Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: A brief introduction to ergodic theory
Speaker: Sharvari Tikekar, TIFR, Mumbai
Date: February 5, 2024
Time: 16:00:00 Hours
Venue: A-369
Abstract: In 1871, while studying the motion of gas particles in a space, Boltzmann hypothesized that, over a long period of time, the amount of time spent by the system in a region of the phase space is roughly proportional to the volume of that region. This eventually came to be known as the Boltzmann Ergodic Hypothesis. The branch of Ergodic Theory, which is the probabilistic study of a dynamical system, was developed exactly in order to mathematically formulate and prove the above hypothesis. In this talk we will study the foundations of ergodic theory. We consider a probability space $(X, \mu)$ and a measure preserving transformation $T$ on $X$. We will define the ergodicity of the measure $\mu$, study its properties, and look at some standard examples of ergodic and non-ergodic systems. We will also discuss the celebrated Poincaré Recurrence Theorem (1890) and Birkhoff's Ergodic Theorem (1931), and if time permits, we will sketch some proofs as well.
Title of Talk: A brief introduction to ergodic theory
Speaker: Sharvari Tikekar, TIFR, Mumbai
Date: February 5, 2024
Time: 16:00:00 Hours
Venue: A-369
Abstract: In 1871, while studying the motion of gas particles in a space, Boltzmann hypothesized that, over a long period of time, the amount of time spent by the system in a region of the phase space is roughly proportional to the volume of that region. This eventually came to be known as the Boltzmann Ergodic Hypothesis. The branch of Ergodic Theory, which is the probabilistic study of a dynamical system, was developed exactly in order to mathematically formulate and prove the above hypothesis. In this talk we will study the foundations of ergodic theory. We consider a probability space $(X, \mu)$ and a measure preserving transformation $T$ on $X$. We will define the ergodicity of the measure $\mu$, study its properties, and look at some standard examples of ergodic and non-ergodic systems. We will also discuss the celebrated Poincaré Recurrence Theorem (1890) and Birkhoff's Ergodic Theorem (1931), and if time permits, we will sketch some proofs as well.