MSA Exhibit: Mathematics in the 20th Century: Major Problems and Advances

	Mathematics in the 20th Century: Major Problems and Advances UCLA Powell Library Rotunda August 7 - 31, 2000 Sponsored by the UCLA Department of Mathematics
	The purpose of this exhibit is to sample some of the important developments in mathematics which have occurred during the century which comes to an end on New Year's Eve, December 31st, 2000. The exhibit covers only a few fields of mathematics, and, even then, not all areas within those chosen fields. The five chosen topics are: Finite Groups Lie Groups Number Theory Mathematical Logic Scientific Computing and Numerical Analysis
The first four of these topics are in Pure Mathematics and the last is in Applied Mathematics, as these terms are commonly understood. That Number Theory is regarded as pure mathematics has not prevented it being applied to real problems, and the same can be said for Lie Groups; this will be explained in those two parts of the exhibit. Also three of the topics have an outstanding result that has been reported in the popular press. For Finite Groups, that result is the classification of the finite simple groups; for Number Theory, it is the positive solution of the Fermat conjecture (Fermat's Last Theorem); and for Mathematical Logic, it is Gödel's Incompleteness Theorem. Outstanding advances have been made in other fields of mathematics as well, for example, in ordinary and partial differential equations, in algebraic geometry, in classical analysis, in topology, in operator algebras, etc. We did not include any of these areas because of the difficulty involved in explaining them to the non-mathematician. One more point: The first three topics are intertwined. Each has had an impact on the other two. For example, Lie theory has produced new finite simple groups, and groups provide an important element of structure in number theory
	(1) Finite Groups The mathematical notion of a group has played an extremely important role in 20th century mathematics. What mathematicians now refer to as groups were first studied in the 19th century. Groups arose in many contexts, among them the study of permutations and the structure of field extensions. You may have learned something about permutations in mathematics. If one takes the set of natural numbers {1, 2,…, n} for some natural number n, then a rearrangement of these numbers is called a permutation of them.
The set of all rearrangements has the following properties: If one takes 2 permutations Sand T, applies T and then applies S, the result is itself a permutation. That result is called the product of S and T and is denoted by ST. For 3 permutations, S, T, U, we have (ST)U=S(TU). That is, the associative law holds for this kind of product. The permutation that moves none of the numbers 1, 2…, n is called the identity permutation. If I denotes the identity permutation, the SI=IS for any permutation S. Finally, for every permutation S there is an opposite permutation T so that ST=TS=I, that is, so that reordering by T and then by S (or vice returns the called the inverse of S.
Mathematicians like to abstract from the concrete. In this case, they have done the following: Let G be ANY set which is endowed with an abstractly defined product; i.e., if a and b are 2 members of G, then there is defined a 3rd member of G denoted by ab. They assume that this product satisfies properties 1, 2, and 3 above. They then prove from these assumptions alone many other properties. In this case, the identity element is usually denoted by e (or sometimes 1), and the inverse of a is denoted by a^-1. Groups are very important in many applications. Perhaps the most important application is to the notion of symmetry. Here are some examples: Consider the square pictured to the right. Let G be the set of all rigid transformations of this square. This consists of the 4 rotations by 0, 90, 180, and 270 degrees, together with the products of these rotations with the reflection across one of the diagonals. G has 8 members and is called a dihedral group. Other dihedral groups arise by considering the rigid transformations (the rotations and reflections) of regular n-sided figures (n-gons). Symmetries abound in mathematics and in nature.
Hermann Minkowski	Examples are the set of all rotations of the sphere, which is given by the equation x² + y² +z²=1 in Euclidean 3-space, or by x₁² + x₂² + … + x_n²=1 in Euclidean space of dimension n. One can look at the set of all symmetries of Minkowski space, which is the 4-dimensional space-time made famous by Einstein's theory of Special Relativity. These groups are infinite and play an important role in atomic physics. Similar examples have been used to classify the elementary particles of physics.
Groups can be either finite or infinite, but while there has been extensive work done on infinite groups, we want to focus on finite groups. Perhaps the best one could hope for would be to be able to list every type of finite group, to characterize every group by a description of its elements and their products. This is a very difficult question. But one result in finite group theory (the Jordan-Hölder Theorem) shows that first one must try to classify the finite SIMPLE groups. To understand simple groups, one must first have an idea of what subgroups are. Effectively, a subgroup is a subset of a group which is itself a group. Simple groups are groups in which the only subgroups that have a certain stability property are the identity element and the whole group itself.
