Mathematics of Dark Matter

Speaker: 

Distinguished Professor Donald Saari

Institution: 

UC Irvine

Time: 

Friday, October 23, 2009 - 2:00pm

Location: 

RH 440R

Dark matter has been a controversial and mysterious topic since 1930s when Zwicky noticed a difference in the amount of mass obtained when computed in different manners. But much of the computations are based on what we knew about the Newtonian N-body problem 70 years ago. In this lecture, more recent results about the dynamics of the Newtonian N-body problem are described; it is shown how these results cast a new "light" on some of the dark matter assertions.

Shadowing in smooth flows and structural stability

Speaker: 

Sergey Tikhomirov

Institution: 

St. Petersburg State University, Russia

Time: 

Tuesday, May 19, 2009 - 3:00pm

Location: 

RH 440 R

The shadowing problem is related to the following question: under which condition, for any pseudotrajectory (approximate trajectory) of a vector field there exists a close trajectory? We study $C^1$-interiors of sets of vector fields with various shadowing properties. In the case of discrete dynamical systems generated by diffeomorphisms, such interiors were proved to coincide with the set of structurally stable diffeomorphisms for most general shadowing properties.

We prove that the $C^1$-interior of the set of vector fields with Oriented shadowing property contains not only structurally stable vector fields. Also, we have found additional assumptions under which the $C^1$-interiors of sets of vector fields with Lipschitz, Oriented and Orbit shadowing properties contain only structurally stable vector fields.

Some of these results were obtained together with my advisor S.Yu.Pilyugin.

Against the generation of infinitely many elementary pieces of dynamics in the partially hyperbolic setting

Speaker: 

Professor Lorenzo Diaz

Institution: 

PUC-Rio, Brazil

Time: 

Tuesday, April 21, 2009 - 3:00pm

Location: 

RH 440 R

Newhouse stated that unfoldings of homoclinic tangencies of a surface $C^2$-diffeomorphisms yield open sets where the diffeomorphisms with infinitely many sinks/sources are locally generic. There is a version of this result for parametrized families of diffeomorphisms. Palis conjectured that the set of parameters corresponding to diffeomorphisms with infinitely many sinks has measure zero. Gorodetski-Kaloshin gave a partial answer to this conjecture.

Motivated by these results, we study a formulation of this result in the partially hyperbolic setting, where sinks/sources are replaced by homoclinic classes and homoclinic tangencies by heterodimensional cycles. Our result is that it is not possible to generate infinitely many different homoclinic classes using a renormalization-like construction.

This is a join work with J. Rocha (Porto, Portugal).

Entropy and the Central Limit Theorem

Speaker: 

Professor Nicolai Haydn

Institution: 

USC

Time: 

Tuesday, February 24, 2009 - 3:00pm

Location: 

RH 440 R

For an ergodic system, the theorem of Shannon-McMillan-Breiman states that for every finite generating partition the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere. In 1962 Ibragimov showed that the distribution of the measure of cylinder sets is lognormally distributed provided the measure is strong mixing and its conditional entropy function is sufficiently well approximable.
Carleson (1958) and Chung (1960) generalised the theorem of SMB to infinite partitions (provided the entropy is finite). We show that the measures of cylinder sets are lognormally distributed for uniformly strong mixing systems and infinite partitions and show that the rate of convergence is polynomial. Apart from the mixing property we require that a higher than fourth moment of the information function is finite. Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function. We also obtain the law of the iterated logarithm and the weak invariance principle for the information function.

Pisot tilings of the line and the discrete spectrum conjecture

Speaker: 

Professor Robert Williams

Institution: 

UT Austin

Time: 

Tuesday, November 18, 2008 - 3:00pm

Location: 

RH 440 R

I will present some of the results by Marcy Barge and Jaroslaw Kwapisz (based on their paper "Geometric Theory of unimidular Pisot substitutions", Amer. J. Math., vol. 128 (2006), no. 5, pp. 1219--1282).

There are two classical ways of studying substitution tilings of the line: symbolic dynamics, and endomorphisms of ``train tracks". The authors give a strikingly new geometric approach and in particular show that if the tiling has a unimodular Pisot matrix of dimension d, then there is a factor onto the d-dimensional torus. In fact, they have a preprint removing the unimodular assumption. I propose to begin defining tilings and the tiling space X of a tiling T. X is a compact metric space that contains all tilings which have the same local patterns as T. In dimension 1 (the subject of this talk) X is similar to a solenoid.

I will not assume any familiarity with tiling theory.

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