Bristol University

Lahars and huaicos are potent natural hazards that threaten lives and livelihoods. They comprise debris-laden fluid that flows rapidly down slopes, bulking up considerably as they progress. Owing to their rapid onset and the significant threat that they pose to communities and infrastructures, it is important to be able to predict their motion in order to assess quantitatively some of the impacts that they may cause. In this seminar I will present mathematical models of these flows and apply them to various natural settings, drawing on examples from Peru and the Philippines.  Along the way I will show some informative, idealised solutions, the susceptibility of these flows to roll wave instabilities, ways to prevent ill-posedness and how to include measured topography in the computation.

In the first part of this lecture, I will discuss the proof of convergence of the Lorentz process, in the Boltzmann-Grad limit, to a random process governed by a generalised linear Boltzmann equation. This will hold for general scatterer configurations, including certain types of quasicrystals, and include the previously known cases of periodic and Poisson random scatterer configurations. The second part of the lecture will focus on quantum transport in the periodic Lorentz gas in a combined short-wavelength/Boltzmann-Grad limit, and I will report on some partial progress in this challenging problem. Based on joint work with Andreas Strombergsson (part I) and Jory Griffin (part II).

Forcing axioms state that the universe inherits certain properties of generic extensions for a given class of forcings. They are usually formulated via the existence of filters, but several alternative characterisations are known. For instance, Bagaria (2000) characterised some forcing axioms via generic absoluteness for objects of size $\omega_1$. In a related new approach, we consider principles stating the existence of filters that induce correct evaluations of sufficiently simple names in prescribed ways. For example, for the properties ‘nonempty’ or ‘unbounded in $\omega_1$’, consider the principle: whenever this property is forced for a given sufficiently simple name, then there exists a filter inducing an evaluation with the same property. This class of principles turns out to be surprisingly general: we will see how to characterise most known forcing axioms, but also some combinatorial principles that are not known to be equivalent to forcing axioms. This is recent joint work in progress with Christopher Turner.

The mapping class group of a surface is associated to its Teichmüller space. In turn, its boundary consists of projective measured laminations. Similarly, the group of outer automorphisms of a free group is associated to its Outer space. Now the boundary contains equivalence classes of arationaltrees as a subset. There exist distinct projective measured laminations that have the same underlying geodesic lamination, which is also minimal and filling. Such geodesic laminations are called `non-uniquely ergodic'. I will talk briefly about laminations on surfaces and then present a construction of non-uniquely ergodic phenomenon for arational trees. This is joint work with Mladen Bestvina and Jing Tao.

Modularity is a function on graphs which is used in algorithms for community detection. For a given graph G, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in $G$. The (max) modularity $q^\ast(G)$ of the graph $G$ is defined to be the maximum over all vertex partitions of the modularity score, and satisfies $0 \leq q^\ast(G) \leq 1$.

We analyse when community structure of an underlying graph can be determined from an observed subset of the graph. In a natural model where we suppose edges in an underlying graph $G$ appear with some probability in our observed graph $G'$ we describe how high a sampling probability we need to infer the community structure of the underlying graph.

Joint work with Colin McDiarmid.

In geometric group theory we study groups by their actions on metric spaces. Although a given group might admit many actions on different metric spaces, on a large scale these spaces will all look similar, and so the large scale properties of a space on which a group acts are intrinsic to the group. One particularly natural example of a large scale property used in this way is the Gromov boundary of a hyperbolic metric space. This is a topological space that can be thought of as compactifying the metric space at infinity. 

In this talk I will describe some constructions of spaces occurring in this way with nasty, fractal-like properties. On the other hand, there are limits to how pathological these spaces can be: theorems of Bestvina and Mess, Bowditch and Swarup imply that boundaries of hyperbolic groups are locally path connected whenever they are connected. I will discuss these results and some generalisations. 

You’re an amateur investigator hired to uncover the mysterious goings on of a dark cult. They call themselves Geometric Group Theorists and they’re under suspicion of pushing humanity’s knowledge too far. You’ve tracked them down to their supposed headquarters. Foolishly, you enter. Your mind writhes as you gaze unwittingly upon the Eldritch horror they’ve summoned… Group Theory! You think fast; donning the foggy glasses of quasi-isometry, you prevent your mind shattering from the unfathomable complexity of The Beast. You spy a weak spot and the phrase `Gromov Hyperbolicity’ flashes across your mind. You peer deeper, further, forever… only to find yourself somewhere rather familiar, strange, but familiar… no, self-similar! You’ve fought with fractals before, this weirdness can be tamed! Your insight is sufficient and The Beast retreats for now.
In other words, given an infinite group, we associate to it an infinite graph, called a Cayley graph, which gives us a notion of the ‘geometry’ of a group. Through this we can ask what kind of groups have hyperbolic geometry, or at least an approximation of it called Gromov hyperbolicity. Hyperbolic groups are quite a nice class of groups but a large one, so we introduce the Gromov boundary of a hyperbolic group and explain how it can be used to distinguish groups in this class.

Two curves in a closed hyperbolic surface of genus g are of the same type if they differ by a mapping class. Mirzakhani studied the number of curves of given type and of hyperbolic length bounded by L, showing that as L grows, it is asymptotic to a constant times L^{6g-6}. In this talk I will discuss a generalization of this result, allowing for other notions of length. For example, the same asymptotics hold if we put any (singular) Riemannian metric on the surface. The main ingredient in this generalization is to study measures on the space of geodesic currents.

We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger,  $L^{2+\epsilon}$ (rather than $L^2$) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.  

Abstract: If A is a finite dimensional algebra, and D(A) the unbounded
derived category of the full module category Mod-A, then it is
straightforward to see that D(A) is generated (as a "localizing
subcategory") by the indecomposable projectives, and by the simple 
modules. It is not so obvious whether it is generated by the 
indecomposable injectives. In 2001, Keller gave a talk in which he 
remarked that"injectives generate" would imply several of the well-known
homological conjectures, such as the Nunke condition and hence the 
generalized Nakayama
conjecture, and asked if there was any relation to the finitistic 
dimension conjecture. I'll show that an algebra that satisfies "injectives 
generate" also satisfies the finitistic dimension conjecture and discuss 
some examples. I'll present things in a fairly concrete way, so most of 
the talk won't assume much knowledge of derived categories.