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Abstract: A generalized polygon is a bipartite graph whose diameter equals half the length of a minimal circuit. Generalized polygons can be viewed as spherical buildings of rank two. Spherical buildings, in their turn, can be viewed as certain edge-colored graphs whose rank is defined to be the number of colors. Spherical buildings were introduced by Jacques Tits. He showed in 1974 that the only spherical buildings of rank at least three are those arising from simple algebraic groups (and certain variations of these groups). This is a beautiful example of a theorem in which a few purely geometrical hypotheses end up implying that the only examples display a high degree of symmetry. Since the class of generalized polygons include the incidence graphs of arbitrary projective planes and these are too numerous to classify, it is not possible to extend Tits' result to arbitrary spherical buildings of rank two. In recent work, however, Tits and the speaker have succeeded in extending Tits' original theorem under the so-called Moufang condition. We will give a brief overview of these results for non-group theorists using only the language of graph theory.
Colloquium - 16:00
Abstract: For a certain class of cake division problems the existence of a solution was shown by topological methods. To be more precise: by cousins of the Borsuk-Ulam theorem for more general group actions.
In a very special case a constructive existence proof is known. Here the relation between the topological theorem and its combinatorial counterpart which yields the alternative proof is well understood.
The talk will concentrate on the same relation in more general cases, i.e., the relation between the cousins of the Borsuk-Ulam theorem and its combinatorial counterparts.