Danny Gorenstein	The finite simple groups have been completely classified in a proof that involved many, many authors and many, many journal pages scattered over many journal articles. (In fact, the original proof is approximately 10,000 pages long!) The mathematician who conceived the final attack on this problem was Daniel Gorenstein (1923-1992) of Rutgers University. Here is the statement of the Classification Theorem: Classification Theorem. The finite simple groups are exactly the groups of prime order, the alternating groups of order at least 60, the simple finite groups of Lie type, and the 26 sporadic simple groups.
A few words of explanation: The order of a group is just the number of members in the group. There is only one group with p elements, where p is a prime number. An alternating group is just the unique subgroup A_n of the group S_n of all permutations of n objects consisting of all the even permutations in S_n for some n. The restriction on the order of A_n just says that n 5. The finite groups of Lie type are finite analogues of the simple Lie groups and were constructed in a uniform manner by Claude Chevalley, Robert Steinberg, Rimhak Ree, and Jacques Tits. The sporadic simple groups consist of 26 groups that don't fit into any of the preceding classes. Five were discovered in the 19th century by Émile Mathieu. In the 20th century four were discovered by Zwonimir Janko; one jointly by Donald Higman and Charles Sims; one each by Dieter Held, Richard Lyons, Jack McLaughlin, Michael O'Nan, Michio Suzuki, and Arunas Rudvalis; three by John Conway; and seven by Bernd Fischer, with important contributions by Robert Griess.
	(2) Lie Groups Two Basic Notions. Lie groups are infinite groups endowed with a special structure of the following kind: They are manifolds; i.e., they are covered by balls in n-dimensional Euclidean space for some fixed n so that , whenever two balls B₁ and B₂ overlap, the Euclidean coordinates in B₁are continuous functions of the Euclidean coordinates in B₂ and vice versa. The group operations of multiplication and taking inverses are continuous in these coordinates.
(Actually, this is not the original definition. It is, however, equivalent to the original definition due to some very difficult theorems of Deane Montgomery, Leo Zippen, and Andrew Gleason in the early 1950s.) Examples of Lie groups are: the group of rotations of the circle; the group of rotations of the sphere in 3-space; the group of rigid transformations of 3-space. A Lie algebra is a vector space in which there is an operation, the Lie bracket, of any two vectors X and Y in the space, which produces another vector [X, Y] in the space, which satisfies the following three conditions: If X is held fixed, the transformation that sends Y to [X,Y] is linear; [X,Y]=-[Y,X]; [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]]=0. The second condition is called anti-symmetry, and the third is called theJacobi identity. Lie algebras and Lie groups are intimately related. Every Lie group gives rise to a unique Lie algebra, and every finite dimensional Lie algebra over the real or complex numbers arises from a Lie group. Lie groups can be studied using their Lie algebras.
Marius Sophus Lie	Early Development. Lie groups and Lie algebras are named in honor of the Norwegian mathematician Marius Sophus Lie (1842-1899). The study of Lie groups actually begins before Lie, with Carl G. J. Jacobi (1804-1851), who worked on the integrability of systems of partial differential equations and introduced what are now called Lie brackets. His work directly inspired Lie, who, with the help of his assistant Friedrich Engel (1861-1941), worked out a classification of "finite groups of infinitesimal transformations" in the plane and did foundational work in other dimensions as well. He introduced two notions mentioned above: Lie algebras and Lie groups. Picking up where the founders of the theory left off, Wilhelm Killing (1847-1923) began the classification of simple Lie algebras over the complex numbers. Although his work was incomplete and very obscure at several points, it paved the way (at least part of it) for the definitive classification obtained by Élie Cartan.
