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

- First Supervisor Name: Edwin Beggs
- Email Address: e.j.beggs@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/e.j.beggs/

**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.

You can look at https://www.springer.com/gb/book/9783030302931 for more details of the background.

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

- First Supervisor Name: Prof. Tomasz Brzezinski
- Email Address: t.brzezinski@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/t.brzezinski/

**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**

- First Supervisor Name: Prof. Tomasz Brzezinski
- Email Address: t.brzezinski@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/t.brzezinski/

**Project Description**: Applications of the Yang-Baxter equation range from the description of physical forces of nature (quantum field theory, integrable Hamiltonian systems in particle and statistical physics), through the 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 5 - Project Title: New approaches to topological data analysis**

- First Supervisor Name: Pawel Dlotko
- Email Address: p.t.dlotko@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/dlotkopt

**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**

- First Supervisor Name: Dmitri Finkelshtein
- Email Address: d.l.finkelshtein@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/d.l.finkelshtein/

**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**

- First Supervisor Name: Grigory Garkusha
- Email Address: g.garkusha@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/g.garkusha/

**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 originated in algebraic topology. This project will investigate the categorical aspects of various triangulated categories of motives.

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

- First Supervisor Name: Jeffrey Giansiracusa
- Email Address: j.h.giansiracusa@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/j.h.giansiracusa/

**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**

- First Supervisor Name: Jeffrey Giansiracusa
- Email Address: j.h.giansiracusa@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/j.h.giansiracusa/

**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 **

- First Supervisor Name: Eugene Lytvynov
- Email Address: e.lytvynov@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/e.lytvynov/

**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**

- First Supervisor Name: Eugene Lytvynov
- Email Address: e.lytvynov@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/e.lytvynov/

**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**

- First Supervisor Name: Vitaly Moroz
- Email Address: v.moroz@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/v.moroz/

**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 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**

- First Supervisor Name: Dr Zeev Sobol
- Email Address: z.sobol@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/z.sobol/

**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**

- First Supervisor Name: Nelly Villamizar
- Email Address: N.Y.Villamizar@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/villamizarny/

**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 **

- First Supervisor Name: Jiang-Lun Wu
- Email Address: j.l.wu@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/j.l.wu/

**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**

- First Supervisor Name: Chenggui Yuan
- Email Address: c.yuan@swansea.ac.uk
- Link to Staff Profile: http://www.swansea.ac.uk/staff/science/maths/c.yuan/

**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 18 - Project Title: Mathematical and Computational Modelling of Brain Evolution, Development and Disease**

- First supervisor: Dr Noemi Picco
- Email Address: noemi.picco@swansea.ac.uk
- Link to Staff Profile: https://mathematicalmedicine.swansea.ac.uk/people/noemi/

The developmental program leading to the formation of the brain is the result of the complex regulation of cellular processes in space and time. To date, brain development has been studied through the analysis of sparse temporal data that may miss crucial information. The project aims to develop novel mathematical and computational approaches that account for both the spatial and temporal aspects of this process leading to the vast array of brain architectures, shapes and sizes that we see in different animal species. Using analysis of dynamical systems and individual-based models, the project aims to map all the possible evolutionary pathways, to highlight the evolutionary principles and fundamental mechanisms of normal brain development shared across species, and to provide insight into disease and malformations.

**DTC MATH 19 - Project title: Stochastic Population Dynamics**

- Email address: c.yuan@swansea.ac.uk
- Link to Staff Profile: https://www.swansea.ac.uk/staff/science/maths/c.yuan/

Project Description: Populations dynamics is a very important topic in our life. In order to better capture the properties of the real population, in this project, we shall establish stochastic population models and study the properties of these models. By using the simulations, we shall give the evidence that our models are more realistic.