# Pure mathematics

## Researchers

 Emeritus Professor Sid Morris Topological groups Associate Professor David Yost Functional analysis, convex geometry Associate Professor Alex Kruger Variational analysis Dr Guillermo Pineda-Villavicencio Graph theory, convex geometry Dr Andrew Percy Algebraic topology Dr Ewan Barker Mathematical logic, combinatorics

## Variational analysis

Variational analysis is a branch of mathematics that extends the methods arising from the classic calculus of variations and convex analysis to more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives (Wikipedia: https://en.wikipedia.org/wiki/Variational_analysis).

Our research is focused on advancing the general regularity theory. We are working on clarifying the relationships between various stationarity and regularity concepts and criteria for typical for variational analysis and optimisation objects: set-valued mappings, collections of sets and (extended) real-valued functions. We also investigate regularity properties of specific mappings arising from typical setting of optimisation problems and develop primal and dual derivative-like tools.

## Combinatorics

Combinatorics is concerned with the study of countable discrete structures and how they can be combined to form increasingly more complex structures. Combinatorics problems are often characterised by the deceptive simplicity of their statements: they are understandable by everyone and yet very difficult to solve. Research in combinatorics has grown substantially in recent years, because of the ubiquity of combinatorics problems in many other branches of mathematics such as algebra, probability, topology and geometry to just name a few. Combinatorics, like number theory, borrows many techniques from a diverse number of branches of mathematics and computer science.

Our research in combinatorics covers problems in graph theory and convex geometry, two of the oldest and most accessible parts of combinatorics. In these two areas our research efforts are directed towards the advancement of the following topics.

• Classification of graphs with given maximum degree and diameter whose order misses the Moore bound by a small number; Tight asymptotic bounds for the maximum number of vertices of a graph in a prescribed class with given maximum degree and diameter; Constructions of large graphs in a prescribed class with given maximum degree and diameter.
• Lower and upper bounds for the number of faces of classes of polytopes; Reconstructability of polytopes from their graphs; Decomposability or otherwise of polytopes with respect to Minkowski sums; Hamiltonicity or otherwise of graphs of simple d-polytopes with d at least 3.

## Functional analysis

Functional analysis is the study of topological vector spaces (which arose naturally as solution spaces for practical problems such as differential equations and the calculus of variations), and the operators between them. It serves as a foundation for many other subjects, including convex analysis, Fourier theory, quantum mechanics and optimisation.

Banach spaces are an important class of topological vector spaces, and contractive projections are an important class of operators on them, which help us understand how they can be decomposed into simpler spaces. Current research in Functional analysis at FedUni revolves around these topics. Particular topics of investigation include:

• Banach spaces which admit large families of contractive projections, which then allow us to establish new properties by transfinite induction
• Finding minimal norm projections on finite dimensional Banach spaces
• Studying conditions under which a given Banach space can embed in a larger space without being the range of any projection thereon, and with the corresponding quotient also being specified in advance

## Topological groups

As the name suggests, a topological group is a group equipped with a topology compatible with the group operations. This includes, for example, any finite group, the homeomorphism group of a topological space, and any topological vector space. They arise naturally in many problems, from geometry to the solution of equations. Additional structure, such as local compactness, connectedness, or Lie structure leads to a richer theory, interactions with other areas such as harmonic analysis, representations, and measure theory, and to a wider range of applications. The development of such structure theories is an ongoing and fruitful research activity.

## Algebraic topology

Algebraic topology has its roots in exporting topological problems into algebraic settings where more tools are available to explore or resolve the problem. The general concept of homotopy can be applied within any model category and so the study has gone beyond topological questions.

Our work centres around the algebra consisting of integral cohomology groups together with all natural n-ary cohomology operations, stable and unstable. Our main areas of study are the generating sets of integral cohomology operations, relations between these operations and the distinction between integral cohomology operations and those over field coefficients. We are also interested in the use of spectral sequences and aspects of category theory.