Speaker #1: Monique Laurent

Affiliation: CWI Amsterdam and Tilburg University

 

Title: Spherical Gram embeddings of graphs

Abstract: We consider geometric realizations of graphs obtained by representing weighted graphs as inner products of vectors in low dimensional spheres.

Given a graph $G=(V,E)$, let $\E(G)$ denote the convex set of all edge weights $x\in \oR^E$ that can be realized as inner products of unit vectors in some space $\oR^d$. Then the {\em Gram dimension} of $G$ is the smallest dimension $d$ allowing to realize all edge weights in $\E(G)$, while the {\em extreme Gram dimension} of $G$ is the smallest dimension $d$ allowing to realize all extreme points of $\E(G)$. These two new graph parameters are motivated by their applications to finding low rank solutions to semidefinite programs, their relevance to matrix completions problems, to the rank constrained Grothendieck constant, and to Colin de Verdi\`ere type graph invariants. Both are minor monotone. We give an explicit forbidden minor characterization of the graphs having small (extreme) Gram dimension as well as structural results.

Based on joint work with M. E.-Nagy and A. Varvitsiotis (CWI Amsterdam).

 

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Speaker #2: Joost Batenburg

Affiliation CWI Amsterdam and University of Antwerp

 

Title: Discrete Tomography for lattice images

 

Abstract: Tomography deals with the reconstruction of images from their projections.

In Discrete Tomography, it is assumed that the unknown image only contains grey levels from a small, discrete set. If one additionally assumes that the image is defined on a discrete domain, we arrive at the field of Discrete Tomography for lattice images. Recently, tomography problems for lattice images have become of high practical relevance, due to their applicability to the reconstruction of nanocrystals at atomic resolution from projections obtained by electron microscopy. Although the reconstruction problem for lattice images appears quite elementary at first sight, it relates to many different subfields of mathematics, ranging from number theory and combinatorics to continuous optimization and analysis. In each of these directions, interesting and sometimes surprising results have been obtained during the past 10 years. In this lecture I will illustrate the links of this tomography problem with different fields of mathematics and highlight some important results, followed by posing some new research questions that are currently unsolved.

 

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Speaker #3: Jan Draisma

Affiliation: TU Eindhoven

 

Title: Bounded-rank tensors

Abstract: The notion of rank for matrices (two-dimensional tensors) has a natural generalisation to higher-dimensional tensors. However, in higher dimensions it is much less well behaved than for matrices. For instance, it is NP-hard to compute, and equations for tensors of bounded rank (the analogue of determinants for matrices) are not known in general. I will present some recent and ongoing work on those equations and on the complexity when the rank is fixed but the dimension of the tensor is allowed to vary.