Towards a Uniform Sampling Procedure for Abstract Triangulations of Surfaces

The study of manifolds is at the centre of geometric topology, where we are interested in qualitative properties which do not change under continuous deformations. Manifolds are often studied by decomposing them into simple pieces. In dimension two, manifolds are surfaces, which can be decomposed into triangles. Such triangulations of surfaces can then be studied using algorithms and software.

The proposed research project aims at developing a uniform sampling procedure for abstract triangulations of surfaces. Being able to sample uniformly from the space of triangulations will solve fundamental problems in the field, such as the dependency of algorithmic methods of a particular choice of triangulation.

We will start with a special class of triangulations, called crystallisations, where this project is very promising, and then work slowly towards more general settings.

Rajan Shankar

The University of Sydney

Rajan is a third-year student majoring in statistics as part of the mathematical sciences program at the University of Sydney. He is looking forward to continuing his studies in statistics as an honours student in 2022.

Rajan possesses prior research experience, having recently completed a supervised project in modelling cattle heat stress as part of the Dalyell Scholars program. He is currently engaged in additional research activities relating to the project. Rajan also enjoys sharing his knowledge with others as a lab demonstrator for first-year data science.

Although his academic interests lie mostly within the realm of probability & estimation theory, Rajan is very keen to take on a vacation research project in pure mathematics to expose himself to more novel areas of research.

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