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A Sampling of Research Accomplishments During the IMA Annual Programs 1989-90 through 1995-96

A SAMPLING OF RESEARCH ACCOMPLISHMENTS IN NONLINEAR WAVES 1989-90

Important research results were obtained by many of the long-term participants in the IMA program, and much significant research and collaboration was initiated by short-term visitors. This research is documented in many IMA technical reports and IMA conference proceedings and other volumes to be published by Springer-Verlag. A brief sample of the research is as follows:

Assessment by James Glimm: The spring quarter emphasized applications of systems of hyperbolic conservation laws in sev eral space dimensions. The goal was to nurture strong and close interdisciplinary interactions among diverse areas of modern applied mathematics including

1. Large (and small) scale computing
2. Asymptotic modeling
3. Qualitative modeling
4. Rigorous proofs for prototype problems
5. Strong interaction of theories from modern applied mathematics with experimental data.

The highlight of the spring quarter was the three-week long workshop during the month of April. One of the striking outgrowths of this meeting was the fact that interdisciplinary interactions using all of the tools of modern applied mathematics mentioned in (1)--(5) above are developing at such a rapid rate that new predictions of phenomena for fluid dynamics, nonlinear elasticity, the behavior of reacting materials, among other applications can be made through a combination of theory and compu tations in regimes which are not accessible readily to experimentalists. Examples presented at the April workshop of this strong interdisciplinary interaction leading to new predictions include: 1) the work of Grove and Menikoff on anomalous shock diffra ction patterns; 2) the work of Glaz, Colella, and collaborators on shock reflections from wedges; 3) the work of Woodward, Majda, and Artola on new phenomena in nonlinear stability for supersonic vortex sheets and jets; 4) the work of Krasny, Shelley, Di perna-Majda, and Caflisch on striking new phenomena in the evolution of thin layers of vorticity in turbulent incompressible flows; 5) the discovery and analysis by T.P. Liu, Roytburd, Hunter, Rosales and Majda among others of diverse types of new resona nt phenomena leading to wave amplification in hyperbolic systems. Another significant announcement concerned the results of Pui-Tah Kan's thesis on existence for conservation laws: an example with an isolated umbilic point.

A special highlight of the workshops was the part of the program held at the Minnesota Super ComputerCenter. Paul Woodward's state of the art'' flow visualization graphical techniques revealed striking new vistas (for the first time for the va st majority of participants) for processing data for documenting and explaining new phenomena in applications.

Many interdisciplinary interactions developed at I.M.A. in 1988-1989 during the spring quarter. These include the following: 1) collaboration by Woodward and J. Glimm's group in merging front-tracking and shock-capturing alogrithms for interface instabilities; 2) a new approach by Rosales and Thomann to explain the experimental phenomena in diffraction of weak shocks by caustics; 3) the development by Embid, Hunter, and Majda of new simplified asymptotics equations to explain the transition to detonation in granular explosives; 4) a new theoretical conjecture by Majda for organizing centers for waves bifurcations in shock diffraction was simulated by computational results, reported by Glaz at the meeting. This conjecture has stimulated Glaz to begin a new round of large scale numerical simulations to test the conjecture.

The Multiphase Flow Workshop produced research results almost immediately. The kinetic theory ideas of Jenkins were new to many people and the idea of ensemble averaging probably was moved to the front during the workshop. The idea of diffusion against a concentration gradient was an important new thought, vital for stability of fluidized beds. Bud Homsy got its first analytical result on this which he will publish in the IMA Worksh op volume. He got this result using the symbol manipulation program MAPLE, which he learned in his IMA office, in about three all night sessions. The multiphase group is surely going to meet again in August 1990 at the Stanford IUTAM conferences. Jenkins and McTigue started a collaboration based on contacts made at the IMA.

Martin Golubitsky's stay at the IMA was most productive as he collaborated on two projects that he would not otherwise have become involved in were it not for this visit. The two pro jects are:

a) Particle sedimentation and time-reversible symmetry, and

b) Arrays of Josephson junctions and bifurcation of mappings with permutation symmetry.

The first project is joint with Martin Krupa and Chjan Lim while the second is joint with Don Aronson and Martin Krupa.

In (a) they study a model, called the Stokeslet model, for the sedimentation of a finite number of particles. This model had been studied by several groups, including one of Caflisch, Lim, Luke & Sangani who showed that periodic solutions ex ist in the three particle model. Golubitsky, Krupa and Lim proved the existence of several families of periodic and quasi-periodic solutions for the n-particle model using a combination of time-reversibility (due to an infinite viscosity limit in the mod el) and spatial symmetries. Their work is based on previous work of Lim (as indicated), Krupa (bifurcation from group orbits of equilibria) and Golubitsky (Hopf bifurcation with dihedral group summetry). Since Lim was a post-doc and Krupa was visiting th e Math department, the IMA presented a unique opportunity to involve the three in collaborative research.

In the second project, Aronson, Krupa and Golubitsky studied period doubling in the presence of permutation summetry classifying, for the types of bifurcations occurring in arrays of identical oscillators, all period two points emanating from this bi furcation. They used the Apollo color graphics to check their abstract results for a system of coupled Josephson junctions. Again the facilities and synergism provided by the IMA was crucial to their research.

There were two main projects James Glimm started as a result of the IMA meetings. Both involve interaction with Paul Woodward and his group. The first (and simplest) is to adapt and use Wo odward's graphics programs. The second is to compare systematically results on Kelvin-Helmholtz instability. Glimm and Woodward have each developed high quality computational algorithms suitable for this problem. The two algorithms are however different in some important ways. The problem is highly unstable, and for this reason is difficult to compute correctly. An ever present question is whether or at what level detail the computations are correct. The presence of two very different methods of comput ation is thus an important check on the validity of the results in this area.

A SAMPLING OF RESEARCH ACCOMPLISHMENTS IN DYNAMICAL SYSTEMS 1989- 90

The theory of dynamical systems was spawned in the attempt to understand mathematical models of physical (and biological) phenomena which change with time. While the early pioneers, such as Poincare and Hill, were motivated by a desire to explain properties of celestial mechanical systems, this theory today has applications in many diverse areas. The program in Dynamical Systems and Their Applications was planned to examine in depth a number of these applications. The objectiv es of the year-long program were to (1) to seek new vistas for the theory of dynamical systems and (2) to encourage researchers to look seriously at areas of application which are ripe for major advances. Important research results were obtained by many of the long-term participants in the IMA program, and much significant research and collaboration was initiated by short-term visitors. This research is documented in many IMA technical reports and IMA conference proceedings and other volumes published b y Springer-Verlag. A brief sample of the research is as follows:

Some of the most important research was done by the post-docs. For example, Chris Gole made progress on his analysis of invariant sets for higher dimensional twist maps. Scott Dumas did excellent work on the relationship between KAM theory and crystal structures. Mohamed Elbialy established the C' regularization of simultaneous binary collision in classical celestial mechanics.

One of the major accomplishments of the IMA Year on Dynamical Systems and Their Applications was a fundamental discovery of the existence of a regular global attractor for the weak solution of the Navier-St okes equation on three-dimensional domains. While this result has now been proven rigorously only for certain this domain in R3, it gives strong evidence that the phenomenon of the occurrence of global attractors may be more the ru le than the exception. The recent history of this discovery is noteworthy because it shows the importance of the IMA program in this development. During the Fall of 1989 Genevieve Raugel and George R. Sell were in residence at the IMA. Their proof of th e global regularity of solutions in the Sobolev space H^1 was completed at that time. In the Spring of 1990, while reading a draft copy of the Raugel-Sell paper at the IMA, Ciprian Foias suggested that the methods of Raugel and Sell may be strong enough to prove the existence of global attractors. Foias was correct, and the results were presented by Raugel at the IMA workshop on Dynamical Theorem of Turbulence.

The theory of Approximation Dynamics is the branch of mathematics which attempts to answer the question of how well is the long-time dynamics of a given equation reproduced by the long-tim e dynamics of a given equation reproduced by the long-time dynamics of approximate equation. The key issue here is the long-time dynamics, as opposed to the finite-time, transient dynamics. This theory addresses fundamental issues of science arising in m athematical modeling and in numerical computations. Victor A. Pliss (Leningrad University), while in residence at the IMA, collaborated with George R. Sell in a study of the behavior of hyperbolic attractors under small perturbations. In their IMA prepri nt on this subject they give a proof of a basic theorem describing the dynamics on such an attractor after a small perturbation.

