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Louis Nirenberg

I just heard the news that Louis Nirenberg died a few days ago, aged 94.  Nirenberg made a vast number of contributions to analysis and PDE (and his work has come up repeatedly on my own blog); I wrote about his beautiful moving planes argument with Gidas and Ni to establish symmetry of ground states […]

Spielman, Meyer, Nirenberg

In my previous post, I briefly discussed the work of the four Fields medalists of 2010 (Lindenstrauss, Ngo, Smirnov, and Villani). In this post I will discuss the work of Dan Spielman (winner of the Nevanlinna prize), Yves Meyer (winner of the Gauss prize), and Louis Nirenberg (winner of the Chern medal). Again by chance, […]

Quantitative bounds for critically bounded solutions to the Navier-Stokes equations

I’ve just uploaded to the arXiv my paper “Quantitative bounds for critically bounded solutions to the Navier-Stokes equations“, submitted to the proceedings of the Linde Hall Inaugural Math Symposium. (I unfortunately had to cancel my physical attendance at this symposium for personal reasons, but was still able to contribute to the proceedings.) In recent years […]

254A, Notes 2: Weak solutions of the Navier-Stokes equations

In the previous set of notes we developed a theory of “strong” solutions to the Navier-Stokes equations. This theory, based around viewing the Navier-Stokes equations as a perturbation of the linear heat equation, has many attractive features: solutions exist locally, are unique, depend continuously on the initial data, have a high degree of regularity, can […]

Yves Meyer wins the 2017 Abel Prize

Just a short post to note that Norwegian Academy of Science and Letters has just announced that the 2017 Abel prize has been awarded to Yves Meyer, “for his pivotal role in the development of the mathematical theory of wavelets”.  The actual prize ceremony will be at Oslo in May. I am actually in Oslo […]

A mathematical formalisation of dimensional analysis

Mathematicians study a variety of different mathematical structures, but perhaps the structures that are most commonly associated with mathematics are the number systems, such as the integers or the real numbers . Indeed, the use of number systems is so closely identified with the practice of mathematics that one sometimes forgets that it is possible […]

Some notes on Weyl quantisation

One of the basic problems in the field of operator algebras is to develop a functional calculus for either a single operator , or a collection of operators. These operators could in principle act on any function space, but typically one either considers complex matrices (which act on a complex finite dimensional space), or operators […]

Asymptotic decay for a one-dimensional nonlinear wave equation

Hans Lindblad and I have just uploaded to the arXiv our joint paper “Asymptotic decay for a one-dimensional nonlinear wave equation“, submitted to Analysis & PDE.  This paper, to our knowledge, is the first paper to analyse the asymptotic behaviour of the one-dimensional defocusing nonlinear wave equation (1) where is the solution and is a […]

Lindenstrauss, Ngo, Smirnov, Villani

As is now widely reported, the Fields medals for 2010 have been awarded to Elon Lindenstrauss, Ngo Bao Chau, Stas Smirnov, and Cedric Villani. Concurrently, the Nevanlinna prize (for outstanding contributions to mathematical aspects of information science) was awarded to Dan Spielman, the Gauss prize (for outstanding mathematical contributions that have found significant applications outside […]

245C, Notes 4: Sobolev spaces

As discussed in previous notes, a function space norm can be viewed as a means to rigorously quantify various statistics of a function . For instance, the “height” and “width” can be quantified via the norms (and their relatives, such as the Lorentz norms ). Indeed, if is a step function , then the norm […]