Archive for PDEs

Martin Hairer gets Breakthrough Prize (and $3M)

Posted in Books, University life with tags , , , , , , , , , on September 14, 2020 by xi'an

Just heard the news that Fields Medallist Martin Hairer (formerly U of Warwick) got the 2021 Breakthrough Prize in Mathematics for his unification theory of stochastic partial differential equations, which he likens to a form of Taylor expansion in the massive Inventiones paper describing this breakthrough. (Looking at the previous winners of the prize, who also made its selection committee, this represents a break from focussing primarily on algebraic geometry! If not from sticking to male recipients…)

We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semi-linear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the \Phi^4_ 3 Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of 3-dimensional ferromagnets near their critical temperature.

ignorami rule

Posted in Kids, Travel, University life with tags , , , , , on May 12, 2016 by xi'an

plane2There has already been many blog entries [incl. Andrew’s] on that story of a plane passenger calling security about a neighbour solving differential equations next seat and many jokes will certainly stem from it. My closest encounter with such a passenger was a while ago, when flying to Manchester for a visit to Lancaster, when the man next to me suddenly asked if I was working on particle physics because he would not tolerate it. Or something like this. As I did not want to get arrested upon arrival I refrained from smashing his head into the seat and muttered something indistinct between a curse and a comment that this was statistics, but I now regret I had not confronted this holier-than-thou (to keep polite) attitude! This story also reminds me of another flight, from Montpelier to Paris, when I was discussing ABC with Jean-Michel Marin and Jean-Marie Cornuet, when an AF flight attendant came by and added an x at random in one of my equations! This did not solve the problem but we had a good laugh and did not end up questioned by security!

Anyway, my reaction to this PDE (or is it ODE?!) scandal is of a more sombre tone: I find the fact that airline personal paid any attention to the complaint deeply worrying. Rather that dismissing the worries of this ignorant (or myopic) passenger [and possibly contacting a psychiatrist], they called security and the PDE had to be produced before the economics professor could resume his seat and the flight take off… This incident shows both (i) a trend in irrationality (if associating maths equations with terrorist threat) or ignorance (if confusing maths equation with Arabic writing), not to mention xenophobia and (ii) a readiness of companies and administrations to pester, detain, question and bother anyone with any exotic characteristics. Including solving PDEs or even trying to. [But what can we expect when bottled water or orange marmalade is treated as a potential threat by security checks?] Beside sticking to writing maths in my notebook when I travel, I think I should start signalling to flight attendants truly irrational behaviours of my fellow passengers, like reading newspapers that seem solely concerned by the anatomy of reality TV shows or muttering prayers to a deity at take-off and landing…