Elie Cartan Hermann Weyl	The Theory Blossoms. Élie Cartan (1869-1951) and Hermann Weyl (1885-1955) were the two towering figures in the development of the theory of Lie groups and Lie algebras in the first half of the 20th century. To give you an idea of the range of their accomplishments, we list some below. Cartan, in his doctoral dissertation, classified all finite dimensional simple Lie algebras over the complex numbers. In subsequent work, he introduced what is now called the Cartan involution for the study of real Lie algebras, gave a proof of the second part of Lie's 3rd theorem (i.e., that every finite dimensional real Lie algebra is the Lie algebra of a Lie group), and studied infinite dimensional Lie algebras. He studied finite dimensional linear representations of finite dimensional Lie algebras, classifying them all when the Lie algebra is simple, whether real or complex. He determined the topological structure of the compact semisimple Lie groups as well as that of the non-compact simply connected Lie groups. He also laid much of the foundation of modern differentiable manifold theory. Hermann Weyl laid the modern foundations of the representation theory of compact topological groups and harmonic analysis on them. He pioneered the application of the representation theory of certain finite dimensional Lie groups to atomic physics. He was the first to use global methods for studying compact Lie groups and to prove that the fundamental group of a compact semisimple Lie group was finite. He obtained his famous character formula for the irreducible representations of compact Lie groups, which has inspired much subsequent research.
Some Later Developments. Claude Chevalley (1909-1984) pioneered the study of semisimple Lie groups over ANY field. (A field is a number system in which one can add and multiply members in a way obeying the fundamental rules governing ordinary addition, subtraction, multiplication and division.) He obtained classification theorems that had been known only for groups over the complex numbers by methods that were new even for the complex case. His methods led to new finite simple groups. These were some of the simple groups of Lie type mentioned in the Classification Theorem at the end of the display on Finite Groups. Important advances in the theory of infinite dimensional representations of finite dimensional Lie groups were made by I. M. Gel'fand, Harish-Chandra, A. A. Kirillov, L. Pukanszky, L. Auslander, and B. Kostant.
Harish-Chandra Murray Gell-Mann	Some Applications. E. P. Wigner and V. Bargmann were physicists who, in the late 1930s and 1940s, classified certain of the infinite dimensional representations of the Poincaré Group, which is the group of symmetries of Minkowski space. This information was then applied to systematize the theory of some of the then known elementary particles of physics, electrons and photons being two examples. R.P. Langlands did fundamental work in Harish-Chandra's area as well as in applications of representation theory to number theory, thereby generating powerful conjectures which are influencing the direction of research in a portion of modern number theory. Applications of Lie groups have also been made in geometry, the theory of ordinary differential equations, and atomic and nuclear physics. Murray Gell-Mann and Yuval Ne'eman independently proposed in 1961 a scheme involving finite dimensional representations of SU(3), the special unitary group of complex 3-space, to deal with the strong interactions of elementary particle physics. The representations were to be used to group certain elementary particles and explain such properties as their masses. For example, the "adjoint" representation, which is 8-dimensional, grouped together and predicted various properties of a set of 8 particles. In 1962, Gell-Mann proposed at a physics meeting that this scheme would do the same for a set of 10 "resonances." Only 9 of these resonances had been observed up to that time, but Gell-Mann predicted the existence of the 10th member of this set which would have negative charge, "strangeness" number -3, and mass of about 1680 MeV. The discovery of this particle was announced in 1964 by a team of physicists working at the Brookhaven National Laboratory, calling it the . Gell-Mann was awarded the Nobel Prize in Physics in 1969.

Pierre de Fermat	(3) Number Theory Background. Number theory goes back at least to the ancient Greeks. Initially, number theory concerned looking for facts - and proofs of those facts - about the ordinary counting numbers N={1,2,3,4,…n,…}. For example, it was recognized that among these numbers were some particularly important ones, namely the prime numbers. Prime numbers are numbers p 2 such that the only numbers n dividing p evenly are n=1 and n=p. Prime numbers are the basic building blocks for the multiplicative structure of N, as a result of the Fundamental Theorem of Arithmetic, which states that every number n 2 can be written uniquely as a product of prime numbers, that is, Unique factorization. If n belongs to N and n 2, then n=p₁p₂p₃…p_k, where k is unique, the p_i are primes, p_i p_i+1 for all i, and this sequence of primes is unique.
