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 xj⁴ ] 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.