# Students

**(MMath) 4th year projects 2022/23** (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
- Large deviations for random integer sequences and their minimisers - an application from MA4L3.
- Large deviations for random partitions and the scaled shape limits of Young tableaux - an application from MA4L3.
- Neural networks and machine learning
- Application in mathematical finance - martingales; Large deviations; Wasserstein gradient flows
- High-Dimensional probability - MA3K0 - concentration inequalities
- Central limit theorems for random walks

**Two examples MMath projects from 2022:**

**Large deviations for symmetries Brownian bridges**(*large system of Brownian motions under symmetrised initial-terminal conditions; empirical path measures; Donker-Varadhan rate function*)- B
**ose-Einstein Condensation on the Complete Graph**(*Markov processes on graphs; Feynman-Kac formulae; loop measures; isomorphism theorem for local times; and an application to Bose-Einstein condensation*)

**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 2 and MA3K0 (High-Dimensional Probability) in term 1. 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) Interacting Brownian motions and applications

(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*