The IMA program on dynamics included major effort in the rapidly evolving area of approximate inertial manifolds. Many of the researchers in this subject were in residence at the IMA and wide-spread collaboration were begun. One of the first fruits of this effort was the proof that, under a reasonable assumption on the dimension, every approximate inertial manifold, for any of a class of equations which include the 20 Navier-Stokes equation, is an actual inertial manifold for an approximate equation. This discovery should have a catalytic effect on future research in Approximation Dynamics.

The IMA program on Mathematical Physiology and Differential-Delay Equation was organized as an attempt to introduce the researchers in dynamical system to the new and challenging problems arising in the bio-medical areas. This was a short one-m onth program, and the main goal was to spawn new research efforts. One of the promising new directions is the role of dynamics in the area of high performance computation. This was initiated to a great extent by the presentation by Raymond Winslow (Depar tment of Physiology and Army High Performance Computing Research Center) on his plans and those of his collaborators to model a cellular network of heart cells on a massively parallel Convection Machine. The study of the dynamics of large cellular networ k is an emerging area of science. It is realistic to predict that future collaboration between dynamical and physiologist will soon lead to important insight into problems related to the onset of and the control of heart diseases in humans.

Victor Donnay was a visitor at the IMA from January to June 1990. While at the IMA, Donnay finished two papers. The first was  Hamiltonians on the Two-Torus for which the Hamiltoni an flow is Ergodic'' in collaboration with Carlangelo Liverani. The IMA invited Liverani for a short visit during which time the authors where able to write the final section of the paper. The second was Using Integrability to Produce Chaos: Billiards with Positive Entropy''. In the first version of the paper, Donnay had conjectured that a half-ellipse was a focusing arc, and hence could be used to build a stadium like billiard table which would have positive entropy, providing the eccentricity of the ellipse was less than \sqrt {2}. While at the IMA, Donnay proved the conjecture. In trying to determine the best line of attack for the problem, Donnay used the computer algebra system Maple. He had just learned how to operate this system at the IMA's w orkshop on Computer Algebra and Dynamical Systems. Donnay solved another problem in billiards during his stay. He found the first example of a C-infinity smooth, strictly convex billiard with chaotic behavior. More explicitly, he showed that a small pert urbation of an ellipse produces a billiard with positive topological entropy. The chaotic behavior in this system happens on a set of measure zero; in his previous examples the chaotic behavior happened on a set of full measure but the billiard was only C-1 smooth. A major advance in the field would be to show that this new example actually has chaotic behavior on a set of positive measure (positive measure entropy). This problem arose out of a late night computer graphics session with Mike Jolly and Cl int Scoval. Jolly had written a beautiful interactive computer program on his Iris to model dynamical systems. Donnay suggested applying the program to the elliptical billiard, which is an integrable system, and hence would provide a good test for the pr ogram. Unexpectedly, the system exhibited zones of instability. Scoval pointed out that these zones where due to the existence of a homoclinic connection and the error due to round-off error. A few days later, Donnay went to a meeting at Northwestern. Ge rhard Kneiper, one of the participants, mentioned a problem about geodesic flow on an ellipsoid. Looking at the analogous question for billiards on an ellipse, Donnay realized that this problem was connected to the phenomena that Jolly, Scoval and he had noticed on the computer.

During his stay at the IMA C. Foias worked in collaboration with several other IMA visitors on several topics concerning infinite dimensional dynamical systems mostly centered on the concept of approximate inertial manifolds (AIM), that is manifolds which rapidly epsilon-attracts all trajectories. Thus together with O.P. Manley and R. Temam he concluded research on an approximate inertial manifold for the space periodic 2D Navier-St okes equations which may describe the correct permanent dynamic behavior of large eddies . Also together with E.S. Titi he showed how some well chosen semi-discrete schemes for the Kuramoto-Sivashinsky equation are the compression of the partial differen tial equation to appropriate AIMS . With M. Jolly and E. Titi he continued by elaborating analytically and computationally on the importance of the preservation of dissipativity in the discretization of a dissipative partial differential equaiton. With A . Eden, B. Nicolaenko and R. Temam he improved a theorem by Man'c concerning the projections in Euclidian spaces which are one-to-one on compact sets of finite fractal dimension. With A. Eden he found a very elementary proof of the 1D Lieb-Thirrring ineq ualities and of their Guidaglia-Marion-Temam generalization. Finally with J.-C. Saut, he completely clarified the algebra of a canonical normal form for the Navier-Stokes equations (and for other similar differential equations) which they discovered in s tudying the asymptotic expansion of decaying solutions. A very fruitful part of his stay at IMA was the scientific interaction with many senior and junior IMA visitors and faculties of the University of Minnesota.

BRIEF SELECTION OF RESEARCH ACCOMPLISHMENTS DURING PHASE TRANSITIONS AND FREE BOUNDARIES PROGRAM 1990-91

The scope of the program was to understand certain types of physical behavior which occur in phase transitions and in phenomena which involve free boundaries. The first half of the year was concerned with phase transition s and concentrated on equilibrium and dynamical problems involving two or more phases, with the transition region a sharp interface or a transition layer. Among the physical problems considered were solidification, dendritic growth, nucleation, spinodal decomposition, solid-solid transitions, crystalline states, shock induced phase transitions, and phase structures.

The second half focused on free-boundary problems and on diffusion problems involving a singular mechanism, such as a degeneracy and free boundaries. Here the physical areas considered included porous flow, jets, cavities, lubrication, combustio n, plasma, coating flows, and the dispersal of biological populations.

Robert V. Kohn made the following extended observations about the fall program: I can hardly contain my enthusiasm about the quality of your fall,1990 concentration on Mathematical Modeli ng of phase transitions. It was my privilege to visit IMA for a total of about 3 weeks (September24--28 and November8--21). I cannot imagine 3 weeks of more intense or more productive mathematical activity. The essence of IMA's success is its ability to bring people together in a focused research environment. A list of specific projects and publications cannot come close to describing IMA's true impact. I hope the following summary captures some of the intangible aspects of IMA's influence, as well as t he more tangible ones.

A) New contacts.

A visit to IMA is always an occasion to create new mathematical connections. This time I was especially pleased to meet many young people whom I had previously known only through correspondence, and whose research interests are linked to my own. I hav e in mind specifically K.Bhattacharya (now finishing Ph.D. work with R.James), C.Collins (a recent student of M.Luskin), P.Pedregal (a recent student of D.Kinderlehrer), G.Zanzotto (a recent student of J.Ericksen), V.Sverak (a powerful young Czech mathem atician, who had not visited the US before), L.Truskinovsky (a talented young mechanician who recently immigrated from the USSR), and L.Ma (a current student of M.Luskin). I also appreciated the opportunity to meet several more senior individuals from th e mechanics and phase transitions communities, including M.Glicksman (whose work on mean field theories for Ostwald ripening makes contact with my own on composite materials) and N.Goldenfeld (whose work on intermediate asymptotics is strikingly and unex pectedly related to mine on blowup of solutions of nonlinear parabolic equations).

B) Things learned.

It would be impossible to list all the things I learned from other visitors while at IMA. However, a few special things stand out in my mind as unexpected and significant advances:

a) P.Pedregal and V.Sverak had both made progress concerning the so-called three-well problem.'' They gave two (different) proofs that if a young-measure limit of gradient is supported on three incompatible matrices, then the associated microstructu re cannot be a laminate.'' N.Firoozye and I had previously tried to prove this, without success.

b) I.Fonseca presented a new framework for the variational modeling of coherent phase transitions in the presence of crystalline defects. Though the results are so far only preliminary in character, this represents a significant advance for the theory .

c) M.Chipot spoke about his work with Collins and Kinderlehrer on the numerical minimization of nonconvex energies. I learned from his talk that one can identify a (mesh-dependent) artificial viscosity'' which behaves almost like a surface energy pe rturbation. This provides an unexpected connection between the numerical simulation of coherent phase microstructures and my work with S.Muller (see below).

C) Things taught.