Using this theorem, Euclid showed that there had to be an infinite number of primes. Moreover, the ancient Greeks showed that the length of the hypotenuse of a right triangle, both of whose sides had length 1, could not be a positive rational number, that is could not be written as the quotient of two numbers in N. In modern terms, this says that the square root of 2 is not rational. It should be remarked that other civilizations were also concerned with matters like these. For example, the ancient Chinese knew that if p is a prime, then p evenly divides 2^p - 2. There were many questions concerning prime numbers, some of which are simple to state. For example: Are there an infinite number of "twin primes"? That is, are there an infinite number of primes p such that p + 2 is also a prime? (Examples: p can equal 3, 5, 11, 17 and so on.) Can every even n 4 be written as a sum of two primes? (This is a conjecture of Christian Goldbach (1690-1764).) Amazingly, the answers to these two questions are not known even today. Before going further, we must extend the notions introduced above. Let Z be the set of numbers n such that n=0, n is in N, or -n is in N. Z is called the set of rational integers, or just integers for short. The set Q of rational numbers consists of exactly the quotients a / b, where a and b are in Z and b 0. Diophantine Problems. Another kind of problem is of the following sort: Find all solutions a of a given equation such that a is in N. Variations ask for all solutions in Z or in Q. Other variations use equations in k unknowns and ask for solutions in N (or in Z or in Q). We have not specified what sort of equations are involved, but will be content to give some examples. These sorts of problems are called Diophantine problems after the 3^rd century Greek mathematician Diophantus. Here are some examples: Find all solutions in N to x² + y²=z². In other words, find all possible right triangles whose sides have positive integral length. Find all solutions in Q of x² + y²=1. That is, find all points on the circle of radius 1 centered at (0,0) with rational coordinates. Find all solutions in N of xⁿ + yⁿ=zⁿ for all n 3. Find all solutions in N of x₁ⁿ + x₂ⁿ + … + x_kⁿ=wⁿ for all 1 < k < n.
Andrew Wiles	The answer to the first of these problems was known to Euclid, and the second problem is equivalent to the first. The third problem is called the Fermat problem after the great French amateur mathematician Pierre de Fermat (1601-1665), who stated that he had a proof that there were no solutions whatever, but did not write the proof down. Leonhard Euler (1707-1783), a Swiss mathematician who worked in Germany and Russia, conjectured that the fourth problem also had no solutions. Note that the fourth problem includes the third problem (just set k=2, which forces n to be at least 3).
The conjecture of Fermat was finally proved to be correct by Andrew Wiles in 1994. Euler's conjecture was shown to be false in a 1988 paper by Noam Elkies, where he produced an infinite number of essentially different counterexamples in the case of n=4, k=3. Wiles's theorem involved a lot of very sophisticated mathematics and built on the work of many mathematicians. It is one of the crown jewels of 20th century mathematics. Applications. There are many applications of Number Theory to other parts of mathematics, as well as to other subjects. One remarkable case was a system for coding and decoding messages in such a way that the coding method is publicly known, but that the decoding method is very, very difficult to uncover. This system was proposed by Ronald Rivest of MIT, Adi Shamir of the Weizmann Institute of Science (Israel), and Leonard Adleman of the University of Southern California in a paper published in 1978. It goes as follows: The coding headquarters chooses two very large primes p and q and sets n = pq. Then it chooses a number c < n so that no prime divisor of (p-1)(q-1) divides c. Headquarters also chooses a number k so that 10^k is less than both p and q. Finally, headquarters computes a number d so that cd-1 is evenly divisible by (p-1)(q-1). This is possible by a theorem of Euler. Headquarters makes both n and c public but keeps p, q, and d secret. Now suppose that a spy in the field wants to send a coded message back to headquarters. The spy converts the message in an agreed upon way into a sequence of digits which is then broken up into packets of k digits each. Let a_i be the number whose decimal expansion is the ith packet. The spy computes the remainder b_i of a_i^c divided by n for each i. Then the b_i are sent back to headquarters. To decode, all headquarters has to do is to compute the remainder of b_i^d divided by n. That remainder will be a_i. What makes this code hard to crack is that factoring the very, very large number n into primes is extremely difficult to do.
	(4) Mathematical Logic A Little Bit of History. Logic originally was the study of correct ways of reasoning. The study of logic goes back at least to the ancient Greeks, but the name most frequently cited is that of Aristotle (384-322 BC). He was a great systematizer of knowledge and an important figure in the history of philosophy. Among his contributions was a list of correct syllogisms in abstract form. A standard example of such a correct syllogistic form is the following: Suppose that whenever property P is true of an object x then property Q must be true of x as well; suppose that P is true of a specific object a; then Q is also true of a. This assertion is true no matter what P, Q, and a are. In other words, this syllogistic form is purely formal; its truth does not depend on any meanings that may be assigned to the letters in italics. An instance of this syllogistic form is the old saw: All men are mortal; Socrates is a man; therefore Socrates is mortal.
