# A Theory of Regularity Structures

Wednesday, January 16, 2013 - 11:30am - 12:20pm

Keller 3-180

Martin Hairer (University of Warwick)

The classical way of measuring the regularity of a function is by comparing it

in the neighbourhood of any point with a polynomial of sufficiently high degree.

Would it be possible to replace monomials by functions with less regular behaviour

or even by distributions? It turns out that the answer to this question has

surprisingly far-reaching consequences for building solution theories for semilinear

PDEs with very rough input signals, revisiting the age-old problem of multiplying

distributions of negative order, and understanding renormalisation theory.

As an application, we build the natural Langevin equation associated

with Phi^4 Euclidean quantum field theory in dimension 3.

in the neighbourhood of any point with a polynomial of sufficiently high degree.

Would it be possible to replace monomials by functions with less regular behaviour

or even by distributions? It turns out that the answer to this question has

surprisingly far-reaching consequences for building solution theories for semilinear

PDEs with very rough input signals, revisiting the age-old problem of multiplying

distributions of negative order, and understanding renormalisation theory.

As an application, we build the natural Langevin equation associated

with Phi^4 Euclidean quantum field theory in dimension 3.

MSC Code:

82C31

Keywords: