A következő extra kombinatorika szeminárium időpontja:

hely:

[Szokatlan IDŐ, szokatlan HELY, talán a 11:00 jelentsen 11:00-t] az előadás:

An arrangement of circles is simply a collection of circles in the plane, that pairwise intersect in exactly 0,1, or 2 points. A {\em pseudocircle} is a simple closed curve in some surface. An arrangement of pseudocircles is supposed to mimic an arrangement of circles: the pseudocircles in the collection pairwise intersect in 0,1, or 2 points, although there exist more general models (such as the one we work with) in which any two curves are allowed to intersect each other in any (finite) number of points. Arrangements of pseudocircles are defined in any surface, not only in the plane. The properties of an arrangement of pseudocircles {C₁,C₂, ...,C_n} in an orientable surface are encoded into an "intersection matrix" in the following way. (Suppose for simplicity that each intersection is a crossing, as opposed to tangential). For the first row we traverse (starting at an arbitrary point) the pseudocircle C₁, and write down the order in which we meet the other pseudocircles, including a + or - sign, depending on whether we cross a curve from the right or from the left. We proceed in the same manner for each curve, thus obtaining an "intersection matrix" with $n$ rows (here matrix is put into quotation marks because maybe not all rows have the same number of entries). Ortner investigated the following question: if we are given such an object (an "abstract" intersection matrix M), how can we tell if it is the intersection matrix of an arrangement of pseudocircles in the plane? He proved that this is the case if and only if each submatrix of M with 4 rows is the intersection matrix of an arrangement of pseudocircles in the plane. He asked if this could be generalized to other surfaces. In this talk we will survey Ortner's results, and will give a sketch of our recent proof that an analogous theorem holds for each orientable surface. Time permitting, we will briefly discuss some very recent work we have done on links arising from arrangements of pseudocircles in the plane.

This is joint work with Carolina Medina and Jorge Ramirez-Alfonsin.

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