# Mathematics Projects 2019/20

MATH 1 - Project Title:  Noncommutative differential geometry and its applications

Project Description: Noncommutative differential geometry is related to algebra, geometry and functional analysis, though in this project we will begin from the idea of differential calculi. It has applications to physics, finite geometries, Hopf algebras and C* algebras among other areas. The principle is to produce as many noncommutative analogues of classical geometrical results and their applications as possible, as well as considering ideas which only make sense in a noncommutative world.

DTC MATH 2 - Project Title: Facets of noncommutative smoothness and foundations of noncommutative geometry

Project Description: Noncommutative geometry combines ideas from geometry and noncommutative algebra and equips noncommutative rings with geometric meaning.  The aim of the project is the study smoothness of algebraic structures and  noncommutative geometric objects encoded by them from various point of view: differential, homological, topological, categorical, etc. By contrasting these points of view not only a deep understanding of noncommutative smoothness will be achieved, but also categorical foundations of noncommutative geometry will be developed.

DTC MATH 3 - Project Title: Trusses

Project Description: Applications of the Yang-Baxter equation range from the description of physical forces of the nature (quantum field theory, integrable Hamiltonian systems in particle and statistical physics), through classification of geometric and topological objects in mathematics (knot theory, group theory) to foundations of mathematics (braided monoidal categories). It has been recently discovered that set-theoretic Yang-Baxter equations are closely linked with an algebraic structure known as a brace, which resembles that of a ring, a most widespread way of entwining of two operations on a given set. Introduced in October 2017 trusses are algebraic structures which encompass both braces and rings; the aim of the project is the detailed study of properties and the nature of trusses, and construction of examples of trusses with application to the Yang-Baxter and related equations.

DTC MATH 4 - Project Title: Singular limits of elliptic and parabolic systems

Project Description: Singular limits of systems of nonlinear elliptic and parabolic partial differential equations (PDE) provide a powerful and deep analytical approach to studying systems of PDE that are often otherwise largely intractable. Such limits yield a rich source of deep mathematical phenomena and open questions, often involving intriguing problems with low regularity, and play an important role in applications ranging from spatial segregation in population dynamics to phase separation in materials. A major current challenge is to understand how disparate diffusivities of the system components can affect the limiting behaviour, and this project will build on exciting recent advances to tackle this issue in the context of segregation and long-time behaviour in a variety of multi-component PDE systems.

DTC MATH 5 - Project Title: New approaches to topological data analysis

Project Description: This project aims at developing new algorithms and their implementation in topological data analysis. It is a problem-driven project aiming at solving real-world problems using rigorous mathematical tools. Good knowledge of mathematics and computer science is required.

DTC MATH 6 - Project Title: Long-time behaviour of complex systems

Project Description: Complex systems are large groups of interacting elements which have the so-called collective behaviour, not peculiar to inner nature of each element itself. The densities and other statistical characteristics of complex systems may be approximated by solutions to (local or nonlocal) partial differential equations. The project expects the study of the long-time behaviour for the solutions and the corresponding densities using some/all of the approaches: probabilistic, analytic, numerical.

DTC MATH 7 - Project Title: Categorical aspects of motivic homotopy theory

Project Description: The primary goal of motivic homotopy theory is to study algebraic varieties from a homotopy theoretic viewpoint. Many of the basic ideas and techniques in this subject originate in algebraic topology. This project will investigate categorical aspects of various triangulated categories of motives.

DTC MATH 8 - Project Title: Tropical geometry and scheme theory

Project Description: Over past few years Giansiracusa and others have initiated a new foundation for the type of polyhedral geometry known as tropical geometry; this foundation is rooted in the combinatorial theory of matroids, the algebra of idempotent semirings, and Grothendieck's theory of schemes. The emerging theory is young and has many open problems and exciting potential applications to algebraic and analytic geometry. In this project, the student will study relations between the topology and geometry of tropical schemes and their projective algebraic invariants such as the Hilbert polynomial.

DTC MATH 9 - Project Title: Topology of homological surface bundles

Project description:  The study of fibre bundles of surfaces and the mapping class group has been an extremely rich subject over the past decades, combining many beautiful ideas from topology, homotopy theory, group theory, and algebra.  The mapping class group sits inside a mysterious larger group made from 3-manifolds with a boundary that has the same homology as a surface crossed with an interval, and there are many open questions about this group.  In this project, we will approach this mysterious group from a novel geometric perspective by studying the topology of the classifying space of homological surface bundles.

