#
Monday Lecture and Colloquium

** Monday, February 3, 2014 **

Technische Universität Berlin

Institut für Mathematik

Straße des 17. Juni 136

10623 Berlin

room MA 01

** Lecture - 14:15**

### Raman Sanyal -
Freie Universität Berlin

### Relative Upper Bound Theorems

*Abstract:*

The Upper Bound Theorem, conjectured by Motzkin and proved by McMullen, is one
of the cornerstones of discrete geometry: Neighborly simplicial polytopes
maximize the number of k-dimensional faces among all d-dimensional convex
polytopes with a fixed number of vertices. Stanley vastly generalized the
Upper Bound Theorem by showing that it even holds for general triangulations
of spheres (and beyond). In this talk I will present a generalization of
Stanley's approach that yields an algebraic framework for treating relative
upper bound problems. As showcases I will present solutions to combinatorial
isometric problems and an Upper Bound Theorem for Minkowski sums. The talk
will be a scenic tour from geometry to algebra and back. This is joint work
with Karim Adiprasito (IHES).

**Colloquium - 16:00**

###
Philipp von Falkenhausen - Technische Universität Berlin

### Quantitative Comparative Statics for a Multimarket Paradox

*Abstract:*

Cournot markets are a classic model in economics. A frequent question is: "What happens when a paramter of the market changes?", e.g. when a subsidy is introduced. Analysis of such parameter changes has so far mostly been restricted to a qualitative assessment of the monotonicity of the effect, for example the paradox "profit may decrease when prices increase" found by Bulow, Geneakoplos, and Klemperer (1985).

We introduce a worst case approach that allows for the first time a quantification of the effect of positive price shocks in the multimarket oligopoly model of Bulow et al. We find that a price increase may diminish a firm's profit as well as the overall market welfare by up to 25%. Moreover, even the market's social surplus, which combines profits of the competing firms with the valuations of the buyers, can decrease by almost 17%. We complement our bounds with concrete examples of markets where these bounds are attained.

Letzte Aktualisierung:
27.01.2014