# Koushik Ramachandran

Reader (F)

## Website

## Research interests

- Complex Analysis
- Potential theory
- Probability

**Complex Analysis and Probability**

My research interests are in analysis and probability, particularly problems which have a flavor of both of them. One of the main objects of my research are* polynomial lemniscates*, which are sets that can be described as {|p| < 1} for some complex polynomial p. Lemniscates are known to exhibit remarkable properties. For instance, any Jordan domain (even a fractal shaped one) in the plane can be approximated arbitrarily closely by a lemnsicate! They also arise in algebraic geometry since the boundary of a lemniscate is an algebraic curve. Given their wide applicability, it makes sense to study various metric and topological properties of these lemniscates. For instance, it is obvious that a lemniscate is a bounded open set. We can ask what is the size of the largest disc that can be fitted inside a lemniscate? Or how many connected components can a lemniscate have? On the other hand, in probability we are more interested in how a *typical object* behaves. So we randomize the previous problem by considering random polynomials (where the roots are chosen i.i.d from a fixed distribution) and ask what is the expected size of the largest disc or the expected number of components of random lemniscates. It turns out that both the determinstic and randomized questions have neat answers, which involve using tools from complex analysis, probability and potential theory. The interested reader can find these and other results at LEM1 and LEM 2 respectively.