**Juha Heinonen**'s parents were Liisa and Vilho Heinonen. He was born in Toivakka, which is a small town about 25 km southeast of the city of Jyväskylä in south-central Finland. Juha's mother single-handedly ran the Toivakka old folks home and it was there that Juha and his sister Maritta were brought up. His father was a lumberjack, an important trade in the heavily forested area, but he was also involved in local politics representing the socialists. While at high school, Heinonen was passionate about athletic pursuits, especially cross-country skiing. At the age of sixteen years he became the National Finnish 5 km Champion in his class. After graduating from high school, Heinonen was conscripted into the Finnish army to serve as an officer for a year.

The first Finnish teachers' training college was founded in Jyväskylä in 1863. In 1934 it became a pedagogical institute that in turn became the University of Jyväskylä in 1966. Heinonen entered this university to study mathematics after serving for a year in the army. He was awarded his Candidate's Degree in 1984 and was appointed as a Research Assistant in the Finnish Academy of Science and Letters while he undertook research at the University of Jyväskylä advised by Olli Martio. In 1985 he spent a semester in the United States as a visiting graduate student at the University of Michigan at Ann Arbor. In 1986 he made a research visit to the USSR Academy of Sciences at Novosibirsk. He completed his thesis on non-linear potential theory in 1987 and in the same year his first publication appeared, namely *Estimates for F-harmonic measures and Oksendal's theorem for quasiconformal mappings* written jointly with his thesis advisor Olli Martio.

Following the award of his doctorate, Heinonen continued to work as a research assistant for the Academy until 1988. He also made a research visit to the Deutsche Forschunsgemeinschaft at Bonn in 1987-88 followed by a visit to the Centre de Recerca Matemática in Barcelona. The year 1988 was a remarkable year for Heinonen in terms of his publications, for no fewer than six of his publications appeared. Three were written jointly with Tero Kilpeläinen: *A-superharmonic functions and supersolutions of degenerate elliptic equations*; *Polar sets for supersolutions of degenerate elliptic equations*; and *On the Wiener criterion and quasilinear obstacle problems*. The others were the single author publications *Boundary accessibility and elliptic harmonic measures* and *Asymptotic paths for subsolutions of quasilinear elliptic equations*, and the paper *On quasiconformal rigidity in plane and space* written with K Astala. In the autumn of 1988 he was appointed to a three-year postdoctoral Assistant Professorship at the University of Michigan [3]:-

In 1992 Heinonen's postdoctoral Assistant Professorship came to an end and he was appointed again as a Research Assistant in the Finnish Academy of Science and Letters. He also accepted a tenure track Assistant Professorship at the University of Michigan. Heinonen's wife Karen, completed her doctorate at the University of Michigan in 1993 and in the following year she was appointed as a Moore Instructor at the Massachusetts Institute of Technology. Heinonen was promoted to Associate Professor at Michigan in 1994 and in that year he and his wife both went to Boston so that she could fill the Moore Instructorship; they remained there for three years. In 1997 both returned to Michigan and in 2000 Heinonen became a full professor at Michigan.When Heinonen originally came to the United States, his intention was to stay for a short period. His plans changed when he met his future wife, Karen Smith, a first-year graduate student of mathematics, who also arrived in Ann Arbor in the fall of1988. They married in1991.

Heinonen's research record was outstanding which was recognised by the award of an Alfred P Sloan Foundation Grant in 1992, and National Science Foundation Grants during 1989-1991, 1991-1993, 1994-1995, and 1996-1999. He was awarded the Excellence in Research Award by the University of Michigan in 1997. Much of his work involved quasiconformal mappings so it is appropriate to quote a few sentences from [1] where Heinonen gives an excellent overview of the topic:-

Heinonen published two important books: (with Olli Martio and Tero Kilpeläinen)Quasiconformal mappings are generalizations of conformal mappings. They can be considered not only on Riemann surfaces, but also on Riemannian manifolds in all dimensions, and even on arbitrary metric spaces. Quasiconformal mappings occur naturally in various mathematical and often a priori unrelated contexts. The importance of quasiconformal mappings in complex analysis was realized by Ahlfors and Teichmüller in the1930s. Ahlfors used quasiconformal mappings in his geometric approach to Nevanlinna's value distribution theory. He also coined the term "quasiconformal" in his1935work on 'Überlagerungsflächen' that earned him one of the first two Fields medals. Teichmüller used quasiconformal mappings to measure a distance between two conformally inequivalent compact Riemann surfaces, starting what is now called Teichmüller theory. ... In the past ten years, it has become known that a full-fledged quasiconformal mapping theory exists in rather general metric measure spaces. This theory has subsequently been applied to new rigidity studies in geometric group theory. There is also a budding theory of quasiconformal mappings in infinitedimensional Banach spaces, based on the concept of quasisymmetry. ... Quasiconformal mappings are fascinating objects in mathematics. They are flexible enough to be ubiquitous, yet they harbour enough subtle analytic and geometric properties so as to be useful in a variety of contexts.

