19th meeting, Edinburgh

Shengding Sun, University of Cambridge

Title: On obtaining convex hull of quadratic inequalities using aggregations 

Abstract: Obtaining tractable description of the convex hull of a set defined by quadratic inequalities is a central task in the field of mathematical optimisation and other fields of mathematics. A particularly simple case is when the convex hull can be described by certain nonnegative linear combinations of the defining inequalities, which we refer to as aggregations. We define the notion of hyperplane hidden convexity (HHC), which is closely related to the classical notion of hidden convexity of quadratic maps. We show that HHC is sufficient for special aggregations to describe the convex hull. We also make several related discussions, especially on the geometry of special aggregations containing the convex hull. Joint work with Greg Blekherman and Santanu S. Dey. 

Tommi Muller, University of Oxford 

Title: Algebraic Constraints on Common Lines in Cryo-EM

Abstract: We revisit the topic of common lines between projection images in single particle cryo-electron microscopy (cryo-EM). We derive a novel low-rank constraint on a certain 2n×n matrix storing properly-scaled basis vectors for the common lines between n projection images of one molecular conformation. Using this algebraic constraint and others, we give optimization algorithms to denoise common lines and recover the unknown 3D rotations associated to the images. As an application, we develop a clustering algorithm to partition a set of noisy images into homogeneous communities using common lines, in the case of discrete heterogeneity in cryo-EM. We demonstrate the methods on synthetic and experimental datasets.

Daniel Windisch, Max Planck Institute for Mathematics in the Sciences, Leipzig (University of Edinburgh)

Title: Algebraic geometry of equilibria in cooperative games

Abstract: The classical notion of Nash equilibria imposes the somewhat unnatural assumption of independent non-cooperative acting on the players of a game. In 2005, the philosopher Wolfgang Spohn introduced a new concept, called dependency equilibria, that also takes into consideration cooperation of the players. Dependency equilibria are, however, much more involved from a mathematical viewpoint.

This talk will give the necessary background in game theory and will show how basic (real) algebraic geometry can be used to study dependency equilibria and game theoretical questions in general. It is based on joint work with Irem Portakal.

Yueqi Cao, Imperial College 

Title: Computational tropical Abel-Jacobi map for metric graphs. 

Abstract: Metric graphs are widely used to model complex real-world data. The problem of how to extract and represent the geometric and topological information from graph data has given rise to various research areas, such as graph representation learning and graph reconstruction. Despite the immense literature in machine learning, the representation of  metric graphs as abstract tropical curves has never been studied previously in computational and machine learning contexts.  In this talk, I will present an algorithm to compute the tropical Abel--Jacobi map for metric graphs, and discuss potential applications of the tropical Abel--Jacobi map to graph embedding and topological data analysis.