Jorn van der Pol (TU Eindhoven)
A matroid is a set system that abstracts the notion of “independence” in various settings. The original motivating examples were vector spaces (linear independence) and graphs (cycle-freeness), but matroids today have strong ties with areas of mathematics as diverse as graph theory, projective geometry, coding theory, and combinatorial optimisation.
We are interested in the statistical properties of a “random” matroid. This is useful, for example, for understanding the expected behaviour of algorithms that take a matroid as input.
The problems start when we consider matroids on a large number of points. The number of matroids grows incredibly fast with the number of points—so fast in fact, that even for sets of 15 points it is infeasible to generate a database of all matroids, let alone obtain statistics on such a database.
We have developed cleverer tools to analyse large random matroids.
Souvik Dhara (TU Eindhoven)
Critical percolation on random networks with prescribed degrees
Random graphs play an instrumental role in modelling real-world networks such as those arising from the internet topology and social networks. Percolation, on the other hand, has been the fundamental model for understanding robustness and spread of epidemics on these networks. From a mathematical perspective, percolation is the simplest model that exhibits phase transition, and fascinating features are observed around its critical point. For percolation on complete graphs, Aldous (1997) derived a groundbreaking result characterizing the scaling limits for the sizes of the connected components at criticality, when the network size grows large. Subsequently, there has seen a surge of activity to understand the critical behavior under more general settings. Conjectures from physics state that the behavior of functionals like the component size, diameter, are not guided by microscopic connectivity structure of the networks, but network statistics such as the degree distribution.
In this talk, we consider percolation on random graphs with prescribed degrees, and describe how three fundamentally different types of critical behavior can emerge depending on the moments of the asymptotic degree distribution, with the scaling limit derived by Aldous being one of them.
This talk is based on joint works with Shankar Bhamidi, Remco van der Hofstad, Johan van Leeuwaarden, and Sanchayan Sen.
Joost Nuiten (Utrecht University)
Deformation problems in derived geometry
Given an algebro-geometric object, such as a ring, a vector bundle or a variety, deformation theory studies infinitesimal families of objects around it. It turns out that over a field of characteristic zero, these infinitesimal families can often be classified explicitly in terms of Lie algebras. This fact first arose in the work of Kodaira and Spencer on deformations of complex manifolds, and has been given a prominent role in work of Deligne and Drinfeld. Recent results by Lurie and Pridham explain this phenomenon in terms of derived geometry: most algebro-geometric objects can be organized into (derived) moduli spaces and the formal neighbourhood of a point in such a moduli space is completely determined by its tangent space, endowed with a Lie bracket.
In this talk, I will discuss a variant of this result, giving a Lie algebraic description of the formal neighbourhood of subvariety inside a moduli space, instead of a single point. This can also be used to study formal deformations of the moduli space itself, possibly into a non-commutative object.