We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under the only assumption of $L^1$ weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler system follows for arbitrary $L^1$ vorticity. Relations with previously known notions of solutions are shown.

# Past Forthcoming Seminars

The physics literature has for a long time posited a connection between the geometry of continuous random fields and discrete percolation models. Specifically the excursion sets of continuous fields are considered to be analogous to the open connected clusters of discrete models. Recent work has begun to formalise this relationship; many of the classic results of percolation (phase transition, RSW estimates etc) have been proven in the setting of smooth Gaussian fields. In the first part of this talk I will summarise these results. In the second I will focus on the number of excursion set components of Gaussian fields in large domains and discuss new results on the mean and variance of this quantity.

In directed algebraic topology, a topological space is endowed

with an extra structure, a selected subset of the paths called the

directed paths or the d-structure. The subset has to contain the

constant paths, be closed under concatenation and non-decreasing

reparametrization. A space with a d-structure is a d-space.

If the space has a partial order, the paths increasing wrt. that order

form a d-structure, but the circle with counter clockwise paths as the

d-structure is a prominent example without an underlying partial order.

Dipaths are dihomotopic if there is a one-parameter family of directed

paths connecting them. Since in general dipaths do not have inverses,

instead of fundamental groups (or groupoids), there is a fundamental

category. So already at this stage, the algebra is less desirable than

for topological spaces.

We will give examples of what is currently known in the area, the kind

of methods used and the problems and questions which need answering - in

particular with applications in computer science in mind.

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of , where 0, is a bounded measurable function, and solves a distribution dependent SDE with initial distribution . As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semi-group property and the invalidity of the formula = , essential difficulties are overcome in the study.

Joint work with Professor Feng-Yu Wang

In recent joint work with Lorenzo Foscolo and Johannes Nordstr\”om we gave an analytic construction of large families of complete circle-invariant $G_2$

holonomy metrics on the total space of circle bundles over a complete noncompact Calabi—Yau 3-fold with asymptotically conical geometry. The

asymptotic models for the geometry of these $G_2$ metrics are circle bundles with fibres of constant length $l$, so-called asymptotically local conical

(ALC) geometry. These ALC $G_2$ metrics can Gromov—Hausdorff collapse with bounded curvature to the given asymptotically conical Calabi—Yau 3-fold as the fibre length $l$ goes to $0$. A natural question is: what happens to these families of $G_2$ metrics as we try to make $l$ large? In general the answer to this question is not known, but in cases with sufficient symmetry we have recently been able to give a complete picture.

We give an overview of all these results and discuss some analogies with the class of asymptotically locally flat (ALF) hyperkaehler 4-manifolds. In

particular we suggest that a particular $G_2$ metric we construct should be regarded as a $G_2$ analogue of the Euclidean Taub—NUT metric on the complex plane.

Our meeting will be a relaxed opportunity to have informal discussions about issues facing minorities in academia and mathematics over lunch. In particular, if anyone would like to suggest a topic to start a discussion about (either in advance or on the day) then please feel free to do this, and it could be a spring board for organised sessions on the same topics in future terms!

Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle. Their moduli spaces carry a natural Hyperkahler structure, through which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention in string theory. After introducing Higgs bundles and the associated Hitchin fibration, we shall look at natural constructions of families of different types of branes, and relate these spaces to the study of 3-manifolds, surface group representations and mirror symmetry.

Transdifferentiation, the process of converting from one cell type to another without going through a pluripotent state, has great promise for regenerative medicine. The identification of key transcription factors for reprogramming is limited by the cost of exhaustive experimental testing of plausible sets of factors, an approach that is inefficient and unscalable. We developed a predictive system (Mogrify) that combines gene expression data with regulatory network information to predict the reprogramming factors necessary to induce cell conversion. We have applied Mogrify to 173 human cell types and 134 tissues, defining an atlas of cellular reprogramming. Mogrify correctly predicts the transcription factors used in known transdifferentiations. Furthermore, we validated several new transdifferentiations predicted by Mogrify, including both into and out of the same cell type (keratinocytes). We provide a practical and efficient mechanism for systematically implementing novel cell conversions, facilitating the generalization of reprogramming of human cells. Predictions are made available via http://mogrify.net to help rapidly further the field of cell conversion.

Plumes are a characteristic feature of convective flow through porous media. Their dynamics are an important part of numerous geological processes, ranging from mixing in magma chambers to the convective dissolution of sequestered carbon dioxide. In this talk, I will discuss models for the spread of convective plumes in a heterogeneous porous environment. I will focus particularly on the effect of thin, roughly horizontal, low-permeability barriers to flow, which provide a generic form of heterogeneity in geological settings, and are a particularly widespread feature of sedimentary formations. With the aid of high-resolution numerical simulations, I will explore how a plume spreads and flows in the presence of one or more of these layers, and will briefly consider the implications of these findings in physical settings.