C5

The Grojnowski-Nakajima theorem states that the direct sum of the homologies of the Hilbert schemes on n points on an algebraic surface is an irreducible highest weight representation of an infinite-dimensional Heisenberg superalgebra. We present an idea to rederive the Grojnowski-Nakajima theorem using Halpern-Leistner's categorical Kirwan surjectivity theorem and Joyce's theorem that the homology of a moduli space of sheaves is a vertex algebra. We compute the homology of the moduli stack of perfect complexes of coherent sheaves on a smooth quasi-projective variety X, identify it as a (modified) lattice vertex algebra on the Lawson homology of X, and explain its relevance to the aforementioned problem.

Protein interaction networks (PINs) allow the representation and analysis of biological processes in cells. Because cells are dynamic and adaptive, these processes change over time. Thus far, research has focused either on the static PIN analysis or the temporal nature of gene expression. By analysing temporal PINs using multilayer networks, we want to link these efforts. The analysis of temporal PINs gives insights into how proteins, individually and in their entirety, change their biological functions. We present a general procedure that integrates temporal gene expression information with a monolayer PIN to a temporal PIN and allows the detection of modular structure using multilayer modularity maximisation.

We will discuss topological and algebraic aspects of splittings of free groups. In particular we will look at the core of two splittings in terms of CAT(0) cube complexes and systems of surfaces in a doubled handlebody.

It is a classical theorem of Magnus that the word problem for one-relator groups is solvable; its precise complexity remains unknown. A geometric characterization of the complexity is given by the Dehn function. I will present joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of Dehn functions, including the Brady--Bridson snowflake groups on which our work relies.

CAT(0) spaces are defined as having triangles that are no fatter than Euclidean triangles, so it is no surprise that under special conditions  you find pieces of the Euclidean plane appearing in CAT(0) spaces. What is surprising though is how weak these special conditions seem to be. I will present some well known results of this phenomenon, along with detailed sketch proofs.

I will talk about the properties of algebraic integers that can arise as stretch factors of pseudo-Anosoc maps. I will mention a conjecture of Fried on which numbers supposedly arise and Thurston’s theorem that proves a similar result in the context of automorphisms of free groups. Then I will talk about recent developments on the Fried conjecture namely, every Salem number has a power arising as a stretch factor. 

We are all familiar with the need for continuum mechanics-based models in physical applications. In this case, we are interested in large-scale water-wave problems, such as coastal flows and dam breaks.
When modelling these problems, we inevitably wish to solve them on a finite domain, and require boundary conditions to do so. Ideally, we would recreate the semi-infinite nature of a coastline by allowing any generated waves to flow out of the domain, as opposed to them reflecting off the far-field boundary and disrupting the remainder of our simulation. However, applying an appropriate boundary condition is not as straightforward as we might think.
In this talk, we aim to evaluate alternatives to so-called 'active boundary condition' absorption. We will derive a toy model of a shallow-water wavetank, and consider the implementation and efficacy of two 'passive' absorption techniques.
 

A theorem of Gromov states that the number of generators of the fundamental group of a manifold with nonnegative 
curvature is bounded by a constant which only depends on the dimension of the manifold. The main ingredient 
in the proof is Toponogov’s theorem, which roughly speaking says that the triangles on spaces with positive 
curvature, such as spheres, are thick compared to triangles in the Euclidean plane. In the talk I shall explain 
this more carefully and deduce Gromov’s result.

I will explain a beautiful link between differential and algebraic geometry, called the Kempf-Ness Theorem, which says that the natural notions of "quotient spaces" in the symplectic and algebraic categories can often be identified. The result will be presented in its most general form where actions are not necessarily free and hence I will also introduce the notion of stratified spaces.