# Honours Thesis Supervisors

### Shirin Boroushaki

###### Nonlinear Analysis and Partial Differential Equations (PDE)

During my PhD, my research primarily focused on the analysis of partial differential equations (PDEs) from a pure mathematical perspective. Specifically, my focus was on proving the existence of solutions to a specific class of PDEs known as Stochastic PDEs.To provide a bit of insight into the existence problem: Traditionally, one of the classic methods used to demonstrate that an elliptic (non-time-dependent) PDE has a solution is through the Euler-Lagrange variational method. In this method, we establish that the solution to an elliptic PDE can be found as the minimum of a functional. This concept resembles finding the optimum value of a function in calculus using derivatives. However, when dealing with PDEs whose solution changes over time, such as parabolic PDEs, and especially Stochastic PDEs, the Euler-Lagrange variational method does not readily apply. In my thesis, I introduced an alternative variational approach known as the self-dual method, which has been developed over the past two decades. This method enables us to obtain solutions to Stochastic PDEs by identifying them as minima of appropriately chosen self-dual functionals. The tools I applied throughout this research encompassed concepts from Calculus of Variations, Functional Analysis, Convex Analysis, and Stochastic Processes. As my mathematical journey has evolved, I've also discovered a passion for teaching math and math pedagogy. My goal is to inspire students' curiosity and courage to step into the journey of mathematical exploration, to share with them the inherent beauty of mathematics, and to help them grasp the concepts.

### Richard Brewster

###### Discrete Mathematics and Theoretical Computing Science

My research sits at the intersection of Discrete Mathematics and Theoretical Computing Science. I am particularly interested in algorithmic aspects of graph theory. Typically, the development of efficient algorithms is guided by the development of mathematical theory and vice versa. A rich area where this happens is graph homomorphisms. Here we see ideas for example from structural graph theory, graph colourings, category theory, and arc consistency checks (from AI). Other topics include graph coverings and packings, and combinatorial reconfiguration.

### Fatma Mahmoud

###### Fuzzy Topological Applications in Information Systems

Topological concepts exist not only in almost all branches of mathematics, but also in many real-life applications ranging from kids’ games to the applications in DNA and modern technology of image analysis. We believe that topological structures and its generalizations are important base for modification of knowledge extraction and processing. Recently, topology is concerned with the problem of ambiguities in the information, since the topological view of the boundary region is the clearest view to treat the area of uncertainty in knowledge, this line implied to the topological facts in many new applications such as structural analysis, in chemistry, physics, and biology. There is a study is done applying topological concepts in information systems. We can generalized this study to Fuzzy topological concepts.

### Sean McGuinness

###### Combinatorics, Graph Theory and Matroid Theory

Broadly speaking my research interests lie in the field of Combinatorics. My specific interests in this area are Graph Theory and Matroid Theory, the latter being more of recent interest to me. Matroid theory is a subject which lies at the crossroads of three subjects: Algebra, Geometry, and Combinatorics. All three contribute a rich diversity of ideas. My interest here has been more on the combinatorial side. I have worked on problems which involve extending known properties of graphs to matroids. I am also interested in various unsolved conjectures, for example, Rota’s basis conjecture and White’s conjecture. Much of my recent research focusses on the properties of bases in matroids. My research interests also touch on subjects tangential to matroid theory. For example, the Alon-Tarsi conjecture for Latin squares. I am also interested in using tools and concepts from Linear algebra, projective geometry and probability.

### Lucas Mol

###### Combinatorics on Words

A word is a sequence of symbols taken from some finite alphabet. In the area of combinatorics on words, we are mostly interested in long words over small alphabets. Which patterns can be avoided, and which patterns must inevitably occur? We use tools from combinatorics, algebra, and computing to answer fundamental questions about words.

### Yana Nec

###### Applied Mathematics and Optimisation

The topics you might work on as an Honours student in my group revolve around applied analysis of partial differential equations, including but not limited to: weak solutions, numerical schemes, asymptotic expansions, fluid flow, mathematical and computational optimisation problems, special functions. Most projects done in my group require a variety of skills: analytical derivations and proofs, coding, visualisation of surfaces or functions. Depending on the project, you might do different amounts of each. To enjoy your work with me you would need to like calculus, algebra, differential equations and numerical analysis. You might not like coding, but you should be able to do it.

### Saeed Rahmati

###### Algebraic Topology and Manifolds

Topological manifolds are generalization of curves and surfaces to higher dimensions; they are topological spaces that “locally’’ look like a Euclidean space. Smooth manifolds are topological manifolds that we can do calculus on them. I’m interested in properties of manifolds that we can express using bridges between Topology and Algebra.

### Yue Zhang

###### Statistical Analysis and Bioinformatics

My research areas primarily focus on statistical analysis and bioinformatics, with a specific emphasis on comparative genomics. I am passionate about exploring the evolutionary relationships and functional implications of genomic variations across different species. Through comparative genomics, I aim to unravel the underlying mechanisms that drive species diversity and adaptation. I utilize various statistical and computational approaches to analyze large-scale genomic datasets, integrating genomic, transcriptomic, and proteomic data to gain a comprehensive understanding of biological systems. I am particularly interested in studying comparative genomics by employing methods of mathematics and statistics and developing computational tools for analyzing large-scale biological datasets. By supervising Honours students, I hope to foster their passion for scientific inquiry and provide them with the necessary guidance to explore comparative genomics and its applications.