*Nonlinear Potential Theory of Degenerate Elliptic Equations*(1993); and

*Lectures on Analysis on Metric Spaces*(2001). A review of the first of these begins:-

A new edition, containing four additional chapters, was published in 2006. The second book mentioned above is described by Christopher Bishop. We quote a couple of sentences from the beginning and the end of his review:-This excellent book is the first monograph dealing with a potential theory of second-order quasilinear elliptic equations of[a certain]type ... It contains results of this theory obtained in the past ten years generalizing the classical theory in a natural way.

In addition to these books, a 44-page pamphletThis very interesting book deals with how various concepts of Euclidean analysis(e.g., gradients, Sobolev spaces, Poincaré inequalities, quasiconformal mappings, etc.)can be extended to more general metric spaces. The book is based on a1996course given by the author at the University of Michigan and it retains a comfortable, conversational style with numerous remarks, comments, references and conjectures. ... The author summarizes and describes an area which has been very active recently and does an excellent job of giving the flavour and main ideas in the field. It is a well-written and enjoyable book containing something for both beginners and experts and deserves to be on the shelf of anyone interested in the interplay of analysis, geometry and topology.

*Geometric embeddings of metric spaces*was published in 2003 being notes from a graduate level course Heinonen gave at the University of Jyväskylä. Matti Vuorinen writes:-

In August 2004 Heinonen lectured on Lipschitz analysis at the 14The purpose of this course was to introduce the students to several aspects of metric geometry. Some of the key notions are doubling spaces, Gromov-Hausdorff convergence, different notions of dimensions and bi-Lipschitz embeddings.

^{th}Jyväskylä Summer School and the notes were published in a 77-page pamphlet.

In 2002 Heinonen gave the invited lecture *The branch set of a quasiregular mapping* at the International Congress of Mathematicians held in Beijing. This was only one of many invited address he gave for, in addition, he gave plenary addresses at the Rolf Nevanlinna Colloquium in Ann Arbor, Michigan in 1993, at the Rolf Nevanlinna Colloquium in Lausanne, Switzerland in 1997, at the Ahlfors-Bers Colloquium at the University of Connecticut in 2001, and at the Ahlfors Centennial in Helsinki, Finland in 2007. He also gave invited addresses at the XVIIth Geometry Festival at the Courant Institute in New York in 2002 and at the American Mathematical Society meeting in Boulder, Colorado in 2003. Another way in which he made a major contribution to mathematics was through being an editor of the *Proceedings* of the American Mathematical Society from 2000. His outstanding mathematical contributions were further recognised in 2004 when he was elected to the Finnish Academy of Science and Letters.

We have already mentioned Heinonen's love of sport and his cross-country skiing achievements while at high school. He kept this sporting interests after going to the United States [2]:-

We have mentioned above that Heinonen married the mathematician Karen Smith in 1991; they had three children, a daughter Sanelma (born in 1998) and boy-girl twins Tapio and Helena (born in 2003). We note that Heinonen's family called him Isi and they lived:-... he travelled around Finland and North America to compete in orienteering, placing in nearly every major US competition he entered. He is widely remembered in US orienteering circles for winning both the US and the North American gold medal in his class in the year2000, but also for his outgoing and upbeat personality that made competing with him a joy.

Let us give a quote to understand Heinonen's character and interests [2]:-... in a beautiful old house built in1910in Ann Arbor, Michigan, less than a kilometre from downtown.

Heinonen died at the age of 47 'after a brief but courageous battle with kidney cancer'. The Department of Mathematics at the University of Michigan established the Juha Heinonen Memorial Graduate Student Fellowship in his honour. An international conference in his memoryJuha was a vibrant, balanced, satisfied person. He loved the outdoors, particularly Michigan autumns and Finnish winters. A devoted father, Juha read animatedly to his children, taught them to bake authentic Finnish rye bread, and shared his love of nature by taking them on hiking trips from the days he carried them on his back. He also enjoyed the company of his children on everyday errands, and was often spotted with a child on the AATA bus or on his bike-seat. His Finnishness remained deeply important to him; he taught his children, and his wife, to love Finland and to speak Finnish. Each time he spoke of his family, the pride was evident through the twinkle in his eye and his broad grin.

*Quasiconformal Mappings and Analysis on Metric Spaces*was organised at the University of Michigan, Ann Arbor in May 2008.

**Article by:** *J J O'Connor* and *E F Robertson*

**Click on this link to see a list of the Glossary entries for this page**