Averaging one-point hyperbolic-type metrics

Speaker: 

Wes Whiting

Institution: 

CSUF

Time: 

Tuesday, April 24, 2018 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

Hyperbolic-type metrics extend the idea of negative curvature to metric spaces, and several well-behaved hyperbolic-type metrics are known on 1-punctured Euclidean space. However, they lose their hyperbolicity on spaces with non-singleton boundary. In this talk, I will discuss the obstructions to hyperbolicity on more general boundaries, and give a recent result which allows hyperbolicity over n-punctured Euclidean space.

Josephson junction, Arnold tongues, and their adjacency points

Speaker: 

Victor Kleptsyn

Institution: 

CNRS, Rennes University

Time: 

Tuesday, March 13, 2018 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

The study of the equation on the 2-torus given by  
x’= sin x + a + b sin t
has been motivated by its relation to the Josephson junction in physics, as well as by purely mathematical reasons. For any values of the parameters a and b, one can consider the time-2\pi (period) map from the x-circle to itself, and study its properties, in particular, its rotation number.

Study of the Arnold tongues corresponding to this family, reveals a miracle: sometimes, their left and right boundaries intersect at a hourglass-type so-called adjacency point. Moreover, the a-coordinates of all these points turn out to be integers. My talk will be devoted to the geometry behind all of this, summarizing the works of many authors: Ilyashenko, Guckenheimer, Buchstaber, Karpov, Tertychnyi, Glutsyuk, Klimenko, Schurov, Filimonov, Romaskevich, Ryzhov, and myself.
 

Groups acting on the circle

Speaker: 

Victor Kleptsyn

Institution: 

CNRS, Rennes University

Time: 

Tuesday, March 6, 2018 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

The talk will be devoted to the study of finitely generated groups acting on the circle. We will start with joint results with A. Navas and B. Deroin: if such an action by analytic diffeomorphisms admits a Cantor minimal set, then this set is of Lebesgue measure zero, and if such an action by free group of analytic diffeomorphisms is minimal, then it is also Lebesgue-ergodic.

If the time permits, we will discuss the new results and state of art of an ongoing joint project with B. Deroin, A. Navas, D. Filimonov, M. Triestino, D. Malicet, S. Alvarez, P. G. Barrientos and C. Meniño, devoted to the further study of such actions, and the different kingdoms of locally discrete and locally non-discrete actions.

Some combinatorial properties of simple Toeplitz subshifts

Speaker: 

Daniel Sell

Institution: 

Friedrich-Schiller-Universität Jena

Time: 

Tuesday, November 7, 2017 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

Toeplitz sequences are constructed from periodic sequences with undetermined positions by successively filling these positions with the letters of other periodic sequences. In this talk, the class of so called simple Toeplitz sequences will be considered. We will describe combinatorial properties, such as the word complexity, of the subshifts that are associated with them. The relation between combinatorial properties of the coding sequences and the Boshernitzan condition will be discussed as well.

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