# Geometry and Topology

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

### Flatification for monoid schemes

A monoid scheme is a geometric object whose functions can only be multiplied, not added - for example, the monoid affine space of dimension d admits only the monomials in d variables as functions. Because these objects are much more rigid than the usual objects studied in algebraic geometry, they can be described combinatorially, and monoid analogues of theorems in algebraic geometry are typically easier to prove. Flatification - usually referred to by its French name "platification par éclatement" - is a crucial theorem in algebraic geometry that should admit a good monoid analogue. In this project, you will learn the language of monoid schemes and attempt to formulate and prove an appropriate flatification result.

Contact: Christian Haesemeyer christian.haesemeyer@unimelb.edu.au

### Configuration spaces and shuffle algebras

This project involves studying the topology of configuration spaces and related objects. Concretely, we want to write down complexes that compute their homology. The configuration space of a manifold is the space of different arrangements of points in a manifold, where no two points can overlap. We can generalise these spaces by adding algebraic labels to the points and allowing points to collide in different ways. These spaces appear geometry, algebra, physics - anytime you want to study points in a system really! - but our particular interest is motivated by counting problems from arithmetic geometry.

Contact: TriThang Tran trtran@unimelb.edu.au

### Enumerative geometry and physics

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 norbury@unimelb.edu.au

### Super hyperbolic surfaces

An ideal triangulation of a cusped hyperbolic surface is a triangulation with no vertices, and all faces triangles. The number of faces and edges is 4g-4+2n, respectively 6g-6 +3n for a surface of genus g with n cusps.  Coordinates of the space of hyperbolic structures are given by positive numbers assigned to edges of the triangulation.  These generalise to super coordinates assigned to edges and faces.  This project will aim to generalise classical results to analogous results with super coordinates

Contact: Paul Norbury norbury@unimelb.edu.au

### Super Hurwitz numbers

Hurwitz numbers count the number of genus g surfaces that cover the two-sphere with prescribed branching. This project will look at a generalisation to super genus g surfaces.

Contact: Paul Norbury norbury@unimelb.edu.au

### Geodesics on cones

Geodesics are curves between two points on a manifold that extremize the length functional. In this project the student will learn about the basics of variational calculus which is concerned with the extremization of functions of functions (such as curves) and apply the techniques to the surprisingly rich question of how to find and describe geodesics on cones. If time permits the project will also involve having a closer look at the properties of the singularity at the tip of the cone.

Contact: Thomas Quella thomas.quella@unimelb.edu.au

### Wave propagation in flat space

In this project we learn about the classical wave equation, the representation formula by spherical means, and its interpretation in different dimensions. The student learns the basics about the formulation of the initial value problem, and derives the asymptotics of general solutions. There are various applications of this material in physics, and also non-linear wave equations can be explored.

Contact: Volker Schlue volker.schlue@unimelb.edu.au

### Topics in general relativity

Contact: Volker Schlue volker.schlue@unimelb.edu.au