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:

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.

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.

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:

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.

 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.

 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.

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.

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.

 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.

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.

 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.

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

3. Stochastic and Adaptive Systems

The workshop on Nonsmooth Analysis and Differential Geometric Methods in Optimal Control, held in February 1993, brought together a number of researchers representing both approaches. Until this workshop, there had been little contact between these tw o subcultures with Optimal Control Theory. During the workshop, many of us were able to engage in extensive discussions with people representing the "other side," and this has led to the birth of new directions of research where both approaches are combi ned. I myself am now actively pursuing one of these directions. Specifically, the visit of Prof. Martino Bardi, from Italy, who gave a couple of lectures on viscosity solutions of first-order partial differential equations and the viscosity solution appr oach to the problem of the characterization of the Value Function in deterministic optimal control, made me renew my interest in this issue, on which I had worked about four years earlier, and about which I had taught a course at Rutgers. Prof. Bardi's l ectures included a list of several important open problems in the theory, such as the question of the characterization of the value function as the unique positive viscosity solution of the Bellman equation for linear quadratic optimal control. It turned out that the techniques I had developed in my Rutgers course made it possible to solve this and other problems. So after extensive discussions with Bardi, I contacted a Rutgers student who had the notes of the course and had them typed at the I.M.A., an d I am now working on a number of papers, some on my own, and some with Bardi, based on these notes. This could only have happened in a setting such as that of the I.M.A., where one had plenty of time for discussion after the regular lectures. In the spe cific case of my conversations with Bardi, it took us several days until the precise correspondence between my techniques and his formulation of the problems and techniques became clear.

 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.

 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.

 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.

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.

 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.

 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

 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.

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.

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.

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.

 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

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.

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.

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.

 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".

 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

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.

 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.'

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