4th year projects 2016 (more information via email to Stefan Adams):
(1.) Large deviations for random walks or Brownian motions (deviation from law of large numbers & connection with entropy): several projects linked to MA4L3.
(2.) Large deviations for random integer sequences and their minimisers - an application from MA4L3.
(3.) Large deviations for random partitions and teh scaled shape limits of Young tableaux - an application from MA4L3.
(4.) Isomorphism theorems for loop measures (on a graph or in $ \R^d$ defined via random walks where a loop is a closed path (finite time horizon)).
(5.) Percolation models (Delaunay graphs, Voronoi graphs, triangle lattice; Euclidean plane) and their critical regime.
(6.) Neural networks
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-standrad CLTs)
(3.) Percolation for interacting Voronoi fields in the Euclidean plane and conformal invariance
(4.) Gaussian free field (discrete) and its scaling to the continuous version
PhD projects (list of examples of possible directions/questions):
(I) Renormalisation group theory via infinite dimensional Laplace integral methods
(II) Large deviations for multi-dimensional random walks under pinning constraints and their percolation behaviour
(III) Gaussian Free Field (real and complex) and Bosonic loop measures
(IV) Continuum percolation for Delaunay models in $ \R^2 $ and other random graph model in two dimensions
(V) Breaking of continuous symmetries - mathematical approach to crystallisation
Current PhD students:
Matthew Dickson (MASDOC) (September 2015- ) - Bose-Einstein condensation phenomena and scaling limits for a class of mean-field models
Quirin Vogel (MASDOC) (September 2016 - ) - Space-time random walks, scaling limits and Bosonic loop measures
Former PhD students:
Owen Daniel (September 2011-September 2015) - Bosonic Loop Soups and Their Occupation Fields
Alexander Kister (September 2015) - 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