Listed on this page are current research projects being offered for the Vacation Scholarship Program.
For more information on this research group see: Pure Mathematics
Riemann surfaces and complex algebraic curves
The theory of Riemann surfaces, or equivalently, complex algebraic curves, sits at the confluence of differential geometry, complex analysis, and algebraic geometry. For this reason, the study of these objects has led to the development of many broader theories with wide-ranging implications across mathematics and broader sciences. This project will focus on gaining an understanding of these objects from one or both of these perspectives with potential projects (depending on interests and background) in rational billiards, enumeration and existence of branched covers, holomorphic differentials and flat geometry, line bundles, divisors, Riemann-Roch theorem, Riemann-Hurwitz theorem, Teichmüller dynamics, projective complex geometry, and the moduli space of Riemann surfaces or algebraic curves.
Contact: Scott Mullane mullanes@unimelb.edu.au
Knot theory
How can we tell whether the trefoil knot is “the same” as a trivial round circle? There are a variety of knot invariants that are up to the task – the simplest ones use basic linear algebra, while more complicated invariants exist which draw on a range of mathematical areas. The precise topic will be chosen based on the student’s interests and background knowledge.
Contact: Arunima Ray aru.ray@unimelb.edu.au
Introduction to Kirby diagrams
This project concerns manifolds: topological spaces which locally resemble Euclidean space. Among these spaces, four-dimensional manifolds are especially mysterious. While they seem highly abstract, they can nevertheless be described concretely using diagrams, the so-called “Kirby diagrams”. This project will include an introduction to manifolds, handle decompositions of them, and a first look at describing 4-dimensional manifolds and surfaces within them. Some background knowledge in geometry or topology will be helpful but not strictly necessary.
Contact: Arunima Ray aru.ray@unimelb.edu.au
Understanding Topological Data Analysis
Topological data analysis (TDA) is a technique incorporating algebraic topology, namely notions of homology to gain greater insight into the shape of large data sets. This project seeks to understand the mathematics and techniques underlying TDA, with an eye towards applications.
Contact: TriThang Tran trithang.tran@unimelb.edu.au and Paul Fijn paul.fijn@unimelb.edu.au