Reinforcement model on graphs and their limit behaviour

Speaker: 

Victor Kleptsyn

Institution: 

CNRS, Institut de Recherche Mathematique de Rennes

Time: 

Tuesday, October 30, 2018 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

    The classical Polya urn process is a reinforcement process, in which there are balls of different color in the urn, we take out a ball at random, and the color that was just out of it gets an advantage for all future turns: we return this ball to the urn and add another one of the same color.   

    However, in this process on every step all the colors are competing. What will happen if on different steps there will be different subsets of competing colors? For instance, if there are companies that compete on different markets, or if a signal is choosing its way to travel?   

    Some questions here have nice and simple answers; my talk will be devoted to the results of our joint project with Mark Holmes and Christian Hirsch on the topic.

Three Fairy Math Stories

Speaker: 

D.Burago

Institution: 

Penn State University

Time: 

Thursday, October 25, 2018 - 2:00pm to 3:00pm

Location: 

RH 306

 Three different math stories in one lecture. Only definitions, motivations, results, some ideas behind proofs, open questions. 

1. One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori remains a great mystery. The main quantitate invariants so far are entropies.  It is easy, by modern standards, to show that topological entropy can be positive. It lives, however, on a zero measure set. We are now able to show that metric entropy can become positive too, under arbitrarily small C^{infty} perturbations, answering an old-standing problem of Kolmogorov. Furthermore, a slightly modified construction resolves another long–standing problem of the existence of entropy non-expansive systems. In these modified examples  positive metric entropy is generated in arbitrarily small tubular neighborhoods of one trajectory. Joint with S. Ivanov and Dong Chen.

2. A survival guide for a feeble fish and homogenization of the G-Equation. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water in the ocean? This is related to the G-Equation and has applications to its homogenization. The G-equation is believed to govern many combustion processes, say wood fires or combustion in combustion engines (generally, in pre-mixed media with “turbulence".  Based on a joint work with S. Ivanov and A. Novikov.

3. Just 20 years ago the topic of my talk at the ICM was a solution of a problem which goes back to Boltzmann and has been formulated mathematically by Ya. Sinai. The conjecture of Boltzmann-Sinai states that the number of collisions in a system of $n$ identical balls colliding elastically in empty space is uniformly bounded for all initial positions and velocities of the balls. The answer is affirmative and the proven upper bound is exponential in $n$. The question is how many collisions can actually occur. On the line, one sees that  there can be $n(n-1)/2$ collisions, and this is the maximum. Since the line embeds in any Euclidean space, the same example works in all dimensions. The only non-trivial (and counter-intuitive) example I am aware of is an observation by Thurston and Sandri who gave an example of 4 collisions between 3 balls in $R^2$. Recently, Sergei Ivanov and me proved that there are examples with exponentially many collisions between  $n$ identical balls in $R^3$, even though the exponents in the lower and upper bounds do not perfectly match. Many open questions left.

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.

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