University College London

The mitigation of bacterial adhesion to surfaces and subsequent biofilm formation is a key challenge in healthcare and manufacturing processes. To accurately predict biofilm formation you must determine how changes to bacteria behaviours and dynamics alter their ability to adhere to surfaces. In this talk, I will present a framework for incorporating microscale behaviour into continuum models using techniques from statistical mechanics at the microscale combined with boundary-layer theory at the macroscale.

We will examine the flow of a dilute suspension of motile bacteria over a flat absorbing surface, developing an effective model for the bacteria density near the boundary inspired by the classical Lévêque boundary layer problem. We use our effective model to derive analytical solutions for the bacterial adhesion rate as a function of fluid shear rate and individual motility parameters of the bacteria, validating against stochastic numerical simulations of individual bacteria. We find that bacterial adhesion is greatest at intermediate flow rates, since at higher flow rates shear-induced upstream swimming limits adhesion.

In 1975, Bollobás, Erdős, and Szemerédi asked the following Zarankiewicz-type problem. What is the smallest $\tau$ such that an $n \times n \times n$ tripartite graph with minimum degree $n + \tau$ must contain $K_{t, t, t}$? They further conjectured that $\tau = O(n^{1/2})$ when $t = 2$.

I will discuss our proof that $\tau = O(n^{1 - 1/t})$ (confirming their conjecture) and an infinite family of extremal examples. The bound $O(n^{1 - 1/t})$ is best possible whenever the Kővári-Sós-Turán bound $\operatorname{ex}(n, K_{t, t}) = O(n^{2 - 1/t})$ is (which is widely-conjectured to be the case).

This is joint work with Francesco Di Braccio (LSE).

A classic result of Chvatál and Erdős (1972) asserts that, if the vertex-connectivity of a graph G is at least as large as its independence number, then G has a Hamilton cycle. We prove a similar result, implying that a graph G is pancyclic, namely it contains cycles of all lengths between 3 and |G|: we show that if |G| is large and the vertex-connectivity of G is larger than its independence number, then G is pancyclic. This confirms a conjecture of Jackson and Ordaz (1990) for large graphs.

Let $N$ be a compact Riemannian manifold of dimension 3 or higher, and $g$ a Lipschitz non-negative (or non-positive) function on $N$. In joint works with Neshan Wickramasekera we prove that there exists a closed hypersurface $M$ whose mean curvature attains the values prescribed by $g$. Except possibly for a small singular set (of codimension 7 or higher), the hypersurface $M$ is $C^2$ immersed and two-sided (it admits a global unit normal); the scalar mean curvature at $x$ is $g(x)$ with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when $g$ is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature.

We study the behaviour of anti-self-dual instantons on $\mathbb{R}^3 \times S^1$ (also known as calorons) under codimension-1 collapse, i.e. when the circle factor shrinks to zero length. In this limit, the instanton equation reduces to the well-known Bogomolny equation of magnetic monopoles on $\mathbb{R}^3 $. However, inspired by work of Kraan and van Baal in the mathematical physics literature, we show how $SU(2)$ instantons can be realised as superpositions of monopoles and "rotated monopoles" glued into a singular background abelian configuration consisting of Dirac monopoles of positive and negative charges. I will also discuss generalisations of the construction to calorons with arbitrary structure group and potential applications to the hyperkähler geometry of moduli spaces of calorons. This is joint work with Calum Ross.

In joint work with Peter Topping we introduce pyramid Ricci flows, defined throughout uniform regions of spacetime that are not simply parabolic cylinders, and enjoying curvature estimates that are not required to remain spatially constant throughout the domain of definition. This weakened notion of Ricci flow may be run in situations ill-suited to the classical theory. As an application of pyramid Ricci flows, we obtain global regularity results for three-dimensional Ricci limit spaces (extending results of Miles Simon and Peter Topping) and for higher dimensional PIC1 limit spaces (extending not only the results of Richard Bamler, Esther Cabezas-Rivas and Burkhard Wilking, but also the subsequent refinements by Yi Lai).
 

We construct maximal surfaces with Neumann boundary conditions in Minkowski space using mean curvature flow. In particular we find curvature conditions on a boundary manifold so that mean curvature flow may be shown to exist for all time, and give conditions under which the maximal hypersurfaces are stable under the flow.

 The Sen conjecture, made in 1994, makes precise predictions about the existence of L^2 harmonic forms on the monopole moduli spaces. For each positive integer k, the moduli space M_k of monopoles of charge k is a non-compact smooth manifold of dimension 4k, carrying a natural hyperkaehler metric.  Thus studying Sen’s conjectures requires a good understanding of the asymptotic structure of M_k and its metric.  This is a challenging analytical problem, because of the non-compactness of M_k and because its asymptotic structure is at least as complicated as the partitions of k.  For k=2, the metric was written down explicitly by Atiyah and Hitchin, and partial results are known in other cases.  In this talk, I shall introduce the main characters in this story and describe recent work aimed at proving Sen’s conjecture.