I spent many hours discussing points of common interest with other IMA visitors. In some cases I think I was able to offer useful insight or direction. Here are three specific examples:

a) I had many discussions with N.Firoozye (my former student, now an IMA postdoc) about modelling the evolution of phase boundaries under the analogue of motion by mean curvature'' for a nonconvex surface energy. Stimulated by the lectures of M.Gurt in, Nick was formulating a research program based on regularization.'' We both began to understand some reasonable conjectures as a result of those discussions.

b) Collins and Luskin have been simulating the formation of microstructure in coherent phase transitions via direct numerical minimization of the nonconvex elastic energy functional. My recent work with S.Muller (see below) strongly suggests that they are reaching local rather than global minima in some cases. I suggested ways to test this numerically; it remains to be seen whether they will get numerical solutions similar to the ones Muller and I found analytically.

c) G.Zanzotto has been working with I.Muller on a model for shape-memory behavior based on a non-convex free energy.'' I was able to show him how the failure of convexity that Muller derives from surface energy'' can actually be obtained from bulk energy considerations alone. I also introduce him to the work of Colli, Fremond, and Visintin, which could represent a useful model for the dynamics of shape-memory alloys.

F) A sense of community.

I have emphasized aspects of my visit that were in some sense specific to IMA. This has the effect of de-emphasizing discussions with other visitors I already knew well, such as J.Ball, R.Pego, R.James, and Y.Giga. It would be wrong to ignore the impo rtance of that aspect, However, Mathematical communities are formed through interpersonal contact; to thrive they require the nourishment of gatherings such as the one which took place at IMA this fall.

Forgive me for going on such length. IMA's fall semester concentration on phase transitions was by every measure a spectacular success. The mix of interdisciplinary activity was just right, as was the choice of a focus that was neither too broad nor too narrow. The impact of this activity on the applied mathematical community will been enormous -- but the most important components of that impact are the intangible ones. I hope this letter will serve as a snapshot'' of this impact, including i ts less tangible components specialized to my own individual experience. Please feel free to share my comments with NSF, other funding agencies, and with your participating institutions.

Steven J. Altschuler was a regular IMA postdoc. He reports as follows: Several years ago, E. Calabi suggested a method for flowing through singularities of plane curves evolving by the cur ve shortening flow. The idea is to lift the plane curve to a family of periodic space curves with lifing parameter. This family of curves has infinite time existence and will converge in infinite time to straight lines. One would like to approximate the plane curve evolution with the parameter very small. One then attempts to get uniform bounds on the family after the time of a singularity in order to define the planar flow as a limit of the space curve flow. In joint work with Matthew Grayson, this app roach was successfully completed in October while the two authors were at the IMA.

For the higher dimensional case of mean curvature flow, very little is known about the formation of singularities. In joint work with S. Angenent and Y. Giga, we were able to get a rather detailed picture of singularitiy formation in the case of compa ct hypersurfaces which are rotationally symmetric. In fact, using the language of viscosity solutions and the level set approach, we were able to extend the flow past the time of a singularity and flow until the surface dissapears to a collection of poin ts. This work was started and completed while all three authors were at the IMA.

Finally, in joint work with John Sullivan at the Minnesota Geometry Center and the Supercomputer Institute, we made extensive use of computer modeling in order to give a complete cataloge of all space curves which can be self similar solutions to the curve shortening flow. In a forthcomming paper, we will give asymptotic descriptions of these curves and describe several of their most interesting features.

David C. Dobson was a industrial postdoc during the 1990-91 year, working with scientists from Honeywell, Inc. His research at the IMA has centered on optical diffraction problems involvin g the scattering of electromagnetic waves from various structures.

The first problem studied involves the scattering of an electromagnetic plane wave incident on an optical substrate into which a doubly periodic (periodic in two orthogonal directions) relief profile has been cut. The Maxwell equations are reduced to a coupled pair of operator equations on the surface of discontinuity. The equations are shown to be equivalent to a Fredholm system, thus establishing existence and uniqueness of solutions for all but a discrete set of parameters. This is joint work with Avner Friedman.

The work described above suggests an obvious numerical scheme to solve the diffraction problem. The integral operator equations on the surface of discontinuity are discretized and solved using a boundary element-collocation procedure. This metho d is now partially implemented; initial test runs have been successful. A brief outline of the method and some preliminary numerical results will appear in an upcoming volume of Proceedings of the SPIE with J. Allen Cox from Honeywell as co-author . A mathematical study of the method, convergence analysis, and more complete numerical results has appeared as an IMA preprint.

Associated with the forward diffraction problem discussed above is the inverse problem of designing a substrate profile which diffracts light in some specified way. In the Fraunhofer approximation, the forward diffraction problem can be approximated b y an ordinary Fourier transform. The inverse problem then reduces to finding the phase of a complex-valued function from knowledge of its magnitude and the magnitude of its Fourier transform. This nonlinear problem can be efficiently solved with a quasi- Newton method. A future IMA preprint will describe this method in the infinite-dimensional setting, prove convergence, and give some numerical results. A code has already been implemented and has been applied to real world'' optical design problems at Honeywel

BRIEF SELECTION OF RESEARCH RESULTS DURING APPLIED LINEAR ALGEBRA PROGRAM 1991--92

The academic year program was divided into three parts (corresponding to the fall, winter and spring quarters), although there was considerable fluidity between the various parts.

1. Discrete Matrix Analysis with emphasis on the mathematical analysis of sparse matrices and combinatorial structure;

2. Matrix Computations with special emphasis on iterative methods for solving systems of linear equations and computing the eigenvalues of sparse, possibly structured matrices;

3. Signal Processing, Systems and Control with emphasis on the matrix analysis and computations that arise in this area of application.

Richard A. Brualdi prepared the following assessment of the program: The IMA provides a very special environment which fosters the cooperative interaction of mathematicians and other scientists. This interaction has several effects. Research interests are broadened as a result of one being exposed to new viewpoints and new applications. Interest in an old research problem may be rekindled because of contact with someone now consider ing the problem or because one learns of a possible new application.

Unfinished work may get finished because of the opportunity to work intensively without the usual academic distractions. Problems, which might otherwise go unsolved, get solved by cooperative work. The person or persons with the right knowledge or rig ht technique becomes newly aware of a problem and a successful attacked is made on the problem. New problems are formulated, new techniques are developed, new contacts are made of which some may have a lasting and decisive effect on one's productivity fo r many years to come. I believe that all of these things happened during the Applied Linear Algebra Program. That so many people from applied linear algebra were together for such a long time (one month, one quarter, two quarters or more) was unprecedent ed. Such an opportunity I expect will not occur again in the near future. The effect and success of the program on Applied Linear Algebra, like all IMA programs, cannot be properly assessed until several years after the calendar says that the formal prog ram is over, since the effects of the program go on for many more years after that.

The first quarter of the program concentrated on sparse matrix analysis, and the application of Combinatorial analysis in matrix problems and the application of linear algebra in combinatorial problems. Here we had an unprecedented mix of some of the best sparse matrix theorists (coming from computer science), like Joseph Liu and John Gilbert, and core matrix theorists and discrete mathematicians. Many of the latter group were exposed for the first time to problems in sparse matrix analysis (of which many require symbolic and therefore combinatorial analysis before implementation). Some instances of problems successfully attacked during the first quarter are: (i) One-quarter visitor M. Fielder at the SIAM meeting on Applied Linear Algebra, conjectur ed that the smallest number of nonzero entries in an orthogonal (or unitary) matrix of order n which does not decompose into two smaller orthogonal matrices is 4(n-1) for all n\geq 2. This problem was solved by LeRoy Beasley, postdoctorate Bryan L. Shade r and Richard A. Brualdi by extracting the right combinatorial property implied by orthogonality and then showing that lower bound of 4(n-1) is a consequence of this property. Shmuel Friedland used powerful analytic techniques to obtain tight bounds on t he real eigenvalues of almost skew symmetric matrices and used these bounds to almost' prove a conjecture of Brualdi and Li(1983) concerning the maximum spectral radius of the class of structured matrices known as tournament matrices. Wayne Barrett, Cha rles R. Johnson and Raphael Loewy used the opportunity of being together at the IMA to solve with Tamir Shalom an old problem of theirs to determine the largest number of diagonal entries of a matrix A of order n with rank A=k<n that have to be pertur bed in order to increase the rank. In addition, Shmuel Friedland, Alex Pothen and Richard A. Brualdi collaborated on some analysis of the NP-hard problem of finding a sparsest basis of the row space of a matrix.

