**S4B1 - Graduate Seminar- Some topics in renormalization group**

Prof. M. Disertori

**There will be an information meeting on Thursday Feb 6 at 11.00 in room 4.045**

**Description. **Description. The word ”renormalization group” denotes a set of mathematical tools that allow to extract ’effective descriptions’ for measures on many variables, the classical example being

e^{−(x,C-1 x) } exp[- λ Σ_{j} xj^{4} ] Π_{j} dx_{j}

where C^{-1} > 0 is a positive matrix. For λ > 0 the integral is well defined,

but cannot be computed exactly. On the other hand, for small λ we expect

this measure to be well approximated by a Gaussian.

*Question: *assume we use the measure to integrate functions of the form

f = f (x' ), with x' = N^{-1 }Σ_{j} x_{j}

How does the effective (or renormalized) measure dμ(x') look like?

Is it still well approximated by a Gaussian?

To answer this question we integrate out step by step degrees of freedom

(short scales). The strategy is very similar to the analysis of dynamical

systems.

In this seminar we plan to learn some basic notions on different RG schemes.

Possible additional (alternative) topics include: variants of the RG scheme

to stuy long time behavior of solutions for nonlinear parabolic partial dif-

ferential equations and/or perturbations of linear self-adjoint operators on

some Hilbert space.

**Prerequisites**. Functional analysis. Some basic knowledge in probability/physics

may be useful but is not necessary.