Title: Distribution of Sequences in Number Theory
Speaker: Sneha Chaubey, Department of Mathematics, IIIT-Delhi
Date: Oct 21, 2019 (Monday)
Abstract: The topic on the distribution of sequences saw its light with the seminal paper of Weyl. This being the motivation of this talk, we begin with a discussion on different types of arithmetic/integer sequences occurring around us. We look at various distribution questions concerning them, for example, questions on uniform distribution and spacings between members of a given sequence. We end the talk with some open problems in this area.

Title: Interval Graphs and the Shipping Problem.
Speaker: Kshitij Gajjar, Visiting Researcher, IIIT-Delhi
Date: Sep 30, 2019 (Monday)
Abstract: Suppose you are the operator of a seaport where ships arrive from all over the world. A cargo container needs to be transferred from port A to port B. Unfortunately, no ship goes from A to B. Instead, a ship from A arrives at your seaport, and another ship destined for B leaves your seaport. Then you can facilitate the transfer of the container from A to B. Now there are many more such demands from other ships to use your seaport as an intermediary. You are given the schedule of all these ships and you want to ensure that these transfers are carried out as efficiently as possible. In this talk, we will model this problem using interval graphs, describe an efficient transfer strategy, and show that this strategy is optimal. This is based on joint work with Jaikumar Radhakrishnan (TIFR).

Title: Turing degrees, intuitionistic logic, and the Muchnik topos.
Speaker: Sankha Basu, Assistant Professor, IIIT-Delhi
Date: Sep 09, 2019 (Monday)
Abstract: Kolmogorov, in 1923, proposed a model for intuitionistic propositional logic called the Calculus of Problems. Although simple and natural, this model was not rigorous. A rigorous version of this was given by Medvedev and Muchnik in the 1950’s and 60’s using the concept of Turing oracles. Thus started the study of mass problems and the reducibility notions between these. Sheaves over topological spaces as models for higher-order intuitionistic logic were studied independently. These models are also examples of elementary topoi. In this work, we have extended the Kolmogorov/ Medvedev/Muchnik line of work to a model of higher-order intuitionistic logic that we call the Muchnik topos. The Muchnik topos may be described in brief as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We have also introduced, within the Muchnik topos, a class of intuitionistic real numbers, different from the Dedekind and Cauchy reals. We call these the Muchnik reals. I will explain the basic notions on the way to the development of the Muchnik topos and the Muchnik reals and hence the talk should be accessible to anyone who is familiar with the concept of a topological space.