# Graduiertenkolleg: Methods for Discrete Structures

Deutsche Forschungsgemeinschaft
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# Monday Lecture and Colloquium

Monday, December 17, 2007

Technische Universität Berlin
Fakultät II, Institut für Mathematik
Str. des 17. Juni 136,
room MA 041

Lecture - 14:15

### Can Biology lead to New Theorems?

Abstract:
This lecture argues for an affirmative answer to the question in the title. In future interactions between mathematics and biology, both fields will contribute to each other, and, in particular, research in the life sciences will inspire new theorems in pure mathematics. This opinion is illustrated by four theorems on discrete structures.

Colloquium - 16:00

### Extremal Linear Extensions

Abstract:
Given a finite poset \$P\$, we consider pairs of linear extensions of \$P\$ with maximum distance. The distance of two linear extensions \$L_1, L_2\$ is the number of pairs of elements of \$P\$ appearing in different order in \$L_1\$ and \$L_2\$. A diametral pair maximizes the distance among all pairs of linear extensions of \$P\$. Diametral pairs are e.g. of interest because their intersection forms a two-dimensional poset close to \$P\$. We are interested in analyzing extremal linear extensions, i.e., those contained in a diametral pair. I will present a conjecture and partial results suggesting a connection between extremal linear extensions and critical pairs of \$P\$. I will also present some proofs, using the concept of the linear extension graph of \$P\$: Every linear extension represents a vertex of this graph, and two of them are adjacent if they only differ by an adjacent transposition. This is joint work with Stefan Felsner.

Letzte Aktualisierung: 21.11.2007