krj@csri.toronto.edu.UUCP (11/28/87)
******************************************************************** | | | INTERNATIONAL MATRIX GROUP ( IMG ) | | -------------------------------------- | | The International Linear Algebra Community | | ------------------------------------------ | | | | E-mail Address: MAR23AA @ TECHNION (bitnet) | | | ==================================================================== 25 November 1987 ------------------------ IMG-NET Message No. 8 ------------------------ CONTRIBUTED ANNOUNCEMENT: FROM: Roger Horn SUBJECT: Thompson's lectures / Conference announcement -------------------------------------- BACKGROUND INFORMATION Research Conference and Lecture Series on MATRIX SPECTRAL INEQUALITIES with principal lecturer PROFESSOR ROBERT C. THOMPSON June 20-24, 1988 DESCRIPTION OF LECTURE SERIES AND CONFERENCE The Department of Mathematical Sciences of The Johns Hopkins Univer- sity will host a conference on Matrix Spectral Inequalities on the Johns Hopkins campus during the week of June 20-24, 1988. The principal speaker will be Professor Robert C. Thompson of the University of California at Santa Barbara. Professor Thompson's ten lectures, two each day, will focus on inequalities for eigenvalues, singular values, and invariant factors, with applications to control theory and functional analysis. Participants are encouraged to offer to contribute a short talk on theory, applications, or history of matrix spectral inequalities and related topics. A limited number of contributed talks will be scheduled. Informal interaction will be encouraged by small group discussions, sessions for presentation and discussion of open problems, and social events. Participants are invited to contribute a paper related to the theme of the lecture series, whether or not they offer to contribute a talk. These papers have been invited for submission to the new SIAM Journal on Matrix Analysis and Applications. Participants are also invited to contribute open problems, which will be compiled and distributed at the conference. Contributed problems should include appropriate history, comments, definitions, and references. If there are enough contributed problems, they will be edited and submitted to Linear Algebra and its Applications for publication. The Conference and Lecture Series are co-sponsored by the Johns Hopkins University Press and the Department of Mathematical Sciences, and possibly by one or more federal agencies, with whom applications for support of travel and subsistence expenses for participants are pending. Following the Lecture Series, Professor Thompson will expand on and fill out the lectures as a monograph for the Johns Hopkins Press Series in the Mathematical Sciences. Participants in the Lecture Series and Conference will be selected from applicants from academia, government and industry. In addition to senior researchers, these will include junior faculty and advanced graduate students, who will receive particular consideration for financial support. Inexpensive accommodations will be available in air conditioned University dormitories on campus. Campus food service and admission to the athletic center will be available to participants. PROFESSOR THOMPSON'S LECTURES The general theme of Professor Thompson's lectures could be stated informally as "the classical eigenvalue inequalities revisited, an updating of Chapter 2 of Gohberg & Krein" (Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society Translations, Vol. 18, 1969). However, the lectures will go far beyond what is in Gohberg and Krein's chapter, and will encounter algebraic structures (such as modules over principal ideal domains), noncommutative scalars, representation theory for the symmetric group, the Schubert calculus of algebraic geometry, Lie theory, and functional analysis. The guiding principle is that the classical eigenvalue and singular value inequalities for matrices are all manifestations of a unifying theme penetrating many more areas than seems apparent at first sight. In the most familiar case of matrices with complex entries, one considers inequalities for eigenvalues of Hermitian matrices or for singular values of general complex matrices, special cases of which are important for applications in numerical analysis. For matrices with polynomial entries, the inequalities being considered take the form of divisibility relations on invariant factors; special cases of these relations are discussed in the recent book of I. Gohberg, P. Lancaster, and L. Rodman, Invariant Spaces, Wiley, 1986 (Section 4.3, see also page 291). For matrices with integer entries, the inequalities are again divisibility relations and they have applications in number theory. For matrices with integral quaternion entries, they are a type of divisibility relation for which applications exist in the modern theory of modular forms (see A. Krieg, Modular Forms on Half Spaces, Springer Verlag Lecture Notes in Mathematics, Volume 1143, 1985). The theory to be developed in the lectures has applications in functional analysis, especially in the context of integral and differential operators. For example, a well-known inequality in operator theory named for the functional analyst N. Aronszajn is a special case of the spectral inequalities to be discussed. The Gohberg-Krein book cited previously focuses heavily on this realm of applications to operator theory. There are also applications at the interface of classical Lie groups/Lie algebras. Specifically, the lectures will present certain new facts concerning the exponential mapping from a Lie algebra to a Lie group, for which applications in partial differential equations have been found by the French Lie theorist F. Rouviere. Many of the spectral inequalities considered have interpretations in the context of group representation theory, especially representations of the symmetric, general linear, or unitary groups, and, more specifically, in the part of representation theory that physicists have found to be central to the theory of elementary particles: the decomposition of a tensor product of irreducible representations into irreducible representa- tions. It is interesting to note that the numbers 2/3, -1/3, etc. that circa 1960 were identified as the quark charges had already appeared in the literature in a 1956 paper of F. Berezin and I. M. Gel'fand (Some Remarks on Spherical Functions on Symmetric Riemannian Manifolds, Tr. Mosk. Mat. Obshch. 