The second and third quarters of the program were dominated by numerical linear algebraists, most from computer science and most with interests in numerical algorithms for problems that arise in queuing theory and Markov processes, signal/image proces sing and control theory, and linearizations of nonlinear problems from applied science. During this time the twice weekly Applied Linear Algebra Seminar was heavily attended, drawing people from the computer sciences department of the University and from the nearby Supercomputer Institute. The results here are too technical to be reported in detail in this short report. But there were significant advances, both theoretical (e.g problems concerned with instability and robustness) and computational, conce rning: computation of eigenvalues of large and structured matrices (the kind that often come up in applications) by various algorithms such as the QR method, symmetric and non-symmetric Lanczos method, subspace iteration, and the Arnoldi algorithm. Some of the long term visitors that contributed to these investigations were Gene Golub, G.W. Stewart, Anne Greenbaum, Robert Plemmons, Roland Freund, Zdenek Strakos, Dianne O'Leary, Adam Bojanczyk) and postdoctorates James Nagy, Walter Mascarenhas and Roy Ma thias. Work on improving (and analyzing) algorithms for the computation of the steady state eigenvector of large and structured matrices that arise in Markov chains and queuing models was done by G.W. Stewart, Dianne O'Leary, Robert Plemmons, Carl Meyer, and Paul Schweitzer. Prompted by queries from two of the developers of the linear algebra package LAPACK , namely James Demmel and Z. Bai, Nicholas Higham obtained new error estimates for the Sylvester equation, including stability analysis. Many of his results will be included in the next release of LAPACK. James Demmel and William Gragg characterized in a combinatorial way (the bipartite graph must be acyclic) those matrices, like bidiagonal matrices, which have the property that small relative pert urbations of the entries result in small relative perturbations of the singular values, independent of the values of the entries of the matrix. G.W. Stewart developed a parallel implementation for an algorithm for updating a URV decomposition (unitary t imes upper triangular times unitary) of a matrix which reveals its effective rank.

Victor Klee returned to the IMA for 1 month during the program. He writes: While at IMA I made some progress on a long-standing joint effort with an economist from Univ. of Alberta who vis ited Minneapolis in order to spend some time with me. Of possibly greater interest is the series of papers in high-dimensional computational geometry that Peter Gritzmann and I started while at IMA in 1987. Just before coming to IMA, we had spent a year together in Seattle, but our research projects started in Seattle were put on hold'' (and are still in that situation) when, at IMA we got some seemingly good ideas for some important problems in computational geometry. We have so far published five jo int papers (some with additional co-authors, one of them being Laurent Habsieger (another IMA-er)) that arose directly from our collaboration during the three months at IMA. Two more such papers have been accepted for publication, including the one towar d which several of the others were leading. It will be 51 printed pages long, in Math. Programming early in 1993. We have several other joint projects under way, and I'll be spending the last three months of 1992 in Trier with Peter in order to pursue th ese.

As both Peter and I have often remarked, concerning that Fall of 1987 at IMA, That was a wonderful time!'' Impressively, our wives feel the same way.

BRIEF SELECTION OF RESEARCH ACCOMPLISHMENTS DURING CONTROL THEORY PROGRAM 1992--93

The academic year program was divided into three parts (corresponding to the fall, winter and spring quarters), although there was considerable fluidity between the various parts.

1. Linear and Distributed Parameter Systems

2. Nonlinear Systems and Optimal Control

Hector J. Sussmann prepared the following assessment of the program: I spent the 1992-93 academic year at the I.M.A., participating in the organization and the activities of the Cont rol Theory program. My own research benefitted tremendously from the possibility of interacting with various long-term and short-term visitors. Naturally, when I look at the research that I did at the I.M.A. and the one I am doing now, it is very hard fo r me to draw a clear line between the work that I probably would have done anyhow because it was a direct continuation of my previous work, and the work that owes its existence to the I.M.A. year. There are, however, a few examples where the decisive rol e of the I.M.A. year is particularly clear, so I shall start by describing these.

Another important example of work that I am currently doing that owes its existence to the I.M.A. year is a paper I am writing on a new version of the Pontryagin Maximum Principle under weaker hypotheses than all previous versions (including the Nonsmooth Analysis version of "the Maximum Principle under Minimal Hypotheses" due to F. Clarke) and with stronger conclusions (including high-order necessary conditions for optimality ). I had been interested in the Maximum Principle for a long time, a nd had used various versions of it in my own work. However, most of my interest arose from specific geometric control problems, such as local controllability or the optimal control problems there was never any difficulty arising from lack of smoothness o f the data. I had thought quite a lot about nonsmooth versions of the Maximum Principle, but I had never been sufficiently motivated to pursue this activity. The I.M.A. year, and in particular the February workshop, provided the motivation. Several exper ts on the Nonsmooth approach ---in particular R.T. Rockafellar and B. Mordukhovitch--- discussed versions of the Nonsmooth Maximum Principle, and this made me aware of two things: (a) that there was a great lack of awareness among the differential-geomet rically oriented practitioners of control theory, such as myself, of the importance of extending our results to Nonsmooth settings, and (b) that among the nonsmooth analysts there existed a misperception that geometric methods were of more restricted app licability because they required more smoothness. Based on my own thoughts on the subject, I became convinced that it had to be possible to extend the very best results of geometric optimal control ---which included things such as high-order conditions t hat could not be handled by Nonsmooth methods--- to general situations where smoothness assumptions were not made. This eventually became a true theorem, whose proof I found in August of 1993. I am now completing a paper where a very general version of t he Maximum Principle under minimal conditions is proved. Although the actual writing of this work is taking place after the end of the I.M.A. control year, the work owes its existence to the I.M.A. year, in particular to the discussion with visitors that took place during the February workshop.

The above discussion of the effect of the I.M.A. control year on my own work provides just one illustration of how much was achieved during the year. Other visitors will tell about their own experiences, and I probably should not speak for them, but from my own conversations with several of these visitors I know that in many cases significant new results were obtained and new directions of research were born. For example, Jan Willems, form Groningen, in the Netherlands, spent three months at th e I.M.A., where he met Karen Rudie, who was there for the whole year as a postdoctoral fellow. Although their areas of interest were in principle quite different, it turned out that the set of concepts that Rudie had been developing in order to formulate a general definition of "discrete event system" was closely related to the ideas of Willems on dynamical systems, and this led to a collaboration in which a new version of Willems' general theory, incorporating discre te events, was developed.

Besides its direct effect on the research of participants such as myself, the control theory year contributed in an important way to our professional activity by helping us widen our knowledge and become acquainted with the c urrent state of the arts in fields of control theory other than our own special research area. This is particularly significant for an area such as control theory, which is mathematically very diverse, so that usually a large investment of time and effor t is required ---even for an experienced member of the control community--- to learn about development in neighboring fields. Among the many events that enabled us to acquire a new perspective of other fields within control theory, I would particularly s ingle out the workshop on Fuzzy Logic and its Applications. As is well known, the evaluation of the applications of Fuzzy Logic ---a large number of which are currently being carried out in Japan--- is a hotly contested subject, on which opinions have be en expressed ranging all the way from extremely enthusiastic to highly critical. The format chosen for the Fuzzy Logic workshop, in which scientists directly involved in specific applications presented their work, and extensive discussions followed, made it possible for most of us to become much better informed about the existing applications and about the controversies surrounding them.

Summarizing, I personally regard the I.M.A. control year as having had a very significant impact on my own research, and I think that its effect on the whole field of control theory has been felt by most specialists in the area. The 1992-93 I.M.A. yea r will be remembered for a long time as a major event in our field.

Scott Hansen was a regular IMA postdoc. He reports: Through interaction with the other postdocs, senior researchers and workshop participants the IMA has provided a rare opportunity to lea rn some of the central research issues in a broad spectrum of control theory applications. This broad exposure is particularly important for several reasons. First, due to the highly interdisciplinary nature of control theory it important to develop and maintain lines of communication between the various disciplines involved. In addition this breadth of exposure will be highly beneficial in designing control theory courses and choosing research problems for graduate students. This type of broad exposure to control theory would have been completely impossible to obtain within one university or through attending conferences.

More directly, my own research conducted over the past year has greatly benefited from interaction with other researchers in the area of distributed parameter control. Mainly, I have been involved in three projects: 1) control of thermoelastic s ystems, 2) control of systems involving point masses and 3) modeling/control of plates. I'll briefly describe each and how the program here at the IMA has benefited the research.

