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Virtual Centre for Random Geometry
  • , Speaker: Aratrika Pandey, IIT Bombay
    4:00 pm IST

    An introduction to Hausdorff measure and Hausdorff dimension

    This talk will introduce the concepts of Hausdorff measure and Hausdorff dimension. Exploring the motivations behind studying this notion of dimension, we will focus on calculating the dimension of some specific sets. Then we will discuss some techniques used for computing the Hausdorff dimension. Finally, if time permits, we will discuss an application of these techniques through an example.

  • , Speaker: Sharvari Tikekar, TIFR
    4:00 PM IST

    A brief introduction to ergodic theory

    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, μ) and a measure preserving transformation T on X. We will define the ergodicity of the measure μ, 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.

  • , Speaker: Biswajit Nag, TIFR
    4:00 PM IST

    A Decomposition Theorem for Groups Acting on Trees

    This talk will introduce Bass-Serre theory, which, given an action of a group on a tree, tells us how to write the group as a sequence of amalgamated free products and HNN extensions of its subgroups. We will go through the relevant notions in the theory, some examples and applications, and if time permits, we will also sketch a proof of the main theorem.

  • , Speaker: Balarka Sen, TIFR
    4:00 PM IST

    An introduction to contact structures on 3-manifolds

    A contact structure on a 3-manifold is a nowhere integrable 2-plane field. These are close cousins of other widely studied codimension 1 structures on 3-manifolds; for instance, foliations. We discuss some key examples, fundamental properties, and why one should care about contact structures. We prove that, just as codimension 1 foliations, every closed oriented 3-manifold admits a contact structure. Time permitting, we will end by stating an outstanding open problem in the field.