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(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

Ph.D. projects (list of examples of possible directions/questions):

(I) Renormalisation group theory & regularity structure (nonlinear SPDEs) via infinite-dimensional Laplace integral methods

(II) Large deviations for multi-dimensional random walks under pinning constraints and their multiple stochastic limits

(III) Gaussian Free Field (real and complex) and loop measures

(IV) Space-time random walks for quantum statistical mechanics

(V) Concentration inequalities for dependent random variables - applications in data science and machine learning


Current Ph.D. students:

Spyridion Garouniatis (August 2021-) - Scattering theory for many Brownian motions

Andreas Koller (September 2020 -) - Renormalisation (multi-scale) for massless models and scaling limits to Gaussian Free Fields

Sotirios Kotitsas (September 2020 -) - Nonlinear scaling limits and machine learning

Alberto Cassar (September 2019 -) - Large deviation theory analysis for integrated random walks (Laplacian) wetting/pinning models

Jason Ly (September 2019 -) - Interacting Brownian motion with symmetrised initial-terminal conditions

Former PhD students:

Shannon Horrigan (September 2020) -Continuum random cluster and Potts with Delaunay interactions

Quirin Vogel (August 2020 ) (now Postdoc at NYU Shanghai) - Geometric properties of random walk loop soups

Matthew Dickson (September 2019) (now Postdoc at LMU Munich) - Interacting Boson Gases and Large Deviation Principles

Owen Daniel (September 2019) - Bosonic Loop Soups and Their Occupation Fields

Alexander Kister (September 2019) - Sample path large deviations for the Laplacian model with pinning interaction in (1 + 1)-dimension

Michael Eyers (January 2015) - On Delaunay Random Cluster Models

William Nollett (December 2013) - Phase transitions and the random-cluster representation for Delaunay Potts models with geometry-dependent interactions