We adopt the paradigm of understanding how the heart develops during pregnancy as a first principal to inform on adult heart repair and regeneration. Our target for cell-based repair is the epicardium and epicardium-derived cells (EPDCs) which line the outside of the forming heart and contribute vascular endothelial and smooth muscle cells to the coronary vasculature, interstitial fibroblasts and cardiomyocytes. The epicardium can also act as a source of signals to condition the growth of the underlying embryonic heart muscle. In the adult heart, whilst the epicardium is retained, it is effectively quiescent. We have sought to extrapolate the developmental potential of the epicardium to the adult heart following injury by stimulating dormant epicardial cells to give rise to new muscle and vasculature. In parallel, we seek to modulate the local environment into which the new cells emerge: a cytotoxic mixture of inflammation and fibrosis which prevents cell engraftment and integration with survived heart tissue. To this end we manipulate the lymphatic vessels in the heart given that, elsewhere in the body, the lymphatics survey the immune system and modulate inflammation at peripheral injury sites. We recently described the development of the cardiac lymphatic vasculature and revealed in the adult heart that they undergo increased vessel sprouting (lymphangiogenesis) in response to injury, to improve function, remodelling and fibrosis. We are currently investigating whether increased lymphangiogenesis functions to clear immune cells and constrain the reparative response for optimal healing.

This half-day research workshop will address issues at the intersection between network science, matrix theory and mathematical physics.

Network science is producing a wide range of challenging research problems that have diverse applications across science and engineering. It is natural to cast these research challenges in terms of matrix function theory. However, in many cases, closely related problems have been tackled by researchers working in statistical physics, notably quantum mechanics on graphs and quantum chaos. This workshop will discuss recent progress that has been made in both fields and highlight opportunities for cross-fertilization. While focusing on mathematical, physical and computational issues, some results will also be presented for real data sets of relevance to practitioners in network science.

R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes showed in their celebrated paper from 1993 that numerous compensated compactness quantities such as Jacobians of mappings in $W^{1,n}(\mathbb{R}^n,\mathbb{R}^n)$ belong the real-variable Hardy space $\mathcal{H}^1(\mathbb{R}^n)$. They proceeded to ask what is the exact range of these nonlinear quantities and in particular whether the Jacobian operator $J$ maps $W^{1,2}(\mathbb{R}^2,\mathbb{R}^2)$ onto $\mathcal{H}^1(\mathbb{R}^2)$.

I present the proof of my recent result that $J \colon W^{1,n}(\mathbb{R}^n,\mathbb{R}^n) \to \mathcal{H}^1(\mathbb{R}^n)$ is non-surjective for every $n \ge 2$. The surjectivity question is still open when the domain of definition of $J$ is the inhomogeneous Sobolev space $\dot{W}^{1,n}(\mathbb{R}^n,\mathbb{R}^n)$. I also shortly discuss my work on T. Iwaniec's conjecture from 1997 which states that for every $n \ge 2$ and $p \in [1,\infty[$ the operator $J \colon W^{1,np}(\mathbb{R}^n,\mathbb{R}^n) \to \mathcal{H}^p(\mathbb{R}^n)$ has a continuous right inverse.

Existence results in large for fully non-linear compressible multi-fluid models are in the mathematical literature in a short supply (if not non-existing). In this talk, we shall recall the main ideas of Lions' proof of the existence of weak solutions to the compressible (mono-fluid) Navier-Stokes equations in the barotropic regime. We shall then eplain how this approach can be adapted to the construction of weak solutions to some simple multi-fluid models. The main tools in the proofs are renormalization techniques for the continuity and transport equations. They will be discussed in more detail.