Chandra Nair Overview

In my early research career I considered random versions of combinatorial optimization problems; my doctoral thesis [17] resolved a long-standing conjecture about the assignment problem and my post-doctoral work resolved another conjecture about the number partitioning problem [6,7]. Upon joining as a faculty member in the Information Engineering department of The Chinese University of Hong Kong, my research focus shifted to fundamental problems in information theory. Engineering motivations: The assignment problem tackles a problem of assigning customers to queues/buffers so as to minimize the total weight. The number partitioning problem tackles a problem of assigning jobs to partitions to minimize the load differential. Both of these problems often arise naturally when looking at load balancing, delay-minimizing, throughput-maximizing type scenarios which arise ubiquitously in many engineering applications. The optimization problems in information theory occur naturally in many wireless settings, with an eye to maximize resource utilization. While my research has been driven primarily by theoretical considerations, the results also provide several practical insights, for example, into structure of optimal (or near optimal) codes, and properties of the extremizers of non-convex problems. The non-convex problems in information theory arose as follows: There were computable regions proposed in the late seventies or early eighties [12, 15], and in the case of broadcast channel there was also an outer bound to the capacity region [15]. It was not known whether these regions were the true capacity region or not, and in the case of the broadcast channel, whether even the two computable inner and outer bounds matched or not. To show the suboptimality or gap between the bounds, one had to evaluate the optimizers of two different non-convex functionals over possibly infinite dimensional probability spaces and show that their values differed. On the other hand, determining the optimality of bounds amounted to showing sub-additivity of certain functionals [8, 28] in probability spaces, which in turn was related to tensorization of certain parameters. The above serendipitous discovery motivated us to study other quantities that were known to tensorize such as the hypercontractivity region, already of considerable interest to probabilists and computer scientists. The research that followed unearthed equivalent characterizations of hypercontractivity in terms of information measures [3, 19]; and in turn implied that the optimization problems that we were dealing with, to compute the regions, were directly similar to computing hypercontractivity parameters for certain joint distributions (hence possibly explaining the difficulty).