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Abstract: As expected, the systems in the title come from a useful generalization of the theory of convexity structures. A closure systems C on a set S is a convex geometry if it is algebraic and (extremally) detachable, meaning that A(x)-x\in C whenever x\not\in A\in C (where A(x) the least member of C containing both A and x). Convex geometries have been characterized by many other interesting properties. Among them are various so-called anti-exchange properties. We establish a scale of anti-exchange properties of increasing strength, characterize them by diverse other properties like certain separation axioms, known from topology, or the generation by extreme points, known from the classical case of convex sets. Whereas the properties in question are quite different in general (as witnessed by various examples), they all collapse for algebraic closure systems. But our primary interest lies in the non-algebraic case, which has interesting applications in other fields like the abstract characterization of intervals in closure systems of complete lattices.
In the second part, we establish abstract lattice-theoretical descriptions of the various classes of anti-exchange closure systems under consideration. Moreover, these abstract characterizations extend to categorical equivalences between suitable categories of complete lattices and of closure spaces, respectively.
Colloquium - 16:00
Abstract: Rational polytopes appear in a lots of different fields of mathematics and applications, and the problem to count the integer ("lattice") points in a polytope is thus basic and important.
We survey an approach to this problem via generating functions and with complex-analytic methods. More precisely, we study the number of lattice points as the polytope gets dilated by an integer factor. This expression is known as the Ehrhart quasipolynomial.
We will show applications of Ehrhart quasipolynomials to number theory (Dedekind sums), combinatorics (the `coin-exchange problem' of Frobenius), and computational geometry (the Birkhoff polytope of doubly stochastic matrices).