Date:

Tuesday 19th May, 2026 

Location:

International Centre for Mathematical Sciences, Edinburgh @ 5th floor of Bayes Centre

Local organiser:

Dimitra Kosta, University of Edinburgh

Speakers:

Carlos Amendola,  TU Berlin

Heather Harrington, University of Oxford and  Director at MPI CBG  Dresden

Alison La Porta, University of  St Andrews & KU Leuven

Vasiliki Petrotou,  Institut de Mathématiques de Jussieu-Paris Rive Gauche

Jessica Sidman,  Amherst College

Louis Theran,  University of  St Andrews

Nelly Villamizar, Swansea University

Schedule (TBC):

9:30 - 10:30  Louis Theran,  Distributed formation control via rigidity theory

10:30-11:00   Coffee break

11:00 - 12:30 Carlos Amendola, The 4-sample theorem on planar graphs

Vasiliki  Petrotou, Parseval-Rayleigh Identities and volumes

Alison La Porta, Coloured equilibrium stresses and maximum likelihood thresholds

12:30-13:30  Lunch break

13:30 - 14:30  Heather Harrington,  Application driven topological data analysis

14:30 - 15:00  Coffee break

15:00 - 16:00  Nelly Villamizar, TBA

18:00  Public Lecture: Jessica Sidman, Frameworks in motion: from theory to design and back again

Carlos Amendola

The 4-sample theorem on planar graphs

The famous 4-color theorem from graph theory says that any planar graph is 4-colorable. The 4-sample theorem from algebraic statistics, proven by Gross and Sullivant in 2018, says that the maximum likelihood estimator (MLE) for the Gaussian graphical model of a planar graph exists with probability one if one has at least four samples. This number of necessary samples, the maximum likelihood threshold (MLT), is a graph invariant not only connected to parameter estimation, but also to matrix completion and rigidity theory. We will revisit this connection along the lines of a recent Oberwolfach snapshot joint with Thomas Kahle, and ask whether the link between algebraic statistics and rigidity theory persists in the case of extremal graphical models. 

Heather Harrington

Application driven topological data analysis

Persistent homology (PH) is a central tool in topological data analysis. PH provides a multiscale geometric descriptor of data that is functorial, stable to perturbations and interpretable, leading to many applications in mathematics and real-world data. Frequently one starts with point cloud data, a finite subset of a metric space (eg Euclidean space) and studies the topology arising from a filtration of simplicial complexes built on the data. While this process yields an interesting nontrivial descriptor of the "shape of data", some theoretical questions remain. In this talk, we will present two directions in application-driven point cloud persistence. First, we will focus on the spaces of point clouds with the same persistence. This inverse problem asks the following: What is the shape of the fiber of the persistent homology map? We use rigidity theory to study this problem. The second problem is motivated by spatial data arising in biology, which includes outliers (eg histology) or dynamic metric spaces (eg collective dynamics). We present statistics for multiparameter persistence and then apply it to complex biological data.  

Alison La Porta

Coloured equilibrium stresses and maximum likelihood thresholds

In this talk, we consider the space of equilibrium stresses that act on a bar-joint framework in arbitrary dimension. The question that we aim to answer is the following: when does a generic framework admit a non-trivial equilibrium stress which has the same value on certain edges? The main motivation of this research derives from an application to statistics. Specifically, such stresses arise naturally when studying structured dependencies in graphical models, where edge constraints encode correlations between variables. In this setting, the stress-equality conditions correspond to symmetries in the underlying statistical structure.

This is work in collaboration with Tony Nixon and Louis Theran.

Vasiliki Petrotou

Parseval-Rayleigh Identities and volumes

One key tool to understand rigidity and higher rigidity of simplicial spheres, and Gorenstein rings more generally, is to study the volume map, that is, the parametrization of the unique top-dimensional linear stress, in terms of the location of the vertices. Describing this explicitly, in terms of differential identities, turned out to be the key of the simpler proof of the g-conjecture in characteristic two. However, in subsequent work on lattice polytopes, we discovered that these differential identities themselves are a special case of a single polynomial identity that resembles Parseval-Rayleigh identities from Fourier analysis and is simultaneously related to Frobenius twists and residue theory. 

I will present this identity, which we first proved for lattice complexes and have since extended to other classes of objects. This includes past and ongoing joint work with K. Adiprasito, E. Katz, R. Oba, and S. Papadakis.

Jessica Sidman

Frameworks in motion: from theory to design and back again

What do your umbrella, a folding gate, and a scissor lift have in common? They all involve frameworks made of rigid parts attached at flexible joints and are designed to move with one degree of freedom. In 1981 architect Santiago Calatrava wrote a PhD thesis, "Concerning the Foldability of Space Frames," containing a systematic exploration of the geometry and design of foldable frameworks. I'll discuss some of his constructions and the clever ways he built them and then use his thesis as a jumping off point to explore the Laman-Pollaczek-Geiringer Theorem.   

Louis Theran

Distributed formation control via rigidity theory

The distance-constrained formation control problem asks for schemes by which a collection of autonomously acting agents can maintain a desired spatial configuration, using only locally sensed information about the distances to a subset of the other agents.  The case in which the sensing is symmetric in the sense that agent pairs sensing each other do so accurately has a classical solution based on a gradient potential structure, and there are many different analyses in the literature.  I will describe a new, larger family of controllers based on lifting an “edge space" flow on the measurement set of the sensing graph to node space.  This gives a new solution to the symmetric problem, and, more interestingly, provides the first general solution to more realistic “directed” variants of the problem, where the sensing is asymmetric. 

Previous work on directed distributed formation control has centered on a combinatorial notion known as “persistence”.  The rigidity-theoretic approach shows that persistence is neither necessary nor sufficient for local exponential convergence of the natural controller.  I will describe an alternative combinatorial proposal called “algebraic admissibility” based on an algebraic relaxation of a semi-algebraic sufficient condition for local exponential convergence.

This is joint work with J Sidman and D Zelazo.

Nelly Villamizar

Title: TBA

Title: TBA