Photo of David Rule

David Rule

Associate Professor, Docent


I am fortunate enough to be able to split my time between what might be seen as the three main tasks of an academic. As a researcher, I am interested in harmonic analysis and its connection to partial differential equations. As a teacher, I am interested in the realisation of mathematics in the mind. And in my union work, I work to preserve and strengthen the role universities play in a healthy democratic society.

Harmonic Analysis

Harmonic analysis is the study of functions, which may represent a variety of physical quantities such as fluids or fields, as the superposition of waves. Central to this analysis is the Fourier transform, which translates a function representing a physical quantity into its constituent frequencies. It also encompasses other ideas, such as the use of averages in the form of so-called maximal functions.

These mathematical tools are developments as extensions of concrete physical processes, such as wave propagation and diffusion, and as such the area has many connections and applications in physics and the modelling of a diverse range of processes.

My main interests are in studying the properties of various operators which arise from solving various partial differential equations, together with the direct study of solutions to these equations.

My work so far has concentrated on study of pseudodifferential operators, Fourier integral operators and oscillatory integral operators, which are closely connected to hyperbolic partial differential equations and also the more direct study of elliptic partial differential equations.


The term learning covers a wide range of processes that humans do, both as individuals and in social settings. It is therefore a central part of a teacher's role to be aware of what kind of learning we want to foster and provide the best environment and tools for a student to do that. Every kind of learning, from rote learning to research, involves incredibly complex and varied biological processes, so we rely on simplified models and accumulated experience to develop an understanding of how best to teach. It is unreasonable to expect definitive answers to pedagogical questions and the right approach will always depend on context, but reflection, self-criticism and even self-doubt are almost certainly preconditions for development as a teacher.

As well as being a mathematics teacher, I also work part-time for Didacticum, a centre at the university which works to improve methods and expertise in teaching and education at the university.

As far as the teaching of mathematics is concerned, the beauty of mathematics is partly found in its power of reasoning and problem solving. The concepts of prove and communication in mathematics are, therefore, central and should always be taught on some level. It is the mathematicians' ability to reason that makes the subject so powerful and makes mathematics a useful subject in so many areas of life and work. It is nevertheless a very unnatural subject for us humans to learn. Just like all other animals, we are fundamentally intuitive creatures rather than logical. It is often a misunderstanding of how this ability can be nurtured, or the belief that it cannot be nurtured at all, that means mathematics is often perceived as a hard subject. One of my main hopes as a teacher is to dispel that myth.


The most obvious way in which I contribute to the collegial life of the university is through my work for SULF  the Swedish Association of University Teachers and Researchers – and Saco – the negotiating body where SULF is represented. I am chairman for the Linköping branch of SULF and am a deputy member of the national board. I also am a representative on Linköping's Saco-S council.

Collegiality enables a university to fulfil its role in a democratic society. Beyond advancing technology and training students, a well-functioning university should test ideas and understanding, inform public debate and speak truth to power. The professional independence of academics is crucial to being able to carry out those tasks. SULF works to improve the working conditions and professional standing of academics and thus enable them to better contribute to an active democracy. Unions' role in academic and work life is an important part of the Nordic economic and social model.

For more information about me and my research please visit my personal webpage



Aksel Bergfeldt, Salvador Rodriguez-Lopez, David Rule, Wolfgang Staubach (2023) MULTILINEAR OSCILLATORY INTEGRALS AND ESTIMATES FOR COUPLED SYSTEMS OF DISPERSIVE PDES Transactions of the American Mathematical Society Continue to DOI


David Rule (2021) Boundedness of multilinear oscillatory integral operators in local Hardy spaces MATRIX-MFO Tandem Workshop/Small Collaboration: Rough Wave Equations, p. 9-10 Continue to DOI
David Rule (2021) Ge svar på tal: en introduktion till matematisk analys
Salvador Rodriguez-Lopez, David Rule, Wolfgang Staubach (2021) Global boundedness of a class of multilinear Fourier integral operators Forum of Mathematics, Sigma, Vol. 9, Article e14 Continue to DOI


Arpan Ghosh, Vladimir Kozlov, Sergey Nazarov, David Rule (2018) A TWO-DIMENSIONAL MODEL OF THE THIN LAMINAR WALL OF A CURVILINEAR FLEXIBLE PIPE Quarterly Journal of Mechanics and Applied Mathematics, Vol. 71, p. 349-367 Continue to DOI


I currently teach the following courses.

  • Introductory Course in Calculus, TATB04
  • Vector Calculus, TATA44
  • Becoming a Teacher in Higher Education, Didacticum


Ge svar på tal

Faced with a proposed cut in the mathematics core requirements from the College of the University of Chicago, mathematics education professor, Paul Sally, retorted that you would not teach Shakespeare's Hamlet in three acts instead of five. 

The book: Answers to numbers by David Rule.Trustees of the British Museum/Jens Martin

This argument was accepted by the College and the cut avoided, but it begs the question, what is a whole play in mathematics? What story are we trying to tell?

Just as the best stories teach us about life, the best university courses equip us with tools we can apply well beyond the university's walls. My book is an alternative to the theory/application dichotomy, which often see theorems and proofs black-boxed and an emphasis placed on calculating numerical answers to standard questions, in an attempt to make mathematics more 'relevant'. Instead I place reason centre stage, encourage discussions about motivation and proof, and talk explicitly about the communication of mathematics.

Powers are a recurring theme in the book. They start as a useful notation for repeated multiplication, then play an important role revealing irrational numbers, and finally connect to logarithms and trigonometry. Above all, I hope that the book shows students what a beautiful subject mathematics can be, and that we can all be mathematicians.

Find out more information about the book on the publisher's website Studentlitteratur.  


  • Docent, Linköping University 2016
  • PhD in Mathematics, University of Chicago, 2007
  • MSc in Mathematics, University of Chicago, 2003
  • MMath in Mathematical Physics, University of Sussex, 2001