Lectures and Colloquia during the semester

**November 26, 2001**

Technische Universität Berlin

Straße des 17. Juni 136, 10623 Berlin

Math building - Room MA 042
- map -

** Lecture - 14:15**

### Konrad Polthier, Technische Universität Berlin

### Discrete Constant Mean Curvature Surfaces and Their Index

*Abstract: *
We define triangulated piecewise linear constant mean curvature
surfaces using a variational characterization. These surfaces are
critical for area amongst continuous piecewise linear variations
which preserve the boundary conditions, the simplicial structures,
and (in the nonminimal case) the volume to one side of the surfaces.
We then find explicit formulas for complete examples, such as
discrete minimal catenoids and helicoids.

As an application we study the index of unstable minimal surfaces,
by numerically evaluating the spectra of their Jacobi operators.
Our numerical estimates confirm known results on the index of some smooth
minimal surfaces, and provide additional information regarding their
area-reducing variations. In other applications show that
these discrete geometries provide a good language for studying
meshes used in computer graphics and mathematical visualization.

** Colloquium - 16:00**

### Julian Pfeifle - Technische Universität Berlin

### Long Ascending Paths on Small Polytopes

*Abstract: *
Suppose we are given a (simple) polytope in *d*-space defined by *n*
linear inequalities, and an "upwards" direction (a linear objective
function). If our goal is to climb towards the top by walking along
edges, we can ask, *"How many vertices might we have to visit in
the worst case?"* Behind this lies the decade-old question about
giving upper bounds for the complexity of the simplex algorithm of
linear programming. One obvious bound is, *"At most as many
vertices as a **d*-polytope with *n* facets could possibly have."
In fact, today this is still the *only* known answer!
McMullen's *Upper Bound Theorem* from 1970 explicitly
evaluates this upper bound, and states that no simple *d*-polytope
with *n* facets can have more vertices than the polar of the cyclic
*d*-polytope *C_d(n)*, which has *O(n^{\lfloor d/2\rfloor})* of them,
for fixed *d*.

In this talk, we will see that for *d=4* and *n=7* (the "first
interesting case"), this bound is tight in some instances, i.e. there
exist realizations of *P=C_4(7)^Delta* together with linear objective
functions that yield strictly ascending Hamiltonian paths on the graph
*G*of *P*. Moreover, we will show that other orientations of *G*
that admit Hamiltonian paths and satisfy two well-known necessary
conditions for realizability actually are *not* realizable, thus
providing the smallest known examples showing that these conditions
are not sufficient.

[home] -
[up] -
[top]