(MMath) 4th year projects 2023/24 (more information via email to Stefan Adams):

• Large deviations for random walks or Brownian motions (deviation from the law of large numbers & connection with entropy): several projects linked to MA4L3; MA3K0; MA4F7 ;MA4F7; MA4A7; MA4L2; MA482
• Scaling limits for random fields (CLT type including large deviation and moderate deviation scales)
• Large deviations for random permutations and the shape of Young tableaux - an application of MA4L3.
• Neural networks (statistical mechanics) and machine learning
• Random chaining (data science) and renormalisation (MA3K0; MA4L3)
• Wasserstein gradient flows as stochastic processes
• High-Dimensional probability - MA3K0 - concentration inequalities
• Central limit theorems for random walks
• Statistical Mechanics for Machine Learning (MA4L3)
• Continuum percolation models in dimension $d=2 $
• Large deviation for queuing and communication networks (MA4L3)
• Space-time random walks

Two examples MMath projects from 2023:

• Large deviations for symmetries Brownian bridges (large system of Brownian motions under symmetrised initial-terminal conditions; empirical path measures; Donker-Varadhan rate function)
• Continuum percolation for random Voronoi cells (Dependent percolation; Voroni tessellations; mesoscopic limits)

Projects can be chosen individually and can be tailored to specific interest (ideally discussions start in June or beginning of term or over the summer). I am going to teach MA4L3 (Large deviation theory) in term 1 and MA3K0 (High-Dimensional Probability) in term 2. Both modules can be used aa a base for a project. For MA4L3 I am going to write a new lecture book on 'Large deviation theory' for undergraduate and beginning postgraduate students which will be used as lecture for MA4L3.

MSc projects (list of examples - more via discussion with Stefan Adams):

(1.) Large deviations for integrated random walks (with different applications)

(2.) Scalings limits for cycle distributions (non-standard CLTs)

(3.) Concentration inequalities - MA3K0

(4.) Gaussian free field (discrete) and its scaling to the continuous version

(5.) Permanental point processes