Time: 15:30

Location: TU Wien, Freihaus, TUForMath room DAEGH18 (Wiedner Hauptstrasse 8-10, close to Karlsplatz)

ISTA Speaker: Elizaveta Streltsova (Wagner group)

Title:  Face numbers of polytopes and levels in arrangements

Abstract: Levels in arrangements are a fundamental notion in discrete and computational geometry and are a natural generalization of convex polytops. In the talk, I will present two new results on the face numbers of levels in arrangements. Collectively, these numbers form the f-matrix (which generalizes the f-vector of a polytope). We determine the affine space spanned by the f-matrices of all arrangements of n hemispheres in S^d. This completes a long line of research on linear relations between face numbers and answers a question posed by Andrzejak and Welzl in 2003. Moreover, we proved a special case n = d + 4 of the long-standing conjecture of Eckhoff, Linhart, and Welzl on the complexity of the (⩽k)-levels, which implies the Harary-Hill Conjecture on the number of crossings of complete graphs for the class of spherical arc drawings. For the proofs, we introduce the g-matrix, which encodes the f-matrix and generalizes the classical g-vector of a polytope. Joint work with Uli Wagner.

VSM Speaker:  Lorca Heeney-Brockett (TU Vienna)

Title: The Ising model and percolation

Abstract: This talk will be about the Ising model, a model of magnetism whose behaviour has proved interesting enough to study for more than 100 years. We will introduce the phase transition of the two-dimensional model and see (with lots of simulations!) the behaviour at and around the critical point. My main goal is to highlight the special relationship between the Ising model and another model, the random-cluster model, and see how this allows questions of understanding the phase transition to be translated into problems of percolation theory. If there is time, we will then consider an ’infinitely zoomed-out’ limit of the Ising model and see how the relationship with percolation extends to a link between the continuum model and certain random fractal geometries.