Lectures and Colloquia during the semester

**May 17, 2004**

Technische Universität Berlin

Straße des 17. Juni 136

10623 Berlin

Math building - Room MA 042
- map -

** Lecture - 14:15**

### Marcel Erné -Universität Hannover

### Convex Geometries and Anti-Exchange Properties

*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**

### Matthias Beck -Max-Planck-Institut für Mathematik (Bonn) and San
Francisco State University

### Integer-point enumeration in polytopes

*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).

[home] -
[up] -
[top]