Math 665: Combinatorial Matrix Theory
Fall 2003
Course meets: TuTh 2:40-4, 3088 East Hall.
Instructor:
Sergey Fomin, 2858 East Hall, 764-6297,
fomin@umich.edu
Course homepage: http://www.math.lsa.umich.edu/~fomin/665f03.html
Level: introductory graduate.
Student work expected: several problem sets.
Synopsis:
This is an introductory graduate course in matrix theory, emphasizing its
algebraic and combinatorial aspects (as opposed to analytic and numerical).
Reference texts (none required):
- A.Björner, M.Las Vergnas, B.Sturmfels, N.White, G.Ziegler,
Oriented matroids,
Cambridge University Press, 1999.
- R.A.Brualdi and H.J.Ryser, Combinatorial matrix
theory,
Cambridge University Press, Cambridge, 1991.
- S.Fomin and A.Zelevinsky,
Total
positivity: tests and parametrizations,
Math. Intelligencer 22 (2000), 23-33.
- F.R.Gantmacher, The theory of matrices, vol.1-2,
AMS, 1998.
- F.R.Gantmacher and M.G.Krein, Oscillation matrices
and kernels and small vibrations of mechanical systems,
AMS, 2002.
- V.V.Prasolov, Problems and theorems in linear
algebra, AMS, 1994.
- B.Sturmfels, Algorithms in invariant theory,
Springer Verlag, 1993.
Lecture topics:
- Overview. Linear-algebraic proof of Hall's marriage theorem.
Combinatorial proof of the Cayley-Hamilton theorem.
- Polynomial identities in matrix algebras.
The Amitsur-Levitzki theorem.
- Combinatorial evaluation of determinants: Vandermonde, Cauchy,
and Smith.
- Compound matrices. Theorems of Binet-Cauchy, Kronecker, and
Sylvester-Franke.
- Jacobi's formula. Muir's Extension Principle and its
applications.
Grassmannians. Plücker embedding.
- First and second fundamental theorems of invariant theory.
Grassmann-Plücker relations and
van der Waerden syzygies.
- The Grassmann-Cayley algebra and its applications in projective
geometry.
Theorems of
Desargues
and
Pappus.
- Matroids. Realizability.
- Matroid stratification of a Grassmannian.
-
Oriented matroids. Pseudoline arrangements.
- Greedy algorithm for a minimal cost basis.
Schubert cells in a Grassmannian.
- The Bruhat order on a Grassmannian. Schubert varieties.
- Bruhat decomposition of a general linear group.
- Jordan form. Nilpotent varieties. The Gansner-Saks theorem.
- Applications to extremal poset theory: theorems of Dilworth and
Greene.
- Stabilizer of two flags. The Robinson-Schensted correspondence.
- Gaussian decomposition. Jacobi's rule.
- Real roots of polynomials. The Sturm-Hermite criterion.
-
Real-rooted polynomials in combinatorics.
The Leverrier-Faddeev method.
- Perron-Frobenius theory.
- Total positivity: first steps.
- Spectra of totally positive and oscillatory matrices.
The Karlin-McGregor-Lindström lemma.
- Rhombus tilings of a hexagon.
Total positivity of Toeplitz matrices.
- Schur functions. Gessel-Viennot proof of the Jacobi-Trudi
formula.
Parametrization of totally positive matrices.
- Total positivity criterion of Gasca and Peña.
Theorems of Loewner and Whitney.
- Pfaffians (guest lecture by A.Barvinok).
- General total positivity criteria.
Totally nonnegative varieties.
Homework assignments:
#1,
#2.