In more modern notation, we can write this form as follows: ((x,P(x) Q(x)) P(a)) Q(a), where means 'for all', P(x) means 'P is true of x', means 'and', and means 'implies'. One can do the same thing for all Aristotelian syllogistic forms. Since the time of Gottfried Wilhelm Leibniz (1646-1716), an extremely important German philosopher, scientist, and mathematician (coinventor with Isaac Newton (1642-1727) of the differential and integral calculus), logicians have dreamed of producing a symbolic 'calculus' of logic that would enable the verification of logical assertions purely formally, that is, by the same methods mathematicians use to check the correctness of a proof: one writes down in symbolic form the axioms of logic and then determines whether the assertion follows from the axioms and from assertions previously proved to be correct, all according to a set of rules for deduction. Many names are associated with this endeavor. Here are a few: De Morgan, Boole, Jevons, Venn, Peirce, Peano, Frege, Whitehead, and Russell. The Mathematization of Logic. We have already seen how it is possible to take a logical statement and turn it into a string of symbols. In this way, we can hope to write down a list of axioms which will define various types of logic. To illustrate the point, let us see how this is done for group theory. The axioms have already been stated informally in the display on Finite Groups. We start with a list of symbols: G, m, a, b, c,… We also have the following logical symbols: ,, , ,,: and=. Here ,, and are as above, means 'there exists', means 'belongs to', : means 'such that', and=means 'equals' or 'is the same as'. Also we must state that m is short for m(. , .), where the dots will be replaced by two symbols each time it is used. m is called a 2-variable function symbol. Here are our axioms: (a,b)(((a G) (b G)) (m (a,b) G)) (a,b,c)(((a G) (b G) (c G)) (m(m(a,b),c)=m(a,m(b,c))) (e) : ((e G) ((a)(( a G) (m(e,a)=m(a,e)=a)))) (a)((a G) ((b : ((b G) (m (a,b)=m(b,a)=e)))) (We have used a great many parentheses in the statements of these axioms, more than are normally used; this was done in order to break the logical pieces into elementary statements.) Group theory would then proceed by deducing statements using the rules of logic from these axioms and from previously deduced statements. In this way, we have a recursive way of arriving at true statements (theorems) about groups from the assumption that the axioms are true. Now what happens when we try to apply this method to logic itself? We must try to write down a list of symbols consisting of the following sorts. Here they are for Propositional Logic: Connectives: , , , Left and right parentheses An infinite list of letters For First Order Logic the symbols are: The same connectives as above The same parentheses as above Quantifiers: , An infinite list of individual variables An infinite list of function symbols An infinite list of relational symbols (Among the connectives, means 'not' and means 'or'.) In each of these cases, one proceeds by stating rules by which one obtains grammatically correct formulas and determines which of them area valid formulas.
David Hilbert	Modern Mathematical Logic. In the 20th century there has been a flood of contributors to mathematical logic and set theory. We will mention some important contributions below. We begin with Axiomatic Set Theory, which attempts to mathematize the naïve notion of a set and the properties of sets. The early names here are Georg Cantor, Gottlob Frege, Bertrand Russell, and Alfred North Whitehead. Whitehead and Russell made improvements on the theories of Cantor and Frege. This became necessary when contradictions arose in the earlier theories. One famous one was the following: Let R be the set of all sets that are not members of themselves. Then R is a member of R if and only if R is not a member of R, which is absurd. Whitehead and Russell dealt with this and similar problems by restricting the formation of sets according to their 'theory of types'. Other kinds of Set Theory accomplished this by other means. The names of David Hilbert, Paul Bernays, Thoralf Skolem, Ernst Zermelo, and Adolf Fraenkel are associated to these efforts. From a well-founded theory of sets, it was hoped that one could build the whole edifice of mathematics.