Control of thermoelastic systems: When I arrived at the IMA I was writing a paper in which it is shown e.g., that by only controlling the displacement at an end of a thermoelastic rod it is possible to exactly control to zero both the tem perature and displacement. Through discussions with Enrique Zuazua, John Lagnese and Vilmos Komornik (who visited the IMA in the fall) I received several valuable suggestions for improvement as well as suggestions for further research in this area. Thus due to this interaction, work on extending this result is ongoing in two different directions. First, working with Bing-Yu Zhang (IMA industrial postdoc), we have been able to show that the same result holds for thermoelastic beams. I expect to have a jo int paper on this sometime in the next year. Secondly, Professor Zuazua has recently made some progress extending the result to several dimensions. Still however, many important questions remain unanswered and I expect to collaborate with Prof. Zuazua on related problems in the future.

Exact controllability of systems involving point masses: Working with Prof. Zuazua, we considered the problem of boundary control of a string (the one dimensional wave equation) having an interior point mass. We were able to prove that a wave travelling along the string is smoothed out one order as it crosses the point mass. We then showed this result is sharp and consequently obtained an exact controllability result. I was fortunate enough to be invited to Universidad de Complutense in Madrid by Prof. Zuazua for the month of July to work on extensions of this result. Some progress was made showing that a one dimensional mass distribution in a two-dimensional membrane exhibits similar smoothing properties. However this problem is much h arder than in the one dimensional case and much work remains in connection with this problem.

Modelling/control of plates: I was interested in modelling dissipation within a plate or beam due to internal friction. Very roughly, the idea is to glue'' two plates together (as a laminated plated) in such a way that some amount of sl ip is possible at the interface, where dissipation occurs through viscous friction (proportional to the slip velocity). I was very fortunate to be able to discuss this problem with Prof. Lagnese who is one on the leading experts on plate modelling. I rec eived some very helpful suggestions on the choice of notation and formulation of the problem. Later, in proving existence and uniqueness results, I received some highly appreciated help from Jiongmin Yong who has been with the IMA all year.

Karen Rudie was a regular IMA postdoc. She reports: Probably the single most important connection I made while at the IMA, resulting in productive collaboration, was with Prof.\ Jan Willems. Our first project was to determine the computational complexity of a decentralized discrete-event systems problem. We found that earlier work on centralized control by Tsitsiklis could be generalized to the decentralized case, proving tha t a particular decentralized supervisory control problem can be solved in polynomial time. These results appear in the Proc. of the 1993 European Control Conference, were presented by me at the IMA workshop on discrete-event systems, and have been submitted to a journal. Our second project focused on exploring whether Jan's behavioral model for systems could be used to describe discrete-event processes. This project has become more of an ongoing discussion. Our collaboration led to Jan inviting m e to spend a month working with him in Groningen (with travel costs being supplied by the Systems and Control Theory Network in the Netherlands).

During my time there, we concentrated on trying to find an appropriate newmodel for discrete-event systems problems. My area of research is only about a decade old and there is as yet no standard model; therefore, the modeling question is key in our f ield. My time in Holland did not result in a paper but was extremely useful in giving me guidance about how to direct my future research. In this respect, I believe that working with Jan (who is experienced and senior in the field) will prove invaluable for my career.

The other person with whom I have worked at the IMA has been Dr. Nahum Shimkin, a fellow postdoc. We have been focusing on modeling a small industrial problem that had been mentioned to Nahum by a friend who works on automating factories. This control problem requires automating a system (using sensors, actuators and a conveyor belt) that feeds cassettes into machines that wind tape onto the cassettes. The problem specifications require that certain timing constraints be met; consequently, we devoted some time to reading literature on timed discrete-event systems and real-time systems. We decided to formalize our industrial problem using the model of Brandin and Wonham. Unfortunately, like all other models of timed discrete-event syst ems, this model suffers from computational state-space explosion. We are currently trying to model the problem using a crude approximation, in an effort to see whether we can get a reasonable solution that is computationally tractable. Our efforts on thi s front are ongoing. ......

While at the IMA, I received many invitations to speak at conferences and universities, some a very direct result of contacts I made here. In particular, I was invited to give four talks while I was in the Netherlands; was invited to give talks at Cal tech (invited by Prof. John Doyle, who had been a long-term visitor to the IMA in the fall) and UC, Santa Barbara (invited by Prof. Roy Smith, who had visited the IMA for a workshop in the fall); was invited (by Prof. P.R.~Kumar, a long-term visitor at t he IMA) to give talks at the IMA tutorial and workshop on discrete-event systems; ........

My year at the IMA has been extremely productive in terms of making professional contacts and exposing my work to others in the field. I have, for instance, received an invitation to give a talk at the University of Quebec at Montreal by Prof. O mar Cherkaoui, whom I met at the IMA workshop on discrete-event systems.

Bo Bernhardsson was a regular IMA postdoc. He reports: I followed the IMA program in control from November -92 to June -93 as a post doc. This time was very rewarding for me. The opportuni ties for interaction with other researchers was great and the many workshops and tutorials have given me a good chance to broaden my interests. I followed most of the workshops and tutorials and also listened to the industrial postdocs presentations.

My own work during the year was concentrated on problems in the areas of linear and robust control. During the spring time I worked closely together with the two persons I shared room with, Anders Rantzer and Li Qiu, both postdocs at IMA. This c ollaboration resulted in a solution to the so called real stability radius problem. This collaboration was a direct consequence of our common stay at IMA.

The problem is, after some work, given by the following linear algebra problem: Given a complex matrix M, what is the smallest (induced 2-) norm of a real matrix A such that rank (I_{n x n} - A M) = n-k . The crux is the condition of realness. I f this is relaxed and complex A are allowed, the answer is given by the inverse of the kth singular values of M. Our work, which was a continuation of Li's work together with his supervisor Ted Davison, showed that the mixed approximation problem above i s closely related to some singular value like numbers. We worked out a formula for the computation of these numbers and showed how this results in the solution of the real stability radius problem. The solution hints on interesting connections with other areas in linear algebra, analysis and operator theory. The work was presented at the IFAC conference in Sydney 1993, and has also been submitted for publication. There are several ideas for generalizations and further studies.

It turned out that Minnesota was a good place also to meet swedish researchers. This year the conference in acoustics and signal processing was held in Minneapolis. During this week I meat half a dozen of the swedish professors in control and signal processing. It is probably hard to find such a concentration of good swedish researchers even in Sweden at any given time.

During the time at IMA I also prepared some work from my PhD thesis for publication. By presenting the material at IMA I had the chance to discuss and polish the material. I here had good help with comments from other visitors. I especially want to mention A. Rantzer, L. Qiu, P. Kumar and J. Doyle.

BRIEF SELECTION OF RESEARCH ACCOMPLISHMENTS DURING EMERGING APPLICATIONS OF PROBABILITY PROGRAM 1993--94

The academic year program was divided into three parts (corresponding to the fall, winter and spring quarters), although there was considerable fluidity between the various parts.

1. Probability and Computer Science

2. Genetics and Stochastics Network

3. Stochastic Models

J. Michael Steele chaired the Organizing Committee and was in residence at the IMA for nine months during 1993-94. Here is his Retrospective on the Special Year in Applied Probabi lity: The bottom line is that one could not have imagined the year to have been more successful. Some of the results first promulgated during the special year seem destined to become the focus of many years of future research and admiration. The two such results that I have in mind are Talagrand's Isoperimetric Inequality and Yuval Perez' new Capacity Theorem (and its applications to points of multiplicity).< /P>

From my perspective, these results are extraordinary, but the success of the year should be measured more broadly. All of the workshops really did work". I had a special appreciation of the Fall workshops, because of having had a good hand in their organization, and again because of this I found them quite inspiring. Not too long ago there was no serious probabilistic theory of trees, but now the subject---led substantially by Aldous---is undeniably rich. Similarly with the new theory of finite Markov chains, where one now systematically exploits connections to differential geometry and PDE. Even two years ago, almost no one could have imagined these connections or their effectiveness on concrete, discrete problems arising in c omputer science.

The workshops of the Spring term are farther from my expertise, but I have no trouble seeing that the workshop of Peter Donnelly and Simon Tavare was an exceptional success. The witches' brew that made this workshop so visibly effective was the blend of theoretical geneticist and of individuals with real field experience. The same combination was also afoot in the workshop on Hidden Markov Models and their Speech Cousins" that was organized by Steve Levinson and Larry Shepp. There were more c ore participants of the mathematical genetics workshop who were in residence at the IMA for a good stretch, and this, I think, added a lot to the overall effectiveness of the genetics workshop.

