Possible topics for postgraduate research

Random growth models

Randomly growing shapes are all around us. Imagine cellular growth in a petri dish, coffee stain on paper or snowflakes gathering on a window. What is the shape of these objects? Macroscopically they look smooth but microscopically they are fractal-like and exhibit interesting phenomenon. Random growth models present increasingly important and applicable mathematical problems that can be analyzed using methods from analysis, combinatorics and probability. They are also closely related to random planar geometry, random permutations, random matrices and asymptotic representation theory.

For instance, consider the question about the length of the longest increasing subsequence in a random permutation of n numbers. The length is asymptotically 2n^1/2 with high probability, and there is a central limit theorem with a non-Normal limiting distribution. Such problems admit exciting new limit theorems that are different from the usual central limit theorem involving Normal distributions. One can explore various questions in these directions, some fundamental ones being the understanding of the shape and fluctuations of random growth models.

Contact: Mustazee Rahman.