MAT 318: Mathematical Models in Biology & Anthropology
26 — 30 September 2005
Old Main 403 Canisius College: 5:00 — 8:00pm Daily
| Instructor: | Per Enflo University Professor Department of Mathematics Kent State University Kent, OH 44242 Email: enflo@math.kent.edu |
On the second and third day I will talk about mathematical modeling in Biology. After giving a survey of how mathematics may be used in some different areas of biology, I will turn to the area in which I am currently doing research: Complex Ecosystems in general, and Lake Erie in particular. I will discuss questions like: How can mathematical models help us to understand the changes that occurred in Lake Erie after the invasion of Zebra mussels in the 1980s? How can we construct a mathematical model from incomplete observations? How can mathematical models help us make predictions?
On the fourth and fifth day I will discuss Population Genetics and Anthropology, another area in which I am currently doing research. After giving a general survey, I will focus on the following problems: What happened in Human Evolution in the last 150000 years? What happened to the Neanderthals? I will show how some new mathematical modeling can be of some help to tackle these problems. On the fifth day I will also, in the end, discuss the modeling project that I handed out on the first day.
Since, in particular on the second and third day, I will have to use ordinary differential equations, it would be useful to have some (small) knowledge of what is a differential equation.
Biographical Information: Per Enflo was born on May 20, 1944, in Stockholm, Sweden. His father was a surveyor, his mother an actress. Per Enflo is one of five children born to his parents. His family has been, and is, very active in music and other performing arts, and this involvement has been a strong influence in his life.
During his school years, the family moved to various places in Sweden, but Enflo enjoyed a stable, happy home life and good schooling. Around the age of eight he became interested in both mathematics and music. These are the two subjects that he was prodigious in and to which he remains most devoted. You are reading about him because of his mathematics, but in fact, Enflo is almost equally a musician and a mathematician.
In music, Enflo has studied piano, composition, and conducting. His first recital was given at age eleven. In 1956 and 1961 he was winner of the Swedish competitions for young pianists. We shall not say much about his music but do mention a few recent activities. He competed in the first annual Van Cliburn Foundation’s International Piano Competition for Outstanding Amateurs in 1999. During the spring of 2000, he played over half a dozen recitals.
Though devoted to both mathematics and music, it is the former that has determined where Enflo has lived. All of his academic degrees have been awarded by the University of Stockholm. Since completing his education in 1970, Enflo has held positions at the University of Stockholm, the University of California at Berkeley, Stanford University, the ´Ecole Polytechnique in Paris, the Mittag-Leffler Institute and Royal Institute of Technology in Stockholm, and at the Ohio State University. Since 1989 he has held the prestigious position of “University Professor” at Kent State University.
Per Enflo is most well known for his solutions, in the 1970s, of the “approximation problem”, the “basis problem”, and the “invariant subspace problem.” These were three fundamental and famous problems from the early days of functional analysis. Since the 1930s, many mathematicians had tried to solve them, but they remained open for about 40 years. The solutions are negative in the sense that they solved by counterexamples; they are positive in the sense that the new methods and concepts have had a great impact on the further development of functional analysis.
Per Enflo’s solution to the approximation problem also gives a counterexample to the basis problem. This work was started in 1967 and completed in 1972 and is a long story of progress and failures and of slowly developing new insights and techniques for final success.
Arguably, his most famous mathematical contribution thus far is his solution to the invariant subspace problem. He constructed a Banach space X and a bounded linear operator T : X -> X with no non-trivial invariant subspaces. The paper containing this example was published in 1987, but it had existed in manuscript form for about twelve years prior to that date. The published paper is 100 pages long, and contains very difficult mathematics. His work on the invariant subspace problem was accomplished during the years 1970–1975, so one can see that the late 1960s to the mid 1970s was a period of remarkable brilliance for Enflo. Enflo’s counterexample, though it gives a complete solution to the invariant subspace problem, left open many doors for future research. For example, determining classes of operators that must have invariant subspaces (in the spirit of Lomonosov’s result) is an active area, and, equally, some of the mathematics developed in his solution to the invariant subspace problem have led him to progress in other areas of operator theory.
The mathematical work discussed in the last few paragraphs might seem particularly abstract, but parts of the associated work have genuine applications. For example, some of the best available software algorithms for polynomial factorizations are based on ideas found in Enflo’s solution to the invariant subspace problem. Also, there are indications that his Banach space work might have good applications to economics.
Enflo’s other important mathematical contributions include several results on general Banach space theory, and also his work on an infinite-dimensional version of Hilbert’s 5th problem.
Enflo’s early career as a musician is an important background for both his originality as a mathematician and for his strong interest in interdisciplinary science. He has done work in biology, on the zebra mussel invasion and phosphorus loading of Lake Erie (work funded by the Lake Erie Protection Fund). In anthropology he has worked on human evolution and has developed a “dynamic” population genetics model that lends strong support for a multiregional theory of human evolution. He has also published work in acoustics, on problems related to noise reduction.”
— Dr Karen Saxe, Beginning Functional Analysis, Undergraduate Texts in Mathematics, 2002 Springer-Verlag New York. Reproduced by kind permission of the author.