Kurt Godel Kurt Godel & Albert Einstein	David Hilbert (1862-1943), one of the giants of 20th century mathematics, proposed that one should be able, within this foundation of mathematics, to prove or disprove every possible mathematical statement. Kurt Gödel (1906-1978) dashed these hopes by showing that, if a logical system was rich enough to be able to express the arithmetic of the rational integers, one could form an arithmetic sentence that one could prove was not provable within the system, yet was obviously true. This Incompleteness Theorem of Gödel opened a whole new era in Mathematical Logic. It was published in 1931. In order to make set theory really useful, one needs to have among its axioms the Axiom of Choice (or equivalent). Let Zermelo-Fraenkel set theory without the Axiom of Choice be denoted by 'ZF' and let ZF plus the Axiom of Choice be denoted by 'ZFC'. Cantor had developed a theory of cardinality (size) of infinite sets and had shown that there were more real numbers than integers. This theory of cardinal numbers led to the Continuum Hypothesis (CH): If ₀ is the cardinal number (size) of the set of counting numbers N, then the cardinal number of the set C of subsets of N is the next largest cardinal number after ₀. The Generalized Continuum Hypothesis (GCH) says that if is the cardinality of an infinite set S, then the cardinality of the set of subsets S is the next largest cardinal number after . In 1938, Gödel proved that if ZF is consistent (free of contradictions), then so is ZFC + CH. Amazingly, in 1963, Paul Cohen proved that if ZFC is consistent, then so is ZFC + CH. Therefore CH is independent of ZFC. He also proved that the Axiom of Choice is independent of ZF. We have left out a great many important advances in mathematical logic in the 20th century. Among them is the theory of computability, which has had a great influence on theoretical computer science. A few of the major contributors to this area are A. Turing, A. Church, A. Tarski, E. Post, and S. Kleene.

	(5) Scientific Computing and Numerical Analysis Brief Introduction. Numerical Analysis concerns methods of obtaining good approximations to answers to numerical problems and analyzing just how good the approximations are. Some precursors to modern numerical analysis are studied in undergraduate differential and integral calculus, for example, methods for evaluating definite integrals via the trapezoidal rule or via Simpson's rule. Another precursor is the method of approximately solving differential equations by the method of finite differences. Modern numerical analysis has found methods of efficiently finding approximate solutions to partial differential equations, often those that arise in the study of physical phenomena. These methods have been used, for example, in the design of airfoils and the simulation of shock phenomena.
Numerical approximation is important not only because one wants to obtain good approximate solutions quickly, but also because exact methods of solving these problems don't yet and probably never will exist. With the advent of high-speed computers and innovative numerical algorithms, we can now simulate real physical phenomena numerically on the computer and visualize them. Of course this is interesting and important in almost all fields of science and engineering (including bioengineering). In computer graphics, using new numerical algorithms and analytic techniques, it is possible to produce realistic visuals of the motions of water, flames, and smoke by simulating the physical laws they obey. Such visuals can be used in motion pictures to provide special effects. The field of mathematics that is useful here is a blend of mathematical modelling, numerical analysis, very nonlinear partial differential equations, and scientific computing. Level Set Method, Ghost Fluid Method, and Vorticity Confinement in Hollywood. The movies displayed on the computer to the left and the stills on the right show simulations of the natural movement of water, fire, and smoke. These simulations were done by Ron Fedkiw and collaborators, including Myunjoo Kang, Xu-Dong Liu, Dave Wasson, and Raymond Yeung, using the following new techniques: The level set method devised by Osher and Sethian (1988) to represent dynamically moving interfaces as the zero level set of a scalar function that evolves in time. The ghost fluid method (GFM) devised by Fedkiw, Aslam, Merriman, and Osher (1998) which simplifies and improves the numerical solutions of two phase flow and related problems such as those involved in the simulation of melting ice. Vorticity confinement devised by Steinhoff (1990) to confine the effect of vorticity to a narrow region. It is used here in the numerical modeling of smoke.
	Image Segmentation of the Brain. Mumford and Shah (1989) obtained a variational method designed to give a multiscale segmentation of images. Chan and Vese (1998), using a variational level set approach, obtained a very simple and versatile implementation of a useful modification of this method based on an active contour model. The new model, based on the variational level set method and the Mumford-Shah multiscale segmentation model, is able to easily detect objects at various scales without using standard gradient dependent techniques. Images at the right show a 2D section of a brain image done by Fedkiw and Kang using a slightly modified version of the Chan and Vese algorithm. 8/9/2000 R. Blattner