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Out of the classical "art of counting" grew modern combinatorics: a network of theories with many connections to pure and applied mathematics. A main focus of our group is the study of intersection patterns of arrangements of submanifolds, with a view towards the topology of the arrangement's complement. Our main tools are matroid theory and combinatorial algebraic topology.
The picture shows an arrangement of lines where every marked point lies on 4 lines and every line passes through 4 marked points. It has 20 points and lines: can you find one with 19?