C5

“The Royal Swedish Academy of Sciences has today decided to award the 2021 Nobel Prize in Physics for ground-breaking contributions to our understanding of complex physical systems”

Last Tuesday this announcement got many in our community very excited: never before had the Nobel prize been awarded to a topic so closely related to Network Science. We will try to understand the contributions that have led to this Nobel Prize announcement and their ties with networks science. The presentation will be held by Erik Hörmann, who has been lucky enough to have had the honour and pleasure of studying and working with one of the awardees, Professor Giorgio Parisi, before joining the Mathematical Institute.

Twisted K-theory is an enrichment of topological K-theory that allows local coefficient systems called twists. For spaces and twists equipped with an action by a group, equivariant twisted K-theory provides an even finer invariant. Equivariant twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the corresponding K-groups to the Verlinde ring of the associated loop group. From the point of view of homotopy theory only a small subgroup of all possible twists is considered in classical treatments. In this talk I will discuss a construction that is joint work with David Evans and produces interesting examples of non-classical twists over the Lie groups SU(n) and over tori constructed from exponential functors. They arise naturally as Fell bundles and are equivariant with respect to the conjugation action of the group on itself. For the determinant functor our construction reproduces the basic gerbe over SU(n) used by Freed, Hopkins and Teleman.

This talk aims to provide a simple introduction on how to probe the
explicit geometry of certain moduli schemes arising in enumerative
geometry. Stable pairs, introduced by Pandharipande and Thomas in 2009, offer a curve-counting theory which is tamer than the Hilbert scheme of
curves used in Donaldson-Thomas theory. In particular, they exclude
curves with zero-dimensional or embedded components.

Ribbons are non-reduced schemes of dimension one, whose non-reduced
structure has multiplicity two in a precise sense. Following Ferrand, Banica, and Forster, there are several results on how to construct
ribbons (and higher non-reduced structures) from the data of line
bundles on a reduced scheme. With this approach, we can consider stable
pairs whose underlying curve is a ribbon: the remaining data is
determined by allowing devenerations of the line bundle defining the
double structure.

In 2013, Bhargava-Shankar proved that (in a suitable sense) the average rank of elliptic curves over Q is bounded above by 1.5, a landmark result which earned Bhargava the Fields medal. Later Bhargava-Gross proved similar results for hyperelliptic curves, and Poonen-Stoll deduced that most hyperelliptic curves of genus g>1 have very few rational points. The goal of my talk is to explain how simple curve singularities and simple Lie algebras come into the picture, via a modified Grothendieck-Brieskorn correspondence.

Moreover, I’ll explain how this viewpoint leads to new results on the arithmetic of curves in families, specifically for certain families of non-hyperelliptic genus 3 curves.

Topology optimisation finds the optimal material distribution of a fluid or solid in a domain, subject to PDE and volume constraints. There are many formulations and we opt for the density approach which results in a PDE, volume and inequality constrained, non-convex, infinite-dimensional optimisation problem without a priori knowledge of a good initial guess. Such problems can exhibit many local minima or even no minima. In practice, heuristics are used to obtain the global minimum, but these can fail even in the simplest of cases. In this talk, we will present an algorithm that solves such problems and systematically discovers as many of these local minima as possible along the way.

A big problem in Riemannian geometry is the search for a "best possible" Riemannian metric on a given compact smooth manifold. When the manifold is complex, one very nice metric we could look for is a Kahler-Einstein metric. For compact Kahler manifolds with non-positive first chern class, these were proven to always exist by Aubin and Yau in the 70's. However, the case of positive first chern class is much more delicate, and there are non-trivial obstructions to existence. It wasn't until this decade that a complete abstract characterisation of Kahler-Einstein metrics became available, in the form of K-stability. This is a purely algebro-geometric stability condition, whose equivalence to the existence of a Kahler-Einstein metric in the Fano case is analogous to the Hitchin-Kobayashi correspondence for vector bundles. In this talk, I will cover the definition of K-stability, its relation to Kahler-Einstein metrics, and (time permitting) give some examples of how K-stability is verified or disproved in practice.

G. Dimitrov and L. Katzarkov introduced in their paper from 2016 the counting of non-commutative curves and their (semi-)stability using T. Bridgeland's stability conditions on triangulated categories. To some degree one could think of this as the non-commutative analog of Gromov-Witten theory. However, its full meaning has not yet been fully discovered. For example there seems to be a relation to proving Markov's conjecture. 

For the talk, I will go over the definitions of stability conditions, non-commutative curves and their counting. After developing some tools relying on working with exceptional collections, I will consider the derived category of representations on the acyclic triangular quiver and will talk about the explicit computation of the invariants for this example.

Recent advances in experimental imaging techniques have allowed us to observe the fine details of how droplets behave upon impact onto a substrate. However, these are highly non-linear, multiscale phenomena and are thus a formidable challenge to model. In addition, when the substrate is deformable, such as an elastic sheet, the fluid-structure interaction introduces an extra layer of complexity.

We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behavior during the early time of impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct simulations designed to give insight into the later times of impact.

We conclude by showing how these methods are complementary, and a combination of both can give a thorough understanding of the droplet impact across timescales. 

Recently, Joyce constructed a Ringel-Hall style graded vertex algebra on the homology of moduli stacks of objects in certain categories of algebro-geometric and representation-theoretic origin. The construction is most natural for 2n-Calabi-Yau categories. We present this construction and explain the geometric reason why it exists. If time permits, we will explain how to compute the homology of the moduli stack of objects in the derived category of a smooth complex projective variety and to identify it with a lattice-type vertex algebra.