Department of Mathematics
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An (n,k) configuration is a collection of n points and n "lines" with k points on each line and k lines through each point. The most familiar Desargues and Pappus theorems are classical examples of configuration theorems. Motivated by the geometric definition of a group law on cubic curves, we define a group realization of a configuration C as an embedding f of C into an abelian group G such that a set of k points P₁, P₂,…, P_k are collinear in C if and only if ∑f(P_i) = 0 in the group G.

One of the open problems in this area is the geometric realizability of combinatorial configurations. There is a classical connection between geometric realizability and a group realization. If a configuration C can be embedded in S₁ × S₁, the direct product of two copies of the circle group, then C is geometrically realizable over the complex projective plane. There is a similar theorem for realizability over the real projective plane. In this talk, we show many examples of configurations having group realizations (including all finite projective planes and some bi- planes). We also give several examples of configurations which have no group realization. Our proofs and counter-examples lean heavily on many fields: classical geometry, combinatorics, commutative algebra, algebraic geometry, number theory and equational logic (with computers).

Some of the results above are obtained in collaboration with my graduate student Eric Ens. Several ideas in this paper were inspired by late Professor N.S. Mendelsohn.

Reference:
Branko Grünbaum, Configurations of Points and Lines, Graduate Studies in Mathematics, AMS, Rhode Island, 2009.