems concerning random matrices, Tao and
Vu introduced the Inverse Littlewood-Offord problem. In the inverse problem,
one would like to give a characterization of the set V, given that rho(V) is
relatively large. In this paper, we introduce a new method to attack the
inverse problem. As an application, we strengthen a previous result of Tao and
Vu, obtaining an optimal characterization for V. This immediately implies
several classical theorems, such as those of Sarkozy-Szemeredi and Halasz. The
method also applies in the continuous setting and leads to a simple proof for
the beta-net theorem of Tao and Vu, which plays a key role in their recent
studies of random matrices. All results extend to the general case when V is a
subset of an abelian torsion-free group and eta_i are independent variables
satisfying some weak conditions
Apollonian circle packings: Dynamics and Number theory
We give an overview of various counting problems for Apollonian circle
packings, which turn out to be related to problems in dynamics and number
theory for thin groups. This survey article is an expanded version of my
lecture notes prepared for the 13th Takagi lectures given at RIMS, Kyoto in the
fall of 2013.Comment: To appear in Japanese Journal of Mat
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