# Geometry & Topology

Listed on this page are current research projects being offered for the Vacation Scholarship Program.

For more information on this research group see: Geometry and Topology

## Exploring triangulations of manifolds in dimensions two and three

(re-posted for 2017-2018, 2014-2015 report)

An n-dimensional manifold is a topological space that is locally modeled on R^n. Two-dimensional and three-dimensional manifolds are particularly nice because they decompose into triangles and tetrahedra, respectively. Although that last fact has been known for quite some time, there are a variety of open questions surrounding the combinatorics and geometry of these triangulations. We are hoping to lead Vacation Scholars to investigate some of these questions. We would be amenable to leading individuals, but would prefer to lead a group project.

**Contact:** Craig Hodgson craigdh@unimelb.edu.au

## Enumerative geometry and physics

(posted for 2014-2015, report 1, report 2)

Mirror symmetry is one of the most important and influential problems in mathematics and mathematical physics. At the simplest level mirror symmetry realises solutions of enumerative problems from mathematical physics in two quite different ways. This project involves concrete calculations related to geometry that give an accessible approach to mirror symmetry for students. It involves techniques from geometry, complex analysis, combinatorics and simple programming.

**Contact:** Paul Norbury P.Norbury@ms.unimelb.edu.au