[ut.na] IMG-NET Message No. 8

krj@csri.toronto.edu.UUCP (11/28/87)

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25 November 1987
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IMG-NET Message No. 8
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CONTRIBUTED ANNOUNCEMENT:
FROM: Roger Horn
SUBJECT: Thompson's lectures / Conference announcement
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                          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

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       E-mail:  krj@csri.toronto.edu     Phone:   416-978-7075