Math 668: Combinatorial Matrix Theory

Fall 2023

Course meets: TuTh 1:00-2:20, 3866 East Hall.

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

Grader: Tuong Le, tuongle@umich.edu.

Course homepage: http://www.math.lsa.umich.edu/~fomin/668f23.html

Level: introductory graduate.

Student work expected: several problem sets.

Synopsis: This introductory course in combinatorial matrix theory will focus on connections between linear algebra and algebraic combinatorics.

Tentative list of topics: Combinatorial techniques in linear algebra. Basic algebraic graph theory. Determinantal identities. Matroids and projective geometry. Grassmannians and Schubert cells. Canonical forms and factorizations. Polynomials with real roots. Totally positive matrices.

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.
• V. V. Prasolov, Problems and theorems in linear algebra, AMS, 1994.
• B. Sturmfels, Algorithms in invariant theory, Springer Verlag, 1993.