Graduiertenkolleg: Methods for Discrete Structures

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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

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

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