M502 Combinatorics II, Spring 2006

Instructor

Dr. A. Betten, room 207, Weber building

When and where

M  W  F     2:10 - 3:00 ENGRG E106

Texts

van Lint / Wilson: Combinatorics, Cambridge

Prerequisites

M501 would be nice but this time we open it up to everybody! (I'll give you a brief review in the beginning anyway)

Topics

M502 is the second course in a 1-year's sequence of graduate intoductory combinatorics. The goal is to show how basic structures can be attacked with combinatorial methods. This will give the student an understanding of the methods with which discrete structures are explored, constructed and classified. The border to actual research is not very far from where we will be.

• Algebraic graph theory (eigenvalues, Bose-Mesner algebra, special classes of graphs and digraphs etc)

• Isomorph free exhaustive generation of graphs and other discrete structures using the method of canonical forms and partition backtracking.

• Hadamard matrices (construction methods, the new Hadamard matrix of order 428 constructed by Kharagani and Tayfeh-Rezaie in 2005)

• Error correcting codes, parameter, bounds, classes of codes, weights, duality, Reed-Muller, BCH and Reed-Solomon codes, codes from curves.

• Projective spaces, in particular finite projective planes, Desargues, Pappus, coordinates, Latin squares, collineations.

• Ovals in Desarguesian projective planes, their Combinatorics, new construction methods using geometric codes.

• Combinatorics of tableaux, the hook formula, symmetric polynomials (elementary, monomial, complete, forgotten, power, Schur) and their Combinatorics. Characters of S_n.

• Designs, Steiner systems, parameter, standard constructions, square designs, construction with prescribed automorphism group, codes and designs.

• Linear Spaces, in particular the recent theory of line transitive point imprimitive ones, their Combinatorics and how to generate them.

• Extremal Combinatorics: distinct representatives (Hall), Ramsey, Sperner, Erdos etc.

Grading Rules

Class participation and homework are essential, as is the final exam. I expect that course participants present small subject topics towards the end of the semester. The mix is: final / subject topic presentation / homework / class participation as to 30 / 30 / 30 / 10.

For further information

Please contact A. Betten:
betten@math.colostate.edu betten@math.colostate.edu

or visit the course webpage at
http://www.math.colostate.edu/ betten/courses/M502/SP06/index.html. http://www.math.colostate.edu/~betten/courses/M502/SP06/index.html