Math 669: Combinatorial Matrix Theory

Winter 2019

Course meets: TuTh 1:00-2:20, 3096 East Hall.

Instructor: Sergey Fomin, 4868 East Hall, 764-6297, fomin@umich.edu

Grader: Shiliang Gao, shiliang@umich.edu.

Course homepage: http://www.math.lsa.umich.edu/~fomin/669w19.html

Level: introductory graduate.

Student work expected: several problem sets.

Synopsis: This is an introductory graduate course in Combinatorial Matrix Theory, emphasizing its algebraic aspects.

Tentative list of topics: Combinatorial techniques in linear algebra. Applications of linear algebra in enumerative combinatorics. Special classes of matrices. Determinantal identities. Matroids and projective geometry. Grassmannians and Schubert varieties. Polynomials with real roots. Total positivity.

Reference texts (none required):

• A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler, Oriented matroids, Cambridge University Press, 1999.
• R. A. Brualdi and D. Cvetkovic, A Combinatorial approach to matrix theory and its applications, Chapman & Hall/CRC, 2008.
• R. A. Brualdi and H. J. Ryser, Combinatorial matrix theory, Cambridge University Press, Cambridge, 1991.
• 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.
• S. Treil, Linear Algebra Done Wrong.