The arc length and topology of a random lemniscate
Speaker 
Dr. Koushik Ramachandran, Oklahoma State University, USA


When 
Jun 06, 2017
from 04:00 PM to 05:00 PM 
Where  LH 006 
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Abstract: A polynomial lemniscate is a curve in the complex plane defined by \({\{z \in\mathbb{C}: p(z) = t\}}\) Erdos, Herzog and Piranian posed the extremal problem of finding the maximum length of a lemniscate \([\Lambda = \{z\in \mathbb{C}: p(z) = 1\}] \) when [p] is a monic polynomial of degree [n.] We study the length and topology of a random lemniscate whose defining polynomial has Gaussian coefficients. When the polynomial is sampled from the Kac ensemble, we show that the expected length approaches a nonzero constant as \([n\rightarrow\infty,] where [ deg(p) =n.]\) We also prove that the average number of components on a random lemniscate is asymptotically [n] and that there is a positive probability (independent of [n]) of a giant component. This talk is based on joint work with Erik Lundberg.