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Special Lectures 2023

Table of Contents
Fall 2023

SL2303: A generalisation of the coupon collector problem #

  • Speaker: Prof. Soumendu Sundar Mukherjee (ISI Kolkata)
  • Abstract: We consider a generalisation of the classical coupon collector problem. We define a super-coupon to be any \(s-\)subset of a universe of \(n\) coupons. In each round, a random \(r-\)subset from the universe is drawn and all its \(s-\)subsets are marked as collected. We show that the time to collect all super-coupons is \(\binom{r}{s}^{-1}\binom{n}{s} \log\binom{n}{s}(1 + o(1))\) on average and has a Gumbel limit after a suitable normalisation. In a similar vein, we show that for any \(\alpha \in (0, 1)\), the expected time to collect \((1 - \alpha)\) proportion of all super-coupons is \(\binom{r}{s}^{-1}\binom{n}{s} \log \big(\frac{1}{\alpha}\big)(1 + o(1))\). The \(r = s\) case of this model is equivalent to the classical coupon collector model.
    We also consider a temporally dependent model where the \(r-\)subsets are drawn according to the following Markovian dynamics: the \(r-\)subset at round \(k + 1\) is formed by replacing a random coupon from the \(r-\)subset drawn at round \(k\) with another random coupon from outside this \(r-\)subset. We link the time it takes to collect all super-coupons in the \(r = s\) case of this model to the cover time of random walk on a certain finite regular graph and conjecture that in general, it takes \(\frac{r}{s} \binom{r}{s}^{-1}\binom{n}{s}\log\binom{n}{s}(1 + o(1))\) time on average to collect all super-coupons.
  • References: On a generalisation of the coupon collector problem – Siva Athreya, Satyaki Mukherjee & Soumendu Sundar Mukherjee
  • Video: Available here.
  • Date and Time: Tuesday, 7th November 2023, 8:30 PM
  • Venue: Online (Google Meet)

SL2302: The Lovász Local Lemma #

  • Speaker: Prof. Jaikumar Radhakrishnan (ICTS-TIFR, Bengaluru & TIFR, Mumbai)
  • Abstract: The local lemma of Lovász provides sufficient conditions to show that the probability of a union of events is strictly less than 1. It forms the basis for several probabilistic existence arguments in discrete mathematics. We will present the lemma and its proofs, and also some of its applications.
  • Notes: Available here.
  • References:
  • Video: Available here.
  • Date and Time: Saturday, 28th October 2023, 11:00 AM - 1:00 PM
  • Venue: 2nd Floor Auditorium, Academic Building

SL2301: Strong unique continuation for second order elliptic operators #

  • Speaker: Prof. Agnid Banerjee (TIFR-CAM, Bengaluru)
  • Abstract: The lecture will focus on a very classical subject: when do the zeros of a solution to a PDE spread?
    • It will start with a brief historic overview, and
    • Then illustrate the two technologies presently available to mankind for addressing the strong unique continuation for \(-\Delta + V\)
  • Video: [TBU]
  • Date and Time: Thursday, 19th October 2023, 5:00 PM - 7:00 PM
  • Venue: 2nd Floor Auditorium, Academic Building