Description

My research covers many areas of applied mathematics and theoretical physics, driven by an unhealthy interest in anything random and unpredictable: noisy dynamical systems, turbulence, coin tossing, finance, the weather, etc.

I'm trained as a physicist but now work mostly at the interface of probability theory, statistics, simulation, optimization, control, and, more recently, machine learning.

My main specialty is the theory of large deviations - a branch of probability theory used to estimate the probability of very rare events arising in random systems as diverse as gases, queues, random walks, information systems or nonequilibrium systems driven by noise and external forces.

I have written a review article on the many applications of large deviation theory in statistical physics. I also have lecture notes and a webpage on the subject.

The following are projects I'm currently working on:

• Understanding how rare events arise in Markov processes (fluctuation dynamics)
  
• Simulation of rare events in Markov processes (splitting and importance sampling)
• Phase transitions in the fluctuations of Markov processes
• Feature and rare event detection in financial time series using machine learning
• Stochastic processes (random walks) on random graphs
• Large scale equivalence of random graphs
  
• Markov processes with reset
• Large-scale equivalence of statistical models (ensemble equivalence problem)
• Applications of large deviation theory in nonequilibrium physics

In previous years, I also worked on Brownian motion with solid friction, studied the physics of information in feedback control, and got distracted by football graphs.

Possible student projects

Suitable for all levels: Honors, MSc, PhD and post-doc. Interested students are encouraged to contact me for more information or to discuss other projects.

Fluctuation dynamics and process conditioning

The study of how fluctuations arise in a stochastic process - e.g., how a queue overflows or how a financial market crashes - is fundamental for predicting and controlling their behavior. In physics, fluctuations are also known to be connected to the response of systems to perturbations and external forces.

The idea of this project is to see how a Markov process 'builds up' fluctuations in time by conditioning (in a probabilistic sense) that process on observing a given fluctuation and by describing this conditioning as a new Markov process, called the effective or fluctuation process.

• Applications: Brownian motion, random walks, nonequilibrium systems, financial time series, etc.
• Keywords: Brownian bridges, Doob transforms, fluctuation paths, pathways or dynamics, large deviation or rare event conditioning
• Subject areas: Theoretical physics, statistical physics, stochastic processes
  

Large deviations for random walks on graphs

This an application of the previous project focused on using conditioning on rare events to find network characteristics of random graphcs, such as connected components, typical regions, central nodes, etc. A related project is to study escape-type problems of random walks evolving on random graphs in the limit of large graphs.

• Applications: All sorts of graphs arising in physical and manmade applications
• Keywords: Random graphs, complex networks, random walks on graphs
• Subject areas: Random walks, graphs theory, numerical simulations
  

Numerical methods for large deviations

Large deviations are difficult to probe numerically as they are, by definition, rarely seen in direct simulations (think of a meteorite hitting earth). The aim of this project is to develop efficient numerical methods, based on Monte-Carlo simulations, to 'accelerate' the sampling of large deviations or rare events in general in Markov processes. This is an active area in physics, statistics, engineering, and financial mathematics.

• Applications: Fluctuations and response of equilibrium and nonequilibrium systems, any other area/topic where rare events are critical
• Keywords: Monte Carlo algorithms, importance sampling, splitting and cloing methods, control theory
• Subject areas: Stochastic processes, numerical simulations, coding (Python, Matlab, C, C++)

Statistical detection of regime changes in time series

The goal of this project is to study and develop statistical tools to detect regime changes (seasonal changes, bubbles, trends, crashes and other non-stationary features) in time series, with applications in finance. This will combine many tools from mathematics (probability, statistics, stochastic processes), physics (statistical mechanics, time series, forecasting) and programming. Coding will be done in Python.

There is a possibility to do this project with NMRQL Research, a private investment fund located in Stellenbosch that uses machine learning for financial forecasting.

• Applications: Financial time series, trading
• Keywords: Statistical analysis, time series, stationarity
• Subject areas: Mathematical finance, statistics

Dynamical phase transitions

Dynamical phase transitions are sudden changes that arise in the fluctuations of stochastic processes or, more precisely, in the mechanisms that create fluctuations. The goal of this project is to study dynamical phase transitions in Markov processes and, especially, in stochastic differential equations. Analogies with phas transitions arising in physics, e.g., in solids and liquids at a certain critical temperature, are established via large deviation theory.

• Applications: Statistical physics, forecasting
• Keywords: Phase transitions in large deviations
• Subject areas: Statistical physics, Markov processes, large deviations
  

Stochastic processes with reset

Markov processes jumping at random from one state to another (depending on their current state) are used in physics to model equilibrium and nonequilibrium systems perturbed by noise. Recently, there has been a lot of interest on a new class of Markov processes, called reset processes, evolving stochastically for some time and then jumping to a fixed 'reset' point. The goal of this project is to develop the theory of these processes and, here again, their large deviation properies.

• Applications: Nonequilibrium systems, population dynamics, random searches, etc.
• Keywords: Reset processes, processes with catastrophes or killings, absorbing processes
• Subject areas: Stochastic processes, probability theory, statistical physics
  

Collaborators

• Raphael Chetrite, Researcher in physics and mathematics at the University of Nice, France
• Frank den Hollander, Professor in Mathematics, Leiden University, Netherlands
  
• Arnaud Guyader, Professor of Statistics at the University of Paris, France
• Grégoire Ferré, PhD student at the CERMICS, École des Ponts of Paris, France
• Daniel Nickelsen, Post-doc at NITheP, Stellenbosc
  
• Rosemary J. Harris, Reader in Applied Mathematics at Queen Mary University of London, UK
• Francesco Coghi, Former visiting student at NITheP, now PhD student at Queen Mary University
  
• Pelerine Tsobgni, Former PhD student at Stellenbosch University
• Janusz Meylahn, Former MSc student at Stellenbosch University, now PhD student at Leiden University
• Sanjib Sabhapandit, Researcher at the Raman Institute, India
• Chris Rohwer, Former post-doc at NITheP Stellenbosch, now post-doc in Stuttgart
  
• Javier López Peña, University College London, UK. Now at Kickdex, a data football company