Approximation by Algebraic Numbers

Speaker: 

Ryan Broderick

Institution: 

UC Irvine

Time: 

Tuesday, May 10, 2016 - 1:00pm to 2:00pm

Location: 

RH 440R

Dirichlet’s approximation theorem states that for every real number x there exist infinitely many rationals p/q with |x-p/q| < 1/q^2. If x is in the unit interval, then viewing rationals as algebraic numbers of degree 1, q is also the height of its primitive integer polynomial, where height means the maximum of the absolute values of the coefficients. This suggests a more general question: How well can real numbers be approximated by algebraic numbers of degree at most n, relative to their heights? We will discuss Wirsing’s conjecture which proposes an answer to this question and Schmidt and Davenport’s proof of the n = 2 case, as well as some open questions.

Sums of two homogeneous Cantor sets II

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, May 3, 2016 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets).

Sums of two homogeneous Cantor sets I

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, April 26, 2016 - 1:00pm to 2:00pm

We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets). We will also discuss the connection of this problem with the question on properties of one dimensional self-similar sets with overlaps.

Random metrics on hierarchical graph models

Speaker: 

Victor Kleptsyn

Institution: 

CNRS, Institute of Mathematical Research of Rennes

Time: 

Tuesday, April 19, 2016 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

In the domain of quantum gravity, people often consider random metrics on surfaces, defined as Riemannian ones with the factor being an exponent of the Gaussian Free Field. Though, GFF is only a distribution, not a function, and its exponent is not well-defined. This leaves open the problem of giving a mathematical sense to this definition (or, to be more precise, of showing rigorously that one of the known regularization procedures indeed converges).

In a joint work with M. Khristoforov and M. Triestino, we approach a « baby version » of this problem, constructing random metrics on hierarchical graphs (like Benjamini's eight graph, dihedral hierarchical lattice, etc.). This situation is still accessible due to the graph structure, but already shares with the planar case the complexity of high non-uniqueness of candidates for the geodesic lines. The behavior of some (pivotal, bridge-type) graphs seems to be a good model for the behavior of the full planar case.

Pages

Subscribe to RSS - Dynamical Systems