Joel Spencer was an organizer for the September, 1993 workshop, and was active at the IMA for three months. He writes:

Talagrand's Inequality

This is really a nice story that shows what a positive effect IMA can have. Let $\Omega=\prod\Omega_i$ be a product probability space, $A\subset\Omega$. Talagrand defines, for $t>0$, a fattening'' $A_t$ containing $A$. To get a rough idea when $\Omega=\{0,1\}^n$ is the Hamming cube then $y\in A_t$ implies $y$ is within Hamming distance $t\sqrt{n}$ of $A$. Talagrand proves $\Pr[A]\Pr[\ol A_t]<e^{-t^2/4}$. Roughly, if $A$ is moderate and $t$ is large then $A_t$ has most all the space.

Mike Steele showed this to a whole group of us. He showed (as Talagrand knew) how to use this to give sharp concentration results for certain random variables. We (Eli Shamir, Svante Janson, Dominic Welsh, myself, and others) started working on it and we (esp. Svante) came up with a general application. Let $h:\Omega \rightarrow R$ be a random variable. Suppose $h(x)$ is not too strongly affected by changing one coordinate of $x=(x_1,\ldots,x_n)$. Suppose further that if $h(x)\geq a$ then ther e is some small'' set of the coordinates $x_i$ that certify'' that $h(x)\geq a$. Then one gets a strong concentration of $h$ around its mean. Talagrand himself then came to a workshop and we had further discussions. We've looked at concentration for a number of classical problems. Consider the job assignment problem where the $i$-th person in the $j$-th job gains a random $a_{ij}$ and $X$ is the gain with the optimal assignment. Then $X$ is strongly concentrated. Consider Minimal Spanning Tree with distances $\rho(x,y)$ being random and $Y$ the size of the MST. Then $Y$ is strongly concentrated. (Janson is working on much more precise results in this case.) I have an IMA Jan '94 preprint that uses these ideas to give an alternate (and I think clean er) proof of the Erd\H{o}s-Hanani conjecture (first shown by R\"odl) on asymptotic packings. I received a preprint by Alan Frieze and Bruce Reed on clique coverings and using these ideas could greatly simplify the proof. It seems clear to me (nothing wri tten yet) that studies of extension statements for random graphs (e.g., every vertex lies in a triangle) will be greatly aided by this idea. Overall, it is a very exciting new concept.

What role did IMA play? The original proof was due to one man (Talagrand) thinking by himself. But the understanding of what the result meant and the dissemination to the general community (which is now fully underway) might not have taken place without the fortuitous grouping of us at IMA and the time available there to explore the potentials.

A problem of Erdos

When Uncle Paul comes to town there are always good problems. Paul asked about the largest induced bipartite graph in a trianglefree graph with $n$ vertices and $e$ edges. With Janson and Luczak we were able pretty much to solve this and it will a ppear in the Proceedings of the Random Structures workshop.

Large Deviations

I benefited tremendously by numerous discussions with Svante Janson on large deviations. For example, let $X$ be the number of empty bins when $m$ balls are thrown randomly into $n$ bins. We saw how to estimate $E[e^{tX}]$ and how to use this to get g ood bounds on the probability of $X$ being far from its mean. I don't think a specific paper will come of this but it was a case of ideas being clarified and distilled.

Ruth J. Williams was one of the organizers of the workshop on Stochastic Networks at the beginning of March, 1994. Here is her report:

INTERACTIONS WITH IMA VISITORS: During her three month stay at the IMA, Professor Williams interacted with visitors in the three areas of diffusion approximations to multiclass queueing networks, passage time moments for reflected diffusions, a nd probabilistic methods for solving nonlinear elliptic equations. These interactions are described in more detail below.

1. Diffusion approximations to multiclass queueing networks. One of the most intriguing open problems for multiclass queueing networks is to determine conditions for the stability of these networks and for approximating them in heavy traffic by r eflected diffusion processes. A related problem is that of determining conditions for positive recurrence of the reflected diffusion processes. Paul Dupuis and Ruth Williams have shown that stability of certain dynamical systems implies positive recurren ce for these reflected diffusion processes. Motivated by this result, Jim Dai has shown that stability of certain fluid models implies positive recurrence of related queueing networks. At the IMA, Professor Williams had discussions with Jim Dai and Gideo n Weiss concerning their recent work on using piecewise linear Lyapunov functions to determine sufficient conditions for stability of fluid models. She also discussed the use of Lyapunov functions to determine conditions for positive recurrence of reflec ted random walks with Michael Menshikov and Vadim Malyshev. Professor Williams is now investigating whether piecewise linear Lyapunov functions might be used to show stability of the dynamical systems of Dupuis and Williams mentioned above. This is part of a larger project to determine concrete conditions for stability of reflected diffusion processes that arise as approximations to multiclass queueing networks.
2. Passage time moments for reflected diffusion processes. At the IMA Professor Williams learned of the work of Michael Menshikov and his coauthors on conditions for finiteness of passage time moments of one-dimensional discrete time semi martingales. These authors applied their results, together with a martingale functional of Varadhan and Williams, to determine conditions for the finiteness of passage time moments of reflected random walks in a quadrant. It is natural to try to extend t hese semimartingale results to continuous time and to apply them to reflected Brownian motions in a quadrant. Professor Williams is collaborating with Michael Menshikov on such a project. To date some progress has been made, but some difficulties still r emain in generalizing the discrete time results to continuous time. Assuming this can be done, it is intended to seek other applications of the semimartingale results, for example to other diffusion processes besides reflected Brownian motions in a quadr ant.
3. Probabilistic methods for solving nonlinear elliptic partial differential equations. Professor Williams is pursuing a mixed probabilistic and analytic approach to solving nonlinear elliptic equations with various boundary conditions in non-smooth domains. Some results that have already been obtained with Z. Q. Chen and Z. Zhao for semilinear elliptic equations were presented at the IMA during the period of concentration on probabilistic methods for solving nonlinear partial differenti al equations. Chen, Williams and Zhao are currently trying to extend these results to include equations with a nonlinear first order derivative term. One of the key ingredients needed for this is a Green function estimate for a linear operator with possi bly singular first order and zero order coefficients, in a Lipschitz domain. Discussions, initiated at the IMA,are continuing with Professor Eugene Fabes on how to derive such an estimate.

INTERACTIONS OF OTHER VISITORS: Jim Dai and Bruce Hajek were good catalysts for discussions on stochastic networks during the three month Winter period. Informal discussions played at least as important a role as formal talks. To give but one example, during a dinner conversation, Bruce Hajek mentioned an open problem for ring networks and the next day Jim Dai presented a solution based on a technique he was developing for determining sufficient conditions for stability of multiclass ne tworks. The four weeks surrounding the stochastic networks workshop were especially lively with the presence of a solid core of experts. Michael Harrison and Frank Kelly were especially instrumental in creating a friendly atmosphere in which they shared their expertise with other participants. The area of stochastic networks is in rapid development. Although I do not know that major open problems were resolved during the IMA period devoted to this subject, I do know that there was progress on understand ing their complexity and subtlety and I expect that we will see some major breakthroughs in this area in the next few years, fueled by the interactions and connections that were made at the IMA.

STOCHASTIC NETWORKS WORKSHOP: The Stochastic Networks workshop held February 28-March 4, 1994, and organized by Frank Kelly and Ruth Williams, was the core for the period of activity on this topic. Stochastic networks are currently used i n studying telecommunications, computer networks, and manufacturing systems. The workshop featured presentations on a variety of open problems for stochastic networks. A notable feature of the workshop was the way in which experts in different areas such as operations research, systems science, and mathematics, converged to discuss problems of common interest from different viewpoints. It is planned that the workshop proceedings volume will convey the current state of knowledge in this rapidly developin g area.

BRIEF SELECTION OF RESEARCH ACCOMPLISHMENTS DURING WAVES AND SCATTERING PROGRAM 1994--95

The academic year program was divided into three parts (corresponding to the fall, winter and spring quarters), although there was considerable fluidity between the various parts.

