Doing a Phd at Heriot Watt and the Maxwell Institute
Heriot Watt and university of Edinburgh together form the Maxwell Institute, a recognised center of excellence in the Mathematical Sciences.
The Maxwell Institute hosts a thriving graduate community.
If you are interested in knowing more about the Maxwell Institute Graduate School and our training programmes
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I am looking for two PhD students to start in September 2025. One in the field of Mathematical Ecology (project joint with CEH), the other in Applied Stochastic analysis.
Below the description of the project with CEH and some sample projects in stochastic analysis. All these positions are fully funded.
PhD project in Mathematical Ecology - joint with the Center for Ecology and Hydrology (CEH)
Landscape management and the pace of nature recovery: multiscale modelling and simulation .
Human activity has a tremendous impact on shaping our landscapes and the habitats within them.
If the rate of landscape change is too fast, for example as a consequence of changing management strategies, species may not be able to adapt, with consequent biodiversity loss.
In this project we will consider this issue for structured, inhomogeneous landscapes, where different patches of land may be used for different purposes
(for example mixed rewilding/agricultural). From a mathematical perspective this will entail the use of (stochastic)
multiscale modelling and analysis, on spatially inhomogeneous models.
The project will be co-supervised by Prof. Michela Ottobre from Heriot Watt, expert in multiscale modelling, and Prof. Christina Cobbold from University of Glasgow,
who is world leading in ecological modelling. Moreover, the project will be in collaboration with Dr. Emma Gardner from the UK Center for Ecology and Hydrology.
Emma Gardner is a leading ecologist and a bridge between research and decision makers. To apply for this project see here .
Sample PhD projects in Stochastic Analysis
The descriptions below should give you an idea of what I work on, but those below are just sample projects. If you are interested in the general area get in touch,
I am happy to discuss more.
The ideal applicant would have a keen interest in developing theory which is motivated by applications.
- Multiscale methods and Multiscale Interacting particle systems
Context: Many systems of interest in the applied sciences share the common feature of possessing multiple scales, either in time or in space, or both.
Some approaches to modelling focus on one scale and incorporate the effect of other scales (e.g. smaller scales) through constitutive relations, which are often obtained empirically. Multiscale modelling approaches are built on the ambition of treating both scales at the same time, with the aim of deriving (rather than empirically obtaining) efficient coarse grained models which incorporate the effects of the smaller/faster scales. Multiscale methods have been tremendously successful in applications, as they provide both underpinning for numerics/simulation algorithms and modelling paradigms in an impressive range of fields, such as engineering, material science, mathematical biology, climate modelling (notably playing a central role in Hasselmann’s programme, where climate/ whether are seen as slow/fast dynamics, respectively), to mention just a few.
More detail. In this project, which is in the field of applied stochastic analysis, we will consider systems that are multiscale in time, with particular reference to multiscale interacting particle systems. We will try to understand how the multiscale approximation interacts with the mean field approximation (produced by letting the number of particles in the systems to infinity so to obtain a PDE for the evolution of the density of the particles). In the context we will consider the systems at hand will have two scales, so called fast and slow scale, each of them modelled by appropriate stochastic differential equations. Classical multiscale paradigms consider the setting in which the fast scale has a unique invariant measure (equilibrium). We will consider the often more realistic scenario in which the fast process has multiple invariant measures. The motivation for this project comes especially from models in mathematical biology, but the applicability of the framework we will investigate is broader. The candidate that will work on this project will be open to investigate both theoretical and modelling aspects – though developing their own preference in time.
For more detail on the project and to apply to work on this topic see here
- Interacting Particle systems and Stochastic Partial Differential Equations
This project belongs to the broad field of applied stochastic analysis.
Context. Many systems of interest consist of a large number of particles or agents, (e.g. individuals, animals, cells, robots) that interact with each other. When the number of agents/particles in the system is very large the dynamics of the full Particle System (PS) can be rather complex and expensive to simulate; moreover, one is quite often more interested in the collective behaviour of the system rather than in its detailed description (e.g. bird flocking). In this context, the established methodology in statistical mechanics and kinetic theory is to look for simplified models that retain relevant characteristics of the original PS by letting the number N of particles to infinity (so called mean field limit); the resulting limiting equation for the density of particles is a low dimensional, (in contrast with the initial high dimensional PS) non-linear partial differential equation (PDE), where the non-linearity has a specific structure, commonly referred to as a McKean-Vlasov nonlinearity. Beyond an intrinsic theoretical interest, such models were proposed with the intent to efficiently direct human traffic, to optimize evacuation times, to study rating systems, opinion formation, etc; and in all these fields they have been incredibly successful.
More detail. In this project we will consider PSs modelled by Stochastic Differential Equations (SDEs) whose limiting behaviour is described by either a deterministic PDE or a stochastic PDE (SPDE). It is indeed important to notice that, depending on the nature of the stochasticity in the PS, the limiting equation can be either a deterministic PDE – and this would be the most classical framework – or a stochastic PDE. Either way, the limiting equation is of McKean-Vlasov type. The overall aim of the project is the comparison of the ergodic and dynamic properties of the particle system and of the limiting PDE/SPDE. These results will help inform modelling decisions for practitioners. From a theoretical standpoint, one of the purposes of this project will be to push forward the ergodic theory for SPDEs.
Keywords for this project are: Stochastic (Partial) Differential equations, McKean Vlasov evolutions, ergodic theory, mean field limits.
Prerequisites: good background in either stochastic analysis/probability or analysis
For more detail on the project and to apply to work on this topic see here