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Abstracts Lecturer series

David Brydges - An introduction to the renormalisation group for probabilists

Abstract: The $\varphi^{4}$ model on the lattice $Z^{d}$ is a variation on the
Ising model: instead of requiring, for every lattice site $x$, that
the spin $\varphi_{x}$ be plus or minus one, $\varphi_{x}$ is real
valued, but concentrated by a weight $\exp \big[-g (\varphi_{x}^{2}-
1)^{2} \big]$ near $\pm 1$. Similarly models where every spin
$\varphi_{x}$ has values in a sphere $S^{n-1}$ have
$(\|\varphi\|^{2}-1)^{2}$ variants where $\varphi$ has values in
$R^{n}$. There is even a precise mathematical definition of the $n=0$
component model which turns out to be a natural model for
self-repelling walk. My colleagues Roland Bauerschmidt and Gordon
Slade and I have recently used a rigorous version of the Wilson
renormalisation group to prove that, on the four dimensional lattice,
some of the critical exponents of these $\|\varphi\|^{4}$ models have
the precise logarithmic corrections predicted by theoretical physics.
In these lectures I will explain the important ideas and some of the
background in physics using the one component $\varphi^{4}$ model on a
special lattice called the hierarchical lattice as the initial example,
and then passing to the models on $Z^d$.

Martin Hairer - Regularity Structures

Abstract: tba

Antii Kupiainen - Renormalization Group and Stochastic PDE's

Abstract: We develop a Renormalization Group (RG) approach to the study of existence and uniqueness of solutions to stochastic partial differential equations driven by space-time white noise. The idea of the RG is to construct the solution of the SPDE scale by scale in space time. One solves the equation for smallest scales which leads to a "renormalized" equation for the remaining scales. Repeating this procedure one gets a sequence of effective equations on successive space time scales which can be viewed as a dynamical system in the "space of equations". In order to control this flow of equations one needs to renormalize some of the parameters in the original equation i.e. to add terms that diverge as a short scale regularization is removed. As examples we discuss the KPZ equation, the \phi^4_3 model and the Sine-Gordon model.

Gordon Slade - Recent applications of the renormalisation group to critical phenomena

Abstract: We discuss recent results obtained using the renormalisation group method. The results include proofs of existence of logarithmic factors in the critical behaviour of 4-dimensional multi-component spin models and weakly self-avoiding walk, and were obtained in collaborations with David Brydges, Roland Bauerschmdit, and Alexandre Tomberg. This provides a context for the course of David Brydges, which will explain some of the ideas in the renormalisation group method used to prove these results.