Even as a child, I told myself that I didn't want to live like everyone else - from morning to afternoon at work, at the machine. I wanted something more. Albert Einstein died just then (in 1955), and I really wanted to understand what he actually discovered, how he came to it. The physics teacher told me that there was no chance of that, that I had to learn many things to understand it. I said to myself, why not? I started to be interested in physics and mathematics.

... a wonderful man and pedagogue. He had a habit that when he took a student to answer, he would put their notebook on the table. When the student gave wrong answers or could not answer at all, he would gradually move the notebook towards the edge of the table until it fell on the floor. It looked quite colourful and dramatic. As far as I remember, my notebook never lay on the floor. After a while, the teacher did not demand anything from me or my brother, we did not even have to do homework. When there was a test, he asked us to leave the room and not to disturb our classmates. Everyone understood this after our successes in the Olympiads. Then we had time to do homework from another subject or prepare for the next lesson, for example Polish.

When we went to university, we promised ourselves that we wouldn't compete with each other, so each of us was to focus on a different field. First, we wanted to sort this out by drawing lots. We decided that each of us would write on a piece of paper what we wanted to be and then we'd see. Of course, it turned out that Henryk and I both wrote that we wanted to be mathematicians. Henryk wanted to work in number theory from the beginning, while I was more interested in the practical application of mathematics. Although at first I was interested in logic.

The authors discuss geometric properties of systems of nonlinear elliptic equations that were introduced by M A Lavrentev. The main ideas are outlined. Detailed proofs will be published elsewhere. It is shown that one part of Lavrentev's definition of strong ellipticity can be derived from the assumption that the system can be solved (globally) for two of the eight variables. A corollary of the main result is that strong ellipticity is invariant under smooth diffeomorphisms. The strongly elliptic system is shown to be essentially equivalent to a non-linear equation of Beltrami type that induces a quasiconformal one-to-one mapping with uniformly bounded dilatation.

A mapping theorem, the analogue of Riemann's conformal mapping theorem, for solutions of strongly elliptic systems was discussed by Lavrentev under the title "the general problem of the theory of quasiconformal mappings". The present authors extend this result under less restrictive assumptions. The nonlinear Beltrami system is transformed into a nonlinear integral equation. The methods indicated for the solution of the integral equation include the Schauder fixed point theorem, the Calderón-Zygmund theorem on singular integral operators and various a priori estimates of the theory of quasiconformal mappings

This article grew out of my dual interest in partial differential equations and quasiconformal mappings. It can be considered as complementary to the lecture of Jussi Väisälä at the Helsinki Congress where direct geometric methods of quasiregular theory were stressed. As a whole, quasiconformal theory develops into a branch of geometric multidimensional analysis with rather broad connections.

There is no point in hiding the fact that I went for the money. At that time, my salary as an assistant professor in Poland was 12 dollars a month.

We express our gratitude to Professor F W Gehring for his constant guidance, insight, and encouragement while we were working on this paper.

I wish to express my sincere thanks to the members of the Courant Institute, especially to Professor Louis Nirenberg, for generous hospitality during the time when this research was done.

Every year since 1990, he has spent several weeks at the University of Naples conducting research; co-authoring articles; and mentoring junior faculty, postdoctoral scholars and graduate students.

This paper arose from a discussion sparked between the authors after the lecture of Louis Nirenberg at the Conference in Naples on 1 June 1995.

In Finland, Iwaniec worked on several different projects related to mathematical analysis. One of them involved Kari Astala, István Prause and Eero Saksman, all prominent professors at the University of Helsinki. Their studies revolved around an age-old problem concerning the Beurling Transform, an integral part of complex analysis and geometric function theory. Saksman, an expert in probabilistic methods in analysis, considers Iwaniec a fount of knowledge. "Tadeusz is always coming up with new ideas, which he scrutinises for hours - sometimes in the middle of the night - before writing them down," he says, adding that Iwaniec leaves "no stone unturned. This makes working with him a pleasure. Ph.D. students and postdocs, in particular, marvel at how nice and approachable he is." Prause echoes these sentiments, adding that Iwaniec, an avowed college basketball fan, is no shrinking violet. "Tadeusz always brought a bit of March Madness to Finland," says Prause, who specialises in harmonic analysis. "We'd often argue at lunch about crucial plays involving the Syracuse University [men's basketball] team. It was fun seeing that side of him."

Another project of Iwaniec's took place at the Finnish Centre of Excellence in Inverse Problems Research. It was there that he interacted with a team of experts on Electrical Impedance Tomography (EIT), a non-invasive imaging technique that helps doctors visualise human organs and body parts, in hopes of better understanding their physiological and anatomical make-up. Iwaniec, who is fluent in the mathematics involving EIT, compares the technique to an electromagnetic wormhole - sort of an invisible tunnel between two points in space. "EIT allows us to infer parts of the body, using surface electrical measurements," he says, drawing comparisons to the "invisibility cloak" and other metamaterial devices. "EIT has major implications for the clinical diagnosis of illness and disease. It's very exciting."

Today I deal with the application of the calculus of variations to mathematical models of the theory of elasticity. Stanisław Zaremba's method is the basis for proofs of the existence of so-called hyperelastic mappings or deformations, i.e. transformations with the lowest energy. Using the method of the calculus of variations we can predict where cracks will occur in elastic materials and how these cracks propagate in these materials. These phenomena are a strong motivation, not only in theoretical mathematics, but also in engineering research. In mathematics there is no end to new questions, which are in effect the key to progress in theoretical science and in applications ...

Mathematicians, like me, have the privilege to enjoy the ingenious ideas and splendid theories imagined and brilliantly developed by previous mathematicians. Their beauty inspires us to ask questions to create our own little theory with grace and prospective applications. And there is never an end to new questions, which in effect is the key to advances in mathematics. But advances come after hours and hours of intense work, trapping and holding our attention for years. This can be a dream, sometimes immense pleasure, sometimes a breath-taking moment when we spot the underlying ideas that are actually relevant to our aspirations. Genuine mathematics does not abide in complexity but, contrary to what one might think, somewhere in the unlimited beauty of applications of sophisticated ideas.