Abstract: In order to solve a precise problem on trigonometric series, “Can a function have more than one representation by a trigonometric series?”, the great German mathematician Georg Cantor created set theory and laid the foundation of the theory of real numbers.

Abstract: Ergodic Theory deals with the study of asymptotic behaviour of a dynamical system, which is either a group action, flow or map on a state space. Dynamical systems can be broadly classified into closed and open systems. In closed systems, the orbit of a point lie in the state space for all time, whereas in open systems, the orbit of a point may escape from the state space through a hole. A classical example of this escape phenomena is in the study of the motion of a billiard ball on the table with a hole (pocket). The first account of open dynamical systems is due to Pianigiani and Yorke in 1979, who were motivated by this example. This talk is a brief historical account of the development of the related theory and some potential questions. This will be explained using some recent results on the period doubling map on the circle with a hole, and generally products of expanding maps on the torus with a hole. Ideas from symbolic dynamics and arithmetic dynamics will be presented in this context.

Abstract: Suppose “n” boys and “n” girls are to find suitable mates of opposite sex from among them. Each boy could indicate his acceptable set of girls among them. Similarly each girl can indicate her acceptable set of boys among them. Suppose we are able to find full matching, with say boy “k” acceptable to girl “k’, and vice versa k=1, 2, n. The main question is to determine the dominating persons between the matched pairs.
It is possible to convert this problem into a cooperative TU game and use either the Shapley value or the nucleolus of this game as a measure of the relative strengths of the person within a matched pair. While the Shapley value has an explicit formula, it is hopelessly difficult to compute it even for these simpler class of TU games. In fact nucleolus is more amenable and we can exploit an alternative characterization of the nucleolus as the lexicographic geometric center as found by Maschler, Peleg and Shapley. While Solymosi and Raghavan gave an algorithm exploiting MPS theorem for the general assignment games, for binary assignment games one can straight away start with the southwest corner core element and one is able to search for the longest path with an associated graph and settle down the relative strengths of mates represented by vertices of the graph along this path. Unlike the algorithm of Solymosi and Raghavan, here Hardwick works on the same digraph by removing all edges whose end vertices are settled (we know their nucleolus values for the individual members of the pairs).
In many Western societies, same sex marriage is becoming a more acceptable mate system and this leads to the problem of measuring the relative strength of mates who are part of the maximal matching on a general graph. While Kuhn’s Hungarian algorithm locates the maximal matching in the bipartite case, Edmund’s blossom algorithm and Berge-Tutte’s theorem on the number of odd components of a graph and the minmax connection is used to split the graph into components containing blossoms and the determination of the nucleolus which works with such a split and via coloring algorithm.
In western societies same sex marriages are on the rise and naturally we can ask for the relative strength of members of matched pairs in a maximal matching.

Abstract: Linear dynamics is a relatively new near area of dynamics which is developing rather rapidly. In this talk we introduce the basic of linear dynamics and show that some operators everyone is familiar with are chaotic in some sense. Then, we introduce new classes of operator with interesting dynamical properties. All concepts will be defined and the talk will be accessible to graduate students. This is joint work with Benito Pires of University of São Paolo, Brazil.