###### Research projects for prospective PhD students

A *polytope* is the convex hulls of finitely many points in R^{d} for some d. Polytopes arise in various contexts, including as sets of feasible solutions in linear programming. The platonic solids are prominent examples of convex polytopes. For background and information on polytopes, refer to [Chapter 1, 1].

A *face* of a polytope *P* is *P* itself (the *trivial face*) or the intersection of *P* with a hyperplane which contains *P* in one of its closed half-spaces (*proper faces*). The *dimension* of a polytope is the dimension of its affine hull. The 0-dimensional faces are *vertices*, and 1-dimensional faces are *edges*.

**Project 1: ****Unlocking the multifaceted structure of convex polytopes**

This project examines the facial structure of polytopes at two levels:

- Establishing relations satisfied by the face numbers of a polytope.
- Identifying properties and constructions that elucidate these relations.

To realise these aims, we focus on questions that capture modern trends in this field, with special attention to prominent class of polytopes, such as *simplicial polytopes*, polytopes whose faces are all simplices*, *and *cubical polytopes,* polytopes whose faces are all cubes.

**Project 2: Graphs ****of convex polytopes**

The *graph of a polytope* is the abstract graph defined on the vertices of the polytope with two vertices being adjacent if they share an edge. This proposal investigates the combinatorial structure of polytopes by studying their graphs. Key questions include the existence of Hamiltonian cycles, minors, and colourings in graphs of polytopes. While these topics are well studied in graph theory, they remain underexplored for graphs of polytopes.

**References**

- Pineda-Villavicencio, G. (2024). Polytopes and graphs, vol. 211, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge.

###### Graduate Students

- Daniel Morales-Silva, 2018, PhD thesis: Assessing healthcare providers’ performance with and without risk adjustment, Federation University Australia. Principal supervisor.

Now data scientist at Medibank. - Ramiro Feria-Purón, 2015, PhD thesis: Large interconnection networks with given degree and diameter, University of Newcastle, Australia. External co-supervisor.

Now iOS developer at Deputy. - Hebert Pérez-Rosés, 2012, PhD thesis: Large networks bounded in degree and diameter, University of Newcastle, Australia. External co-supervisor.

Now invited lecturer at Universitat Rovira i Virgili, Spain.