DTC MATH 10 - Project Title: Probability measures on infinite dimensional spaces and umbral calculus

Project Description: The classical umbral calculus is a study of Sheffer polynomial sequences on the real line. This theory has numerous connections with algebra, analysis, probability theory, mathematical physics, topology etc. In this project, you will study umbral calculus in the infinite dimensional setting and probability measures on infinite dimensional spaces with respect to which Sheffer polynomial sequences defined on an infinite dimensional space are orthogonal.

DTC MATH 11 - Project Title: Quasi-free states on the algebra of the anyon commutation relations

Project Description: Anyon statistics are intermediate statistics between Fermi and Bose statistics and they have been used by physicists to describe the quantum Hall effect. Mathematically anyon statistics are described by operator algebras generated by certain creation and annihilation operators satisfying the q-commutation relations with q complex of modulus one. You will study representations of the q-commutation relations related to quasi-free states on the anyon algebra.

DTC MATH 12 - Project Title: Mathematical analysis of Thomas-Fermi theory of charge screening in graphene

Project Description: Mathematically, graphene could be considered as a two-dimensional object. Two-dimensionality of graphene leads to a number of remarkable physical and mechanical properties which also give rise to new challenging mathematical problems. The aim of the project is to study from the rigorous mathematical perspective of nonlinear PDEs a Thomas-Fermi type model of electron's screening in a single layer graphene and to find a mathematical proof of the remarkable screening behaviour proposed by physicists.

DTC MATH 13 - Project Title: Heat kernel for Schroedinger operator

Project Description: The subject is two-sided estimates for the integral kernel of the semigroup of operators generated by the Schroedinger operators, with applications to NSE and spectral analysis.

DTC MATH 14 - Project Title:  Algebraic spline geometry methods in approximation theory

Project Description: The aim of the project is to explore algebraic geometry tools for the study of approximation methods based on piecewise polynomial functions, also known as splines. Splines are a powerful tool widely used in different challenges arising in computational mathematics, such as the numerical approximation of partial differential equations (the finite element method), and the parametrization of shapes in geometric modelling and computer-aided geometric design. This project will focus on algebraic aspects of splines, its ring structure, and its consequences in concrete computational problems such as the construction of suitable spline spaces for approximation.

DTC MATH 15 - Project Title: Analysis of stochastic evolution equations

Project Description: This PhD project will investigate random field solutions of parabolic partial differential equations perturbed by random noises, including stochastic heat equations and parabolic Anderson models as prototype examples. Analysis of partial differential equations perturbed by random noises is a very hot topic in probability and statistics with deep links to diverse disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions (including polymer science), fluid dynamics, as well as mathematical finance. The project is suitable for a student who has a solid foundation in probability theory and functional analysis, with a moderate knowledge of partial differential equations.

DTC MATH 16 - Project Title: Delay McKean-Vlasov stochastic differential equations and applications to finance

Project Description: In this project, we shall investigate a class of delay McKean-Vlasov stochastic differential equations (SDEs), also known as mean-field delay SDEs. We shall study the properties of these equations such as existence and uniqueness of the solution, optimal control and portfolio optimization.

DTC MATH 17 - Project Title: Spreading speeds and travelling waves

Project Description: Front propagation is ubiquitous throughout science and beyond, from the invasion of species in population dynamics to the switching properties of liquid crystals to the spread of rumours, and is often modelled mathematically using initial-value problems and travelling waves for systems of reaction-diffusion equations and their non-local variants. This very active field enjoys a rich interplay between challenging interdisciplinary questions that lead to novel and deep mathematical results. In this project, we will investigate spreading speeds and travelling waves in models arising in a variety of topical applications using state-of-the-art techniques in the analysis of nonlinear partial differential equations.

## How to apply for a Mathematics Research Degree Project

Candidates must have a minimum of an upper second class honours degree or equivalent in a relevant subject, or an appropriate Master’s degree (with Merit). Informal enquiries are welcome by emailing the project supervisor.

Please send the following to science-scholarships@swansea.ac.uk and include the reference number of the reference number of the project in the email subject line (eg DTC MATH 1):

• A comprehensive CV to include:
• Details of qualifications, including grades
• Details of any current and relevant employment or work experience
• A covering letter stating why the project you are applying for particularly matches your skills and experience and how you would choose to develop the project