1. Theory and Computation for Wave Propagation

2. Inverse Problems in Wave Propagation

3. Singularities, Oscillations, Quasiclassical and Multiparticle Problems

Margaret Cheney spent the entire 1994-95 year in residence at the IMA. She reports: Spending my sabbatical at the IMA, for its special year on waves and scattering, was very constructive f or me. I learned a great deal by attending most of the talks at most of the tutorials and workshops. I had many useful conversations with visitors to the IMA. I worked on the following problems.

Finishing the Layer-Stripping Paper. I finished up my paper with David Isaacson on a layer-stripping approach to reconstructing a perturbed dissipative half-space. This paper appeared in the IMA preprint series, and was published in Inverse Pro blems. While at the IMA, I gave a number of talks about this work.

Consulting for Endocardial Solutions. The presence of both myself and Eric Voth at the IMA led me to a formal consulting arrangement with Endocardial Solutions, a small start-up company. This company is attempting to develop an electrical system for treating cardiac arrhythmias without open-heart surgery. They do this by passing into the heart a probe covered with electrodes. The success of the system depends on using the probe to s ense the location of conduction abnormalities. This involves the following three main mathematical problems:

1. Given voltage measurements on the probe due to current sources on the probe, determine the location of the endocardium;
2. Given the location of the endocardium, map the naturally occuring electrical potentials on the endocardium;
3. Use the probe to determine the location of an additional ablation catheter.

These problems are all ill-posed inverse problems reminiscent of the electrical impedance imaging problem I work on at Rensselaer.

Modeling an Anesthesia Breathing System. I worked with Paul Bigeleisen, a staff anesthesiologist at St. Paul--Ramsey Medical Center, in developing a mathematical model of the breathing system used to ad minister anesthesia. This problem is very much in the spirit of the IMA Industrial Mathematics program, in that it involved: 1) a practical, real-world problem, 2) close collaboration between physician and mathematician, 3) modelling work, and 4) experiments. The problem is described in the following excerpt from the introduction to our paper.

A critical time in the administration of anesthesia is the period called induction when the patient has received a drug dose sufficient to paralyze the muscles of respiration, but before tracheal intubation and mechanical ventilation has commenced. So that the anesthetist will have the most possible time to accomplish intubation before the patient becomes hypoxic, the patient should start with the greatest possible oxygen resevoir. In particular, the nitrogen normally in the lungs should be replaced with oxygen before induction. This process, called de-nitrogenation, is accomplished a few minutes before the anesthetic drugs are given by having the patient breathe pure oxygen through a mask connected to the breathing circuit. This circuit is also con nected to the anesthesia vapor machine and is used to administer anesthetic gases after induction and intubation.

There has been considerable debate within the anesthesia community as to the most efficient method to accomplish denitrogenation. In 1981 Gold, using blood gas samples, showed that four deep breaths from a standard circle system produced an oxygen con centration in the blood equivalent to that accomplished by breathing normally for five minutes from the same circle system. Thereafter, it became standard practice to have the patient take four deep breaths of pure oxygen prior to the induction of anest hesia. More recent studies, using pulse oxymetry and in line mass spectrometry, have suggested that four deep breathes are not as efficient at denitrogenation as normal tidal breathing for three minutes. The latter authors suggest, without evidence, that greater rebreathing of nitrogen from the circle system when large breaths are taken, is the cause for inefficient denitrogenation. With rebreathing, the patient is actually recycling part of his/her own breath, including nitrogen, through the circle sys tem. This effect slows down the elimination of nitrogen from the lungs.

In order to determine the most efficient method of denitrogenation, we developed a model of the circle system and compared it to experimental data using the authors as subjects.

Our mathematical model consisted of a system of 8 coupled differential-delay equations. We implemented this model in Matlab. We have nearly completed a paper based on this work.

Processing of Radar Data. I discussed with a number of visitors my ideas for using a geometrical optics expansion to recover the surface electrical properties of a material from the first reflection of a radar pulse. Greg Beylkin suggested that a certain matrix in my scheme might be ill-conditioned, which led me to a method for overcoming this difficulty. Ingrid Daubechies gave me some advice on windowing techniques for signal processing.

Wave Propagation in Random Media. While listening to talks on wave propagation in random media, I started wondering if the method of multiple scales could be used to study the propagation of electromagnetic waves in sea ice. This is a dif ficult wave propagation problem, because sea ice contains a multitude of pockets of brine and air, which are strong scatterers of electromagnetic waves. I discussed this problem with a variety of visitors, who uniformly found the problem very interesting but did not know how to solve it. I am continuing to think about this problem back here at RPI, and am discussing it with Julian Cole.

Other Connections with Industry. Doug Huntley, one of the industrial postdocs, arranged for me to give a talk at 3M. This eventually led to an arrangement between 3M and my impedance imaging research group at Rensselaer, in which 3M suppl ies us with electrodes. I discussed with Keith Kastella, a staff scientist at Unisys, the possibilities for using electrical impedance imaging to determine the extent of ground contamination at waste disposal sites. Our discussions led to an improvement in the algorithm he was developing. I met with representatives of Renaissance Technologies to discuss similarities between the Rensselaer impedance imaging device and their IQ system for assessing cardiac function. The IQ system uses the bulk electrical resistance of the torso, together with sophisticated signal processing techniques, to recover information about cardiac function.

Vladimir Malkin of the Courant Institute visited IMA from September 1 to December 31 in 1994. He reports: During this period I published at two preprints: "On self-focusing of short laser pulses" (IMA #1272, with George Papanicolaou) and "Kolmogorov and nonstationary spectra of optical turbulence" (IMA #1283). Apart from this, I gave a cycle of lectures on model making in nonlinear mathematical physics, PDE seminar on singularities for no nlinear Shroedinger equations and lecture on turbulence theory at IMA, 2 seminars and 2 colloquiums at Mathematical Departments of CWRU and University of Michigan, Ann Arbor. I also gave lectures on instabilities of self-similar collapse and on regulariz ation of singularities of nonlinear wave fields at Mathematical Departments of Chicago University and University of Illinois, Chicago, and a colloquium at Mathematical Department of IUPUI. As a result of the IMA Letter with brief description of my lectur es on model making in nonlinear mathematical physics, I was offered to write a book of Lecture Notes on Nonlinear Mathematical Physics for the World Scientific Publishing.

I had many useful informal contacts at IMA and in participating Universities. In particular, I met in IMA Prof. Tom Spencer from IAS (Princeton) and have considered with him some problems of weak turbulence theory. He suggested me to work out th ese problems during the next academic year at IAS, and I accepted his offer. I also was pleased to consider with Prof. Howard Levine from Iowa State University possible extensions of his famous theorem on blow-up for nonlinear wave equations to a more ge neral and very important for applications class of mixed nonlinear Schroedinger and wave equations. (The possibilityof so far reaching extensions follows from my recent results published in Preprint IMA #1272). I appreciate considerations of self-focusin g of ultra-short powerful laser pulses with Prof. Jeffrey Rauch from University of Michigan, Ann Arbor. It was useful for me to get some impression about industrial mathematics from the "first hands".

BRIEF SELECTION OF RESEARCH ACCOMPLISHMENTS DURING MATERIALS SCIENCE PROGRAM 1995--96

The academic year program was divided into three parts (corresponding to the fall, winter and spring quarters), although there was considerable fluidity between the various parts.

1. Phase Transitions, Optimal Microstructures and Disordered Materials

2. Thin Films, Particulate Flows and Nonlinear Optical Materials

3. Numerical Methods and Topological/Geometric Properties in Polymers

Richard D. James chaired the organizing committee. He reports on the IMA year as follows: The goal of year on Mathematical Methods and Materials Science was to actively encourage mathemati cal research on materials and particularly an exchange of information between mathematicians and materials scientists. This certainly occurred, with particular success in bringing mathematicians in contact with serious materials scientists.

The breadth of coverage of the program can be partly judged based on the variety of materials discussed. The following materials were treated seriously from the point of view of both behavior and mathematical modeling: composites, foams, sea ice, mart ensitic materials, microelectronic materials, nonlinear optical materials, shape-memory materials, soft magnetic films, giant magnetoresistive materials, giant magnetostrictive materials, magnetic sensors, biomaterials, self-assembled composites, porous rocks, disordered conductors, colloidal suspensions, rubber, gels, granular materials, magnetic particles in bacteria, polymers with various degrees of cross linking, optically active materials, liquid crystals (nematic and smectic), piezocomposites, f erroelectrics, magnetic fluids, materials that undergo diffusional phase transformations, silicon, paper, high strength structural alloys, diamond films, brittle ceramics, functionally graded materials, electro-rheological fluids and optical fibers! In m any cases there were several talks on the class of material, and analysis of the appropriate mathematical models.

