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
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HT 2016