Svenska matematikersamfundets höstmöte, 2014

For more than two centuries, partial differential equations have been an indispensable tool for scientists and engineers for modeling and understanding a wide range of physical phenomena. The solution of a partial differential equation is often a very challenging and time-consuming task, involving mathematical analysis, numerical analysis, and the development of sophisticated computer programs.

In this talk, I demonstrate how the solution process can be completely automated, to the point where complicated systems of nonlinear partial differential equations can be solved with ease in a web browser.

This work is a joint effort between research groups at Chalmers, University of Chicago, University of Cambridge, KTH and Simula Research Laboratory (in order of appearance) as part of the FEniCS Project (http://fenicsproject.org), a collaborative project for the development of innovative concepts and tools for automated computational modeling, with a particular focus on automated solution of differential equations by finite element methods, based on domain specific languages, code generation, and high-performance parallel computing.

The resolution of the Kato square root conjecture in 2002 by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian was a momentous effort that brought together techniques from functional calculus and harmonic analysis. This problem, along with many other problems in harmonic analysis, was considered from a first-order perspective in the seminal work of Axelsson (Rosén), Keith and McIntosh in 2005. Here, the Kato Square Root problem on manifolds was considered for the first time and resolved in compact case. This was followed by the resolution of this problem on Euclidean submanifolds by Morris in 2010. We will present some recent results where this problem is resolved in the noncompact setting and where no embedding is assumed. Instead, we will show that the solution only makes assumptions on intrinsic curvature which signifies progress towards understanding the core geometric features of this problem. Furthermore, we will describe some geometric consequences, including applications to the study of manifolds with low-regularity metrics and possible connections to curvature flows.

A derivation on an algebra is a linear operator satisfying the Leibniz rule, i.e. the familiar product rule for derivatives in elementary calculus. The set of all derivations of an algebra forms a Lie algebra. By viewing the algebra as a module over its Lie algebra of derivations, one can use representation theory to understand its structure, in particular if the Lie algebra is in some sense large. Work in this direction has been done for division algebras and for composition algebras, two well-studied families of algebras.

Over the real numbers, the finite-dimensional algebras which are both composition algebras and division algebras are precisely those which are absolute valued, i.e. endowed with a multiplicative norm. These only exist in dimensions 1, 2, 4 and 8, and in dimension 8 their classification problem is far from being solved.

In this talk we give an introduction to the subject, and describe an approach to finite dimensional absolute valued algebras whose derivation algebra is non-abelian. Our approach in fact solves their classification problem by systematically reducing most of it to the study of actions of rotation groups on 3-spheres.

What are the dynamics of general singular solutions to the Einstein equations near singularities? Characterizing such behavior is one of the main outstanding problems in classical general relativity. In this talk I will present so called Fuchsian methods for obtaining solutions to singular symmetric hyperbolic PDE. These methods allow us to prove the existence of large familes of singular solutions to the Einstein equations. As a product of the method we also obtain information on the dynamics near the singularity.

Hausdorffdimensionen är en generalisering av dimensionsbegreppet från mängder som kurvor och ytor till godtyckliga delmängder av R^d. Iden är att dimensionen av en mängd ska mäta hur mängdens volym ändras när mängden skalas. Ett sätt att visa en undre begränsning av hausdorffdimensionen av en mängd är att hitta ett mått som är koncentrerat på mängden och vars fouriertransform avtar polynomiellt. Detta ger upphov till den så kallade fourierdimensionen, vilken mäter den optimala exponenten för avtagandet som går att uppnå. Även om fourierdimensionen heter "dimension" så saknar den många egenskaper som man vanligtvis förknippar med dimensioner, t.ex. är den inte stabil under ändliga unioner.

I will survey recent research on the TASEP (totally asymmetric simple exclusion process) on a ring with several classes of particles. This is a Markov chain on all permutations of length n where at each step we sort the elements at positions i and i+1 modulo n (putting the smaller letter at i). The stationary distribution of this process has been studied both from combinatorial and physical viewpoints. There is an elegant description of the stationary distribution in terms of so-called multiline queues, discovered by Ferrari and Martin.