All sessions will take place in room LMB/031, within the Law & Management Building, University of York

Louis Theran (University of St Andrews): Higher order rigidity and pre-stress stability

A framework (G,p) is a placement of the vertices of a finite graph in a Euclidean space.  We say (G,p) is rigid if any (G,q) sufficiently close to (G,p) that has the same edge lengths is congruent to (G,p). One of the most fundamental questions in rigidity theory is whether a specific framework is rigid.  While this can be addressed by using methods from computational semi-algebraic geometry, this approach is infeasible in most cases.

In practice, sufficient conditions for rigidity, such as "infinitesimal rigidity" and "prestress stability" are used instead, since they can be checked with linear algebra or semidefinite programming, respectively.  

However, certain structures arising in applications such as biophysics and statistical mechanics are known not to be amenable to this kind of first order analysis, motivating interest in sufficient conditions for rigidity that incorporate higher order information.  Various proposals for such tests have been made in the structural engineering and robotics literature, but all of them are known to have theoretical and practical limitations, so finding a correct notion of higher order rigidity has been an open problem.

I will describe an energy-based approach that assigns to each rigid framework a numerical “rigidity order” that quantifies how rigid it is, and give some examples of how it can be used on practical problems. The rigidity order is universal over all “stiff bar” energies, a class that includes all the energies commonly encountered in applications, including spring energies and the Lennard-Jones potential. It generalises the notions of first and second order rigidity, and it can be characterised in terms of “higher order flexes”. In certain specific cases, the rigidity order can be computed efficiently. There is also a variant of rigidity order for “prestressed energies” that generalises prestress stability.

This talk is based on joint work with S J Gortler and M Holmes-Cerfon.

Yizhi Wang (University of York): GIT Stability and Connectivity of Helmke Systems

Helmke systems were introduced as generalisations of classical linear dynamical systems. It is well understood that Helmke systems of a given type can be realised as the space of representations of a marked quiver, with an action of a complex reductive group induced by changes of basis at the marked vertices. A choice of character of this group determines corresponding GIT stable and semistable loci.

In this talk I will first describe a general theorem valid for GIT quotients of affine space that determines when the homotopy groups of the stable locus vanish. This gives nontrivial results for choices of character where the stable and semistable loci do not coincide. I will then describe a control-theoretic application to Helmke systems, where (for particular choices of character) the stable locus corresponds to the systems that are both controllable and observable, and the theorem can be used to compute the connectivity of this space of systems.

Alex Milner (University of Edinburgh): Distance Reduction in Toric Ideals of Graphs

From a statistical standpoint, Markov bases provide moves for constructing connected Markov chains on the sample spaces arising in conditional tests for discrete exponential families. However, from an algebraic point of view, a Markov basis is simply a generating set of a toric ideal. The distance-reduction property ensures that a Markov basis provides moves which make progress through the sample space with respect to a convenient metric (the 1-norm). Given any two states in the sample space, we ask whether one can always apply a move from the Markov basis to one of them so as to decrease the distance between the two. If this is always the case, then the Markov basis is called distance-reducing.

In this talk, we will look at distance reduction in the context of toric ideals coming from graphs. Toric ideals of graphs are particularly nice for two reasons: they are homogeneous, meaning that the distance-reduction condition significantly simplifies, and also, primitive elements of the toric ideal can be represented as even, closed walks on the graph, allowing questions about distance-reduction to be translated into graph-theoretic ones. Finally, we will see that when the toric ideal of a graph is a complete intersection, a minimal Markov basis is distance-reducing if and only if it distance-reduces the circuits (and we also discuss evidence suggesting that the complete-intersection hypothesis may not in fact be necessary).

Maximilien Gadouleau (Durham University): Semirings of formal sums and injective partial transformations

The semiring of partial transformations is an algebraic way to model modularity in deterministic systems. This semiring is rather difficult to work with, where even the division problem seems computationally hard. As such, we use the framework of semirings of formal sums to simplify problems and highlight interesting algebraic properties. In this talk, we will focus on injective partial transformations, and formal sums thereof over the binary field. In terms of algebra, we discover structures, such as Boolean algebras ; we characterise the extended Green’s relations ; we classify idempotents, units, and regular elements; etc. In terms of algorithms, we show that the division problem is straightforward in this setting. This is joint work with Marianne Johnson, University of Manchester.

Eliana Duarte (University of Porto): An algebraic approach to context-specific conditional independence via toric ideals

In this talk I will give a brief introduction to the study of conditional independence (CI) and context-specific conditional (CSI) independence through the use of commutative algebra. I will explain how algebra is useful to better understand the set of all CI and CSI statements that are implied by a set of such statements. The case when the ideals that arise are toric is combinatorially and algebraically very appealing. I will then present results concerning the toric ideals of certain context-specific independence models. This is joint work with Yulia Alexander and Julian Vill. arXiv:2210.11521

Qiquan Wang (Queen Mary University of London): A Topological Lens on Biology and AI

Topological methods offer a way to study complex data through their underlying shape and structure. This talk explores how one such tool, persistent homology, can be used to quantify multiscale topological structure and uncover informative signatures in two seemingly disparate settings: cancer morphology and artificial intelligence. In acute myeloid leukaemia, persistent homology captures structural abnormalities in bone marrow vascular architecture, distinguishing between control, early, and late stages of disease progression. These topological summaries are integrated into stage-dependent Gaussian mixture models, enabling inference on morphological changes and supporting prediction. In large language models, persistent homology characterises how adversarial perturbations reshape internal representation spaces. Across models and attack mechanisms, adversarial influence induces a consistent signature of topological compression, whereby diverse small-scale structures give way to fewer dominant large-scale features. Together, these case studies illustrate how topology can provide new perspectives on the structure and evolution of complex systems, highlighting the breadth of topological methods in applications spanning biology and AI.

Brett Kolesnik (University of Warwick): Coxeter tournaments

A tournament is an orientation of the complete graph. In this talk, we will introduce Coxeter analogues of tournaments and discuss connections with algebra, combinatorics, and geometry. 

Roan Talbut (Durham University): Uniqueness of Fréchet Means for Polytope Norms

Fréchet means are a popular type of average for non-Euclidean datasets, defined as those points which minimise the sum squared distance to a set of datapoints. In this talk, we discuss the behaviour of sample Fréchet means on normed spaces whose unit ball is a polytope, a setting with direct applications to phylogenetic tree data analysis via tropical geometry. This setting is rarely covered by existing literature on Fréchet means, which focuses on smooth spaces or spaces with bounded curvature. We study the geometry of the set of Fréchet means over polytope normed spaces, with a focus on dimension and probabilistic conditions for uniqueness. In particular, we study the threshold sample size at which Fréchet means have a positive probability of being unique.