# 19th meeting, Edinburgh

## Speakers (confirmed):

Yueqi Cao, Imperial College

Tommi Muller, University of Oxford

Yue Ren, Durham University

Shengding Sun, University of Cambridge

Victoria Schleis, Durham University

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

## Date:

July 11-12, 2024

## Location:

In person at Bayes Centre, University of Edinburgh, 5th floor, room: 5.46 Bayes Centre .

## Local organiser:

Dimitra Kosta, University of Edinburgh

## Registration :

If you wish to attend the meeting, please register by sending an e-mail 📬 to Dimitra Kosta (D.Kosta@ed.ac.uk).

## Schedule:

Thursday 11th July 2024

10:30-11:00 Welcome and coffee

11:00-12:00 Yue Ren, Tropical homotopies two ways

12:00-13:00 Victoria Schleis, Linear maps in tropical geometry

13:00-14:30 Lunch

14:30-15:30 Shengding Sun, On obtaining convex hull of quadratic inequalities using aggregations

15:30-16:30 Tommi Muller, Algebraic Constraints on Common Lines in Cryo-EM

18:30 Dinner

Friday 12th July 2024

9:30-10:00 Morning Coffee

10:00-11:00 Yueqi Cao, Computational tropical Abel-Jacobi map for metric graphs.

11:00-12:00 Daniel Windisch, Algebraic geometry of equilibria in cooperative games

12:30 Lunch

## Title and Abstract collection

Yue Ren, Durham University

Title: Tropical homotopies two ways

Abstract: Polyhedral homotopies were originally introduced by Huber and Sturmfels nearly 30 years ago, and have since become a staple strategy for solving polynomial systems. Main topic of the talk is a generalisation thereof.

Building on ideas of Jensen, Leykin, and Yu, we will discuss two distinct types of tropical homotopies: First, we will discuss how to use tropical points to construct homotopies for solving systems of polynomial equations. Second, we will discuss how to compute tropical points using homotopies for intersecting systems of balanced polyhedral complexes.

Centerpiece of the talk are systems of parametrized polynomial equations, and we will focus on two main cases: Vertically parametrized polynomial systems are systems in which parameters are shared between equations but always bound to the same monomial. These are for example the steady state equations of chemical reaction networks or they arise in the computation of ED or ML degrees.

Horizontally parametrized polynomial systems are systems in which parameters are shared between monomials but always bound to the same equation. These are prominently studied using the theory of Khovanskii bases and Newton Okounkov bodies.

Victoria Schleis, Durham University

Title: Linear maps in tropical geometry

Abstract: Tropical linear spaces and valuated matroids can be used in various applications, for instance to compute generic root counts of different polynomial systems, or to study phylogenetic trees in computational biology.

In this context, it can be helpful to understand linear maps between linear spaces. In this talk, I will give an overview of different candidates for linear maps of tropical linear spaces: (affine) morphisms of valuated matroids, tropical matrix multiplication, and multiplication by valuated bimatroids. We will discuss their properties, advantages and drawbacks from both a mathematical and a computational perspective.

Time permitting, we will additionally construct quivers of tropical linear spaces, which we can use to describe more complicated systems of tropical linear spaces. Through their parameter spaces, we will connect these a priori combinatorial objects to algebraic geometry and draw a connection to quiver representation theory. The talk draws from multiple collaborations, including works with A. Borzi, G. Iezzi, and J. Giansiracusa-F. Rincon-M. Ulirsch.

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.

## Sponsors:

We are grateful for the financial support from the Isaac Newton Institute, the Glasgow Mathematical Journal Learning and Research Support Fund, from the Edinburgh Mathematical Society and the London Mathematical Society.