and probability theory. In particular these are:

- Bose-Einstein condensation and spatial random permutations
- Study of particles coupled to a quantized field, and
functional integrals

- Systems of interacting Brownian motions, in
particular in

connection with modelling molecular chains - Quantum dynamics of molecules, in particular
dynamics at

avoided crossings of energy bands

The problem of Bose-Einstein condensation can be reformulated in

terms of a model of random spatial permutations. The phase transition

of the quantum system then corresponds to the emergence of infinite

cycles in the permutations. While the model corresponding to the

full interacting Bose gas is still too difficult to treat, several natural

simplifications are of independent mathematical interest and exhibit

the same phase transition behaviour. Together with Daniel Ueltschi,

I am investigating these models, with the ultimate goal of understanding

the phase transition in the fully interacting Bose gas.

Spatial random permutations are, however, very interesting objects in

their own right. In suitable parameter regimes, they have connections to

curve shortening flow, via a close connection to the low temperature Ising

model on the one hand, but also more generally. In a different limiting

regime and in two space dimensions, they are closely related to the double

dimer model, and I expect fractal or even SLE properties due to this connection.

For a bit more information and some nice pictures see the picture gallery!

Relevant publications: [18], [15], [14], [13], [8].

This is the topic that I started with for my PhD. There, I investigated

the Nelson model using the representation where the imaginary time

is propagator of the quantum field is an infinite dimensional Ornstein-

Uhlenbeck process. In collaboration with Jozsef Lorinczi, Fumio Hiroshima,

Robert Minlos and Herbert Spohn, I investigated the structure of the

emerging Gibbs measures relative to Brownian motion, and used them

to study ground state properties of the quantum model. More recently,

I have been interested in the quantum statistical mechanics version of

the model. In a joint work with Domenico Castrigiano, we study in

detail the effective density of states of a quantum oscillator in contact

with a radiation field. In the book [A], we give a systematic account of the

theory.

Relevant publications: [A], [16], [7], [4], [3], [2], [1].

A standard method in molecular dynamics to take into account

thermal fluctuations is to model the particles as interacting Brownian

motions. Together with my student Michale Allman, we were interested

in what this method yields when taken seriously and applied to a

chain (e.g. a polymer) under strain. One natural question is where

the chain breaks when different it is pulled apart, depending on the

speed of pulling and the level of the noise. Together with Martin Hairer,

we extended these investigations to the case where the particles have

mass and are modelled by the Langevin equation.

Relevant publications: [19], [10].

Non-adiabatic (or, radiationless) transitions of molecules are one of

the most important but also most difficult problems in quantum chemistry,

due to the breakdown of the Born-Oppenheimer approximation. Together

with Ben Goddard and Stefan Teufel, I try to understand the problem in

the context of exponential asymptotics, using superadiabatic representations.

Relevant publications: [20], [12], [11], [9], [6], [5].