Subhajit Ghosh receives “Honourable Mention” for his PhD thesis

Dr. Subhajit Ghosh has received honorable mention for his thesis titled

On the number of components of polynomial lemniscates: Deterministic and random

from the TIFR Alumni Association (TAA) as part of the TAA-Harish Chandra Memorial Awards in Mathematics for 2025. The citation from the committee states:

for novel results on lemniscates of polynomials, examining both deterministic and random cases, and using both potential theory and probability.

About the thesis work:

If p(z) is a polynomial in one complex variable, its lemniscate is defined to be the open set in the plane where |p| < 1. The interest in lemniscates goes back a long way. The computation of the length of lemniscates gave rise to the theory of elliptic integrals. Also of importance is Hilbert's theorem, that any reasonable domain in the plane can be approximated to an arbitrary degree by a polynomial lemniscate. Lemniscates are also the simplest algebraic objects which lend themselves to amenable analysis. In this regard, an important paper by Erdos, Herzog, and Piranian (1958) investigated several metric properties of lemniscates (area, length, in radius, diameter etc) and posed a plethora of open problems. 

One of these was: How many connected components does a lemniscate have if the polynomial p is monic, and has all its zeros in a fixed compact set K? It is known by an elementary argument that if p is of degree n, then one can have at most n connected components. But can it be much less in general? In Subhajit's thesis this question and a related randomized problem were both tackled in detail. Subhajit showed that if the zeros were constrained to lie in K, a compact set of capacity (which is a measure of the size of K in a potential theory sense) less than 1, then one can have at most t(K)n many components. Here t(K) is a positive constant, strictly less than 1, that only depends on K. On the other hand if K had capacity more than 1, then he showed that one can reach n or nearly n many components. He also handled the most interesting case when capacity of K equals 1 and showed that in certain cases, here also n is obtainable. The randomized problem studies the lemniscate of random polynomials whose zeros are uniformly distributed on the unit disk, and asks for the average number of components in the limiting sense. Subhajit showed that this average is of the order of square root of n. So this means that in a crude sense, most polynomials of degree n, with zeros on the unit disk have much less number of components than the generic bound of n.