Research on mathematical problems in materials science is an area of rapid growth in applied mathematics. Some of the workshops were devoted to summarizing the evolving state of research. The workshop on "Phase Transformations, Composite Materia ls and Microstructure" was of this type. It became clear at that workshop that a great many diverse approaches to the homogenization of composites could be related, and streamlined, under the framework of weak convergence methods, primarily compensated c ompactness. There were also new methods revealed which do not fit this framework, based on expansions (Bruno) and careful study of the differential equations (Nesi). Similarly, the workshop on "Disordered Materials and Percolation" summarized improvement s on the calculation of critical exponents near percolation. But new directions emerged on two fronts. First, there were a host of new disordered materials --- from sea ice to fractured ceramics --- that infused this field with new problems. Second, the mathematical implications of conformal invariance of the equations were revealed. The workshop on "Particulate flows: Processing and Rheology" offered an up-to-date summary of problems of the flow of fluids containing particles. These problems are impor tant in industry and also embody exceedingly challenging computational issues. These arise from the sheer complexity of flows with many particles, and from the fundamental difficulty --- still far from resolved --- of computing non-Newtonian flows at hig h Deborah number.

In addition to those workshops that summarized and energized an existing area, new fields of applied mathematics were begun during the year, some unanticipated to the organizers. These are summarized below.

1. The serious study by mathematicians of the schemes currently used by physicists to compute atomic forces and atomic configurations. Physicists at the workshop, "The Mechanical Pro perties of Materials from Angstroms to Meters" described their large scale computations of atomic configurations at various levels of sophistication, and indicated the hopelessness of computing the structure of even the finest microstructural feature of interest in materials science. Mathematicians summarized the success of weak convergence methods and change-of-scale in other areas. For the first time it seems that there were suggested alternatives to the use of conventional computing methods on larger and larger systems. These are under development. As a byproduct of this discussion, some mathematicians began a serious mathematical study of density functional theory. A book by G. Friesecke based on lectures (organized on the fly) and published throug h the IMA will summarize these developments.

2. New theories for the growth and behavior of thin films. During the program NSF/DARPA requested that the IMA organize a workshop on the growth of thin films, with the goal of bring ing new mathematical methods to bear on the growth of films of interest in microelectronics. This preceded another workshop, "Interfaces and Thin Films", that had been organized earlier as part of the year long program. The NSF/DARPA workshop sparked a f lurry of interest in the community, and this was translated into serious mathematical modeling in the workshop on Interfaces and Thin Films. This seems to be a good area for mathematics, involving questions at the boundary between continuum and atomic th eory. It appears that mathematical modeling can make a real impact on the practice of thin film research in materials science, and a variety of mathematicians are now involved.

3. New approaches to the statistical mechanics of polymers, via topological methods. The central idea here is to account for entanglements of long chain polymer molecules in statisti cal mechanical calculations, which were previously not included by DeGennes' theory of reptation or older treatments of rubber elasticity. This has led to discrepancies with experiment on polymers of modern interest. Ideas in this direction had previousl y been initiated by Whittington and Sumners and their collaborators. The workshop exposed a much larger variety of people to these ideas, and generated substantial interest.

4. The first interaction between people doing large-scale computations of turbulence and those doing large-scale computations of microstructure. Two groups of people, similar difficulties with small scales, and severe limitations of complexity which rules out many problems of practical interest. The workshop on "The effect of Small Scales in Computations of Microstructure and Turbulence" brought these two groups together. Combined analyti cal/numerical methods involving weak convergence methods to incorporate the effects of the small scales emerged as a theme.

5. Nonlinear optical materials. While there is a relatively long history of mathematical research on nonlinear optical phenomena, involving deep connections with nonlinear dynamics, and al so on diffraction phenomena, the organizers could not find any mathematician who was doing research on the nonlinear optical material itself. Nevertheless, the workshop attracted a variety of applied mathematicians knowledgeable on optical phenomena, and research on optical materials was begun.

On a personal note, theories of microstructure for martensitic materials have evolved from models that accurately define the energy wells of the material. During the year I pondered the form of the energy wells for a hypothetical material that would combine two types of phenomena: a ferromagnetic transition and a martensitic transformation. Thinking about the properties of this energy well structure led to a strategy for seeking thi s class of materials. The strategy has been put into practice (Joint work with Manfred Wuttig, a materials scientist and IMA visitor), and this class of materials has been realized (based on Fe_3Pd). They currently exhibit the largest known magnetostrict ive effect.

Stefan Mueller (now a director of the MAX-PLANCK INSTITUT F. MATH IN THE NATURAL SCIENCES IN DRESDEN) was a three month visitor at the IMA. He writes: One of the fundamental problems in complex materials is to understand the interaction of different scales (atomic, mesoscopic, macroscopic). Recent progress in the understanding of fine microstructures in advanced materials through continuum models raises important new questions about the foundations of the continuum theory. I used my stay at the IMA to learn about the hierachy of microscopic models (N-body quantum mechanics, density functional theory, embedded atom method, pair potentials) and to continue joint efforts with G. Alber ti, A. De Simone, G. Dolzmann, R.D. James, R. V. Kohn, S. Spector and V. Sverak to develop mathematical tools for the analysis of microstructures and the passage between different scales. I also greatly benefited from discussion with many other scientist s including K. Bhattacharya, D. Dahlberg, D. G. Pettifor, R. Phillips, D. Schryvers, A. Sutton, L. Truskinovsky and J. G. Zhu.

My research mainly focused on elastic and on magnetic materials. In collaboration with Scott Spector I obtained new results on cavitation (spontaneous formation of voids) in elastomers. A joint manuscript with J. Sivaloganathan is finished and w ill be submitted for publication shortly. Jointly with R. D. James I studied the relation between lattice and continuum models of elasticity. We had previously obtained results relating lattice and continuum models for magnetostatic interaction but the g eneralization to elastic materials presents new difficulties. With A. De Simone, G. Dolzmann, R. D. James and R. V. Kohn I discussed the outline of a program for a rigorous derivation of lower-dimensional theories for magnetic materials such as thin film s. This program is at its beginnings. and only the future will tell how successful it will be. An interaction of formal and rigorous asymptotic analysis, analysis of experimental data and numerical computation in this area seems, however, very promising to me. The atmosphere at the IMA which encourages exchange and discussion, as well as the opportunity to meet experimentalists such as D. Dahlberg and researchers involved in computation such as J. G. Zhu was an important boost for our efforts.

In joint work with G. Alberti I made progress in finding a suitable mathematical description for microstructures that involve more than one small scale. A paper which summarizes our results is now almost finished. Applications are so far to some what academic problems but I expect that similar tools will be important for the understanding of crystal microstructure and the multiple scales that arise in micromagnetics.

I also finished a joint paper with G. Dolzmann and N. Hungerbuhler on degenerate elliptic equations which has been accepted in Math. Zeitschrift.

I was also involved in the organization of the workshop Mechanical response of materials from Angstroms to meters' at the beginning of the special year. Following after a week of excellent tutorials by R. V. Kohn, A. Sutton and V. Sverak, it be came a very good forum for exchange of ideas between people from very diverse fields such as {\it ab initio} calculations, electronic structure theory, continuum mechanics, microstructure and variational methods. Many researchers met at this workshop for the first time, and the groundwork was laid for a very fruitful interaction. Of the many interesting contributions I would just like to mention the approach by R. Phillips (and co-workers) which combines the models of continuum mechanics and atomic theo ry in a single numerical code to simulate dislocation nucleation.

As another result of the Angstroms to meters' workshop and the surrounding workshops and tutorials G. Friesecke started to work on a mathematically rigorous exposition of density functional theory (DFT). This is the most common method for large scale ab initio simulations. It is very successful, but a deeper mathematical understanding of its success (and its occasional failures) and thus a way of systematic improvement is almost completely lacking. Friesecke taught a course during the fall se mester at IMAand is now writing a book (to appear in the IMA series) which will make DFT accessible to a large number of researchers whose background is traditional in continuum mechanics and partial differential equations. I think this is one of many ex amples of new and fruitful interactions that were initiated during the IMA year on `Mathematical methods in materials science.'