Mathematical Sciences Colloquium | Department of Mathematics

Mathematical Sciences Colloquium

Prof. Ravindra B. BapatSeptember 08, 2022A glimpse of spectral graph theory.This talk introduced the elements of spectral graph theory and presented some key results highlighting the interplay between the spectrum of an adjacency matrix and the properties of a graph.
Professor Niladri ChatterjeeOctober 20, 2022Decision-Making using Rough Sets.This talk demonstrated the role of Rough sets in decision-making problems. Solutions to certain real-life problems were also demonstrated.
Prof. TSSRK RaoNovember 24, 2022An invitition to the geometry of Banach spaces.In this talk, a seminal idea of J. Lindenstrauss on the `intersection properties of balls' is explained and shown how these ideas help in understanding and determining the properties of Banach spaces.
Prof. Pradeep DubeyDecember 15, 2022Game Representations & Extensions of the Shapley Value.

It is shown that any transferable utility game can be “represented” by an assignment of costly facilities to players, in which it is intuitively obvious how to allocate the total cost of all facilities amongst the players in an equitable manner. The equitable solution in the representation turns out to be the Shapley value of the game and thus serves as an alternative justification of the value.

This approach enables us to extend the Shapley value to games where not all coalitions can form, provided those that can constitute a semi-algebra of sets (i.e., contain the grand coalition and are closed under complements) or, more generally, form a "hierarchy" or have "full span". Some examples of how such games might arise in practice are given.

Prof. T. E. S. Raghavan

Feb. 06, 2023

Cones in real Banach spaces and the spectral theory of positive operators via Game theory

While ( zero-sum two-person) matrix games with say positive entries have unique positive values, the same matrix has a positive eigenvalue with a positive eigenvector. This eigenvalue is indeed the spectral radius. A clear connection between the two can be made possible via theorems of Kaplanski and David Blackwell. The theory has a very natural generalization to real Banach spaces when we observe that positive matrices map the positive orthant into itself. We can consider convex cones in general and study the spectral properties of linear operators leaving invariant a cone in real Banach spaces.  The profound contributions of MG Krein and MA Rutman on compact operators and operators in reflexive Banach spaces etc can as well be seen also through game theory by replacing von Neumann's minimax theorem with Ky Fan's general minimax theorem.