5, 1956, 311-351), which had as a central purpose the study of the eigenvalues of a sum of Hermitian matrices and the singular values of a product of general matrices. Furthermore, the inequalities that Gohberg, Lancaster, and Rodman discuss in their recent book have precisely the same structure as the Lidskii-Berezin-Gel'fand inequalities for singular values and eigenvalues. The reason for this striking similarity is now under- stood, at least in part, and will be described in the lectures. The explanation involves representations. Finally, spectral inequalities have applications in probability theory. An example occurs in a paper by M. Eaton and M. Perlman (Reflec- tion Groups, Generalized Schur Functions, and the Geometry of Majorization, Annals of Probability 5, 1977, 829-860). The group theory in their paper is not quite the same as in the Berezin-Gel'fand context, involving finite reflection groups instead of representations of the symmetric group, but it isn't far removed, either. The subject of spectral inequalities, properly viewed, pulls together a wealth of ideas involving some of the most fundamental mathematical structures. Since this is not widely known, a major objective of the lectures and the ensuing monograph will be to explain and illustrate this point of view without becoming too technical or requiring too much specialized knowledge. Informal titles for the ten lectures, and working titles for the ten chapters in the monograph to follow, are 1. Nonelementary facts about the exponential function. 2. The eigenvalues of a sum of Hermitian matrices. 3. The eigenvalues of a product of unitary matrices. 4. Eigenvalues and the exponential function. 5. Eigenvalues and the Schubert calculus. 6. Invariant factors of matrix products. 7. Invariant factors of matrix sums. 8. Invariant factors, eigenvalues, and representation theory. 9. Quaternions and invariant factors. 10. Major facts about minors. Each chapter will include a historical perspective for the topics under discussion. Professor Thompson is eminently qualified to present these lectures, as his own research in the area is extensive, highly innovative, and extremely influential. He is, moreover, highly regarded as a lecturer, writer, and editor. SUGGESTED BACKGROUND READING I. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfad- joint Operators, American Mathematical Society Translations, Vol. 18, 1969, Chapter 2. R. C. Thompson, Principal Submatrices IX: Interlacing Inequalities for Singular Values of Submatrices, Linear Algebra Appl. 5 (1972), 1-12. R. C. Thompson and L. Freede, On the Eigenvalues of a Sum of Hermitian Matrices, Linear Algebra Appl. 4 (1971), 369-376. R. C. Thompson and S. Therianos, The Singular Values of Matrix Products, Scripta Math. 29 (1973), 99-110. ---, The Eigenvalues of Complementary Principal Submatrices of a Positive Definite Matrix, Canad. J. Math. 24 (1972), 658-667. ---, On a Construction of B. P. Zwahlen, Linear Mult. Alg. 1 (1974), 309- 325. R. C. Thompson, On the Eigenvalues of a Product of Unitary Matrices I, Linear Mult. Alg. 2 (1974), 13-25. ---, Singular Values, Diagonal Elements, and Convexity, SIAM J. Appl. Math. 32 (1977), 39-63. ---, Convex and Concave Functions of Singular Values of Matrix Sums, Pacific J. Math. 66 (1976), 285-290. ---, Interlacing Inequalities for Invariant Factors, Linear Algebra Appl. 24 (1979), 1-31. ---, The Smith Invariants of a Matrix Sum, Proc. Amer. Math. Soc. 78 (1980), 162-165. ---, Singular Values and Diagonal Elements of Complex Symmetric Matrices, Linear Algebra Appl. 26 (1979), 65-106. ---, The Case of Equality in the Matrix Valued Triangle Inequality, Pacific J. Math. 82 (1979), 279-280. ---, An Inequality for Invariant Factors, Proc. Amer. Math. Soc. 86 (1982), 9-11. ---, Proof of a Conjectured Exponential Formula, Linear Mult. Alg. 19 (1986), 187-197. R. C. Thompson and M. Newman, Numerical Values for the Goldberg Coeffi- cients in the Series for log(exey), Math. Comp. 44 (1987), 265-271. R. C. Thompson, Invariant Factors of Integral Quaternion Matrices, Linear Mult. Alg. 22 (1987), 385-407. PREVIOUS LECTURE SERIES AND MONOGRAPHS The first Mathematical Sciences Lecture Series and Conference was held in June, 1975. Each summer Lecture Series and Conference lasts a week, features a principal speaker or speakers, includes invited participants who report on their own research, and results in a manuscript for a major research monograph by the principal speaker(s), which is considered for publication by The Johns Hopkins University Press. Lecture Series and Conference topics are chosen not only for their timeliness and scholarly value, but also for their consonance with the research interests of the department's faculty. The 1988 lectures by Professor Robert C. Thompson will be the twelfth in the series. Books by Series lecturers that have appeared to date in the Johns Hopkins University Press Series in the Mathematical Sciences include: Nonparametric Probability Density Estimation by R.A. Tapia and J.R. Thompson Matrix-Geometric Methods in Stochastic Models by M.F. Neuts Matrix Computations by G. Golub and C.A. van Loan Traffic Processes in Queueing Networks by R.L. Disney and P.C. Kiessler Additional Series volumes are now in various stages of preparation by the following previous Series lecturers: P.J. Bickel and J.A. Wellner: adaptive statistical inference P.E. Greenwood and A.N. Shiryayev: inference for stochastic processes of semi-martingale type C.R. Johnson: combinatorial aspects of matrix analysis P.C. Fishburn: nonlinear preference and utility theory R.M. Karp: probabilistic analysis of combinatorial algorithms ============================================================================ -- RE-POSTED by Prof. Ken Jackson, Computer Science Dept., Univ. of Toronto. Toronto, Ontario, Canada, M5S 1A4. E-mail: krj@csri.toronto.edu Phone: 416-978-7075