Title: Parametric Shortest Paths in Planar Graphs
Speaker: Kshitij Gajjar, Research Scholar, Tata Institute of Fundamental Research (TIFR), Mumbai
Date: July 10, 2019 (Wednesday)
Abstract: Suppose you have a road network with n traffic signals such that the amount of traffic on each road varies as a linear function of time. Then the shortest path from a source s to a destination t might be different at different points of time. It was well known (Gusfield, 1980) that the number of different shortest s-t paths is at most n^(O(log n)). In this talk, we will show that there exists a planar road network for which the number of different shortest s-t paths is at least n^(Ω(log⁡〖n)〗 ), refuting a conjecture of Nikolova (2009). This is based on joint work with Prof. Jaikumar Radhakrishnan (TIFR).

Title: Repdigits in Narayana's cows sequence and its consequence
Speaker: Pranabesh Das, postdoctoral fellow, Department of Pure Mathematics,University of Waterloo, Canada
Date: June 6, 2019 (Monday)
Abstract: In this talk we introduce Narayana's cows sequence which is a third order linear recurrence sequence. Repdigits (i.e., numbers with only one distinct digit in its base b expansion) in number sequence has been well studied in literature. In a joint work with Bravo and Guzman, we studied b-repdigits which are sums of two Narayana numbers. As an illustration we explicitly determine these numbers for the bases 2 < b ≤ 100. Results about the existence of Mersenne prime numbers, 10-repdigits and numbers with only one distinct block of digits of length 2 in its decimal expansion in the Narayana sequence are also deduced. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in Diophantine approximation.

Title: A model theorists tale about the square-free integers
Speaker: Neer Bhardwaj
Date: May 20, 2019 (Monday)
Abstract: The talk will be based on joint work with Chieu Minh Tran wherein we used the generic distribution of the square-free integers to prove certain 'tameness' properties of 4 natural (first-order) structures. I will primarily discuss the number-theoretic phenomenon we established and employed, and finish by stating the model-theoretic consequences which it entailed. No prior knowledge of first-order logic will be assumed, and the concept of a structure and its first-order theory will be defined. The material should be accessible to anyone with a basic undergraduate mathematical background.