Pure mathematics

Supervisor: Professor Xuan Duong

Topic description

The theory of singular integrals is a central topic of modern harmonic analysis. We aim to study boundedness of certain singular integrals whose kernels are not regular so that they do not belong to the standard class of Calderon-Zygmund operators.

These operators arise from some practical problems in physics and engineering, for example the Schrodinger operators, Kolmogorov operators, hence form a foundation for further applications in applied and industrial mathematics.

Supervisor: Dr The Anh Bui

Topic description

This project focuses on the study of function spaces arising in harmonic analysis, including Hardy, Besov, Triebel–Lizorkin and BMO-type spaces associated with differential operators.

The aim is to develop their structural theory and investigate their connections with singular integral operators, partial differential equations and spectral theory. Applications to Schrodinger operators, non-Euclidean settings and other problems in modern analysis will also be explored.

Supervisor: Associate Professor Ji Li

Topic description

This project investigates specific model domains, their boundaries and the associated group structures that are fundamental to the analysis of singular integrals, such as Cauchy-Szegő projections. Additionally, the research will explore applications to complex analysis within these domains.

Supervisor: Professor Xuan Duong

Topic description

In this project, the student will study some topics of modern harmonic analysis which aim to estimate different types of singular integrals on various function spaces. Some parts of this topic are related to the current research of harmonic analysis and partial differential equations.

Supervisor: Associate Professor Ji Li

Topic description

The student will study some recent developments and topics in modern harmonic analysis, such as singular integrals and the related function spaces on:

1. domains arisen in the complex setting
2. associated to differential operators arisen from partial differential equations.

Supervisor: Dr Bregje Pauwels

Topic description

Representation theory in prime characteristic is complicated but incredibly rich and successful approaches to this subject often use sophisticated category theoretical or geometrical methods.

The representation theory of a finite group in prime characteristic need not be semi-simple. In particular, computing the dimensions of irreducible representations of the symmetric group remains one of the largest open problems in representation theory.

Inspired by the DS functor in supersymmetry, One Tree Island (OTI) functors were introduced recently to study algebraic groups as well as finite groups in prime characteristic. OTI functors have explicit, computable definitions, are compatible with categorical and geometrical structures and arise from any ‘categorical action’.

In this project we may use OTI functors to study the representation theory of the symmetric group, categorical Heisenberg actions, and other interesting tensor categories in prime characteristic.

Supervisor: Dr Jean-Simon (JS) Pacaud Lemay

Topic description

Category theory is well-established as the primary toolbox to provide the mathematical foundations in computer science.

In particular, the theory of differential categories has been successful in providing the foundations of differentiation throughout computer science. For example, differential categories provide the categorical semantics of differential linear logic, while Cartesian differential categories provide the categorical semantics for the differential lambda-calculus and differentiable programming.

Most recently, the theory of reverse differential categories have been introduced to provide the categorical foundations of the reverse differential operator used in automatic differentiation and machine learning. In fact, reverse differential categories have been very successful, having found applications in automatic differentiation and machine learning. Applications of differential categories in computer science is an active area of research and there are many interesting research directions including:

• finding novel applications of (reverse) tangent categories in computer science
• further developing the theory of reverse differential categories
• formalising automatic differentiation and machine learning algorithms in a reverse differential category
• looking into the differential lambda calculus given by cofree Cartesian differential categories.

If you have your own potential idea for a research project involving differential/tangent categories and their applications in computer science, in particular related to automatic differentiation and machine learning, feel free to reach out.

Supervisor: Dr Jean-Simon (JS) Pacaud Lemay

Topic description

Not every matrix has an inverse, but every matrix has various kinds of generalised inverses, such as the famous Moore-Penrose inverse or Drazin inverse. Generalised inverses have many important applications since they often allow one to provide solutions to non-invertible operations.

Recently, there has been renewed interest in studying generalised inverses in category theory. Indeed, maps in a category need not be invertible, but they can have a generalised inverse. This is a relatively new area of research and there are many interesting questions worth exploring including:

• abstracting other kinds of generalised inverses in arbitrary (dagger) categories
• formalising other applications of generalized inverses, especially the Moore-Penrose inverse or the Drazin inverse, in categorical frameworks
• generalised inverses of functors
• studying the notion of parallel sum in Moore-Penrose dagger categories, generalising the application used for electrical circuits
• taking a closer look at the Gauss Construction, which uses Moore-Penrose inverses to connect dagger categories to probability theory.

If you have your own potential idea for a research project involving generalised inverses in categories and their applications, feel free to reach out.

Supervisor: Dr Jean-Simon (JS) Pacaud Lemay

Topic description

The theory of differential categories uses category theory to provide the foundations of differential calculus. There are three main areas:

1. differential categories, which capture the algebraic foundations of differentiation
2. Cartesian differential categories, which capture the foundations for multivariable differential calculus over Euclidean spaces
3. Tangent categories, which capture the foundations of differentiation geometry over smooth manifolds.

The theory of differential categories has been very successful in formalising important differentiation related concepts throughout various areas of mathematics including algebra, operads, differential geometry and algebraic geometry. The theory of differential categories is active area of research and on the rise and continues to find new applications in new areas. As such, there are many interesting new research directions for differential categories including:

• finding new connections for differential/tangent categories in novel areas, such as algebraic topology, probability theory, analysis, etc
• develop the theory of trigonometric functions in Cartesian differential categories
• study and find applications for cofree Cartesian differential categories and tangent categories
• study in detail certain models of tangent categoris, such as coalgebras, Rota-Baxter algebras, differential algebras, etc

If you have your own potential idea for a research project involving differential/tangent categories in any areas of mathematics, feel free to reach out.

Supervisor: Associate Professor Adam Sikora

Topic description

The proposed research will analyse L^p thresholds for Riesz transforms associated with a broader class of Grushin-type operators. It will combine Lie-group methods, heat-kernel bounds, resolvent estimates and counterexample constructions to distinguish bounded from unbounded regimes.

The central aim is to determine whether the admissible range of p is controlled by topological dimension, degeneracy order or an effective dimension generated near the singular set.

The project will also examine endpoint phenomena, including weak type (1,1), and assess how far existing results extend beyond finite-dimensional Lie-algebraic frameworks. It continues earlier work on Riesz transforms for manifolds with ends.

Supervisor: Dr The Anh Bui

Topic description

One of the major themes of harmonic analysis is the study of singular integrals on certain function spaces which rely on the theory of Calderon-Zygmund operators.

We emphasise that this theory plays an important role in analysis and has a strong influence on other branches of mathematics including the study of partial differential equations. However, in practice there are a number of important settings which do not fall within its scope.

The main aim of this project is to study the boundedness of singular integrals beyond the Calderon-Zygmund theory and applications in partial differential equations.