Spectral Inequalities

Recommended Prerequisites

Linear Algebra, Commutative Algebra, Linear Analysis.

Teaching Methods

Oral exposition, problem solving, paper reading, discussion.

Learning Outcomes

To revisit material from Linear Algebra, Commutative Algebra and Linear Analysis. To introduce students to important research topics in which there has been recent progress.

Work Placement(s)

No

Syllabus

Eigenvalues of Hermitian matrices and selfadjoint compact operators on Hilbert spaces. Singular values of nonselfadjoint compact operators. Invariant factors of integer matrices and modules.

Extremal characterizations and approximation properties of eigenvalues, singular values and invariant factors.

Inequalities for eigenvalues of sums and for singular values of products. Divisibility relations for invariant factors of matrix products and for extensions and quotients of modules.

Assessment Methods

Assessment
Exam: 40-60% and Synthesis work: 40-60% (including oral presentations): 100.0%

Bibliography

I. C. Gohberg, M. G. Krein, Introduction to the theory of linear nonselfadjoint operators in Hilbert space, American Mathematical Society, 1969.

B. Hartley, T. O. Hawkes, Rings, modules and linear algebra, Chapman & Hall, 1970.

Morris Newman, Integral matrices, Academic Press, 1972.

I. M. Ljubic, J. I. Glazman, Finite-dimensional linear analysis, The MIT Press, 1974.

N. Young, An introduction to Hilbert space, Cambridge University Press, 1988.

Béla Bollobás, Linear analysis: an introductory course, Cambridge University Press, 1999.

A. P. Santana, J. F. Queiró, E. Marques de Sá, Group representations and matrix spectral problems, Linear and Multilinear Algebra 46 (1999), 1-23.

W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. AMS 37 (2000), 209-249.

R. Bhatia, Linear algebra to quantum cohomology: the story of Alfred Horn’s inequalities, Amer. Math. Monthly 108 (2001), 289-318.