L4

I will discuss an ongoing joint work with Luigi Appolloni and Andrea Malchiodi concerning the above objects. Minimal surfaces are critical points of the area functional, which is analytic in this case, so we should expect critical points (minimal surfaces) to be either isolated or to belong to smooth nearby minimal foliations. On the other hand, the flat plane of multiplicity two in $\mathbb{R}^3$ can be (in compact regions) approximated by a blown-down catenoid, which will converge back to the plane with multiplicity two in the limit. Hence a plane of multiplicity two cannot be thought of as being isolated, or belonging solely to a smooth family, because there are “nearby” minimal surfaces of distinct topology weakly converging to it. We will nevertheless prove that, when the ambient manifold is closed and analytic, this type of local degeneration is impossible amongst closed and embedded minimal surfaces of bounded topology: such surfaces, even with multiplicity are either isolated or belong to smooth families of nearby minimal surfaces.  

We consider a class of Lagrangian sections L contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: L is isomorphic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. On
surfaces and threefolds, under more restrictive assumptions, this result can be used to show a precise relation with Bridgeland stability, as predicted by Joyce. Based on arXiv:2505.07228 and arXiv:2508.17709.

I will discuss the paper "Construction of Gross-Neveu model using Polchinski flow equation" by Pawel Duch (https://arxiv.org/abs/2403.18562).

Everything you have been taught about Turing patterns is wrong! (Well, not everything, but qualifying statements tend to weaken a punchy first sentence). Turing patterns are universally used to generate and understand patterns across a wide range of biological phenomena. They are wonderful to work with from a theoretical, simulation and application point of view. However, they have a paradoxical problem of being too easy to produce generally, whilst simultaneously being heavily dependent on the details. In this talk I demonstrate how to fix known problems such as small parameter regions and sensitivity, but then highlight a new set of issues that arise from usually overlooked issues, such as boundary conditions, initial conditions, and domain shape. Although we’ve been exploring Turing’s theory for longer than I’ve been alive, there’s still life in the old (spotty) dog yet.

We survey the life of a Turing pattern, from initial diffusive instability through the emergence of dominant spatial modes and to an eventual spatially heterogeneous pattern. While many mathematically ideal Turing patterns are regular, repeating in structure and remaining of a fixed length scale throughout space, in the real world there is often a degree of irregularity to patterns. Viewing the life of a Turing pattern through the lens of spatial modes generated by the geometry of the bounded space domain housing the Turing system, we discuss how irregularity in a Turing pattern may arise over time due to specific features of this space domain or specific spatial dependencies of the reaction-diffusion system generating the pattern.

Normal heart function relies of the fine-tuned synchronization of cellular components. In healthy hearts, calcium oscillations and physical contractions are coupled across a synchronised network of 3 billion heart cells. When the process of functional isolation of rogue cells isn’t successful, the network becomes maladapted, resulting in cardiovascular diseases, including heart failure and arrythmia. To advance knowledge on this normal-to-disease transition we must first address the lack of a mechanistic understanding of the plastic readaptation of these networks. In this talk I will explore coupling and loss of synchronisation using a mathematical model of calcium oscillations informed by experimental data. I will show some preliminary results pointing at the heterogeneity hidden behind seemingly uniform cell populations, as a causative mechanism behind disrupted dynamics in maladapted networks.

Specialist species thrive under specific environmental conditions in narrow geographic ranges and are widely recognized as heavily threatened by climate deregulation. Many might rely on both their potential to adapt and to disperse towards a refugium to avoid extinction. It is thus crucial to understand the influence of environmental conditions on the unfolding process of adaptation. I will present a PDE model of the eco-evolutionary dynamics of a specialist species in a two-patch environment with moving optima. The transmission of the adaptive trait across generations is modelled by a non-linear, non-local operator of sexual reproduction. In an asymptotic regime of small variance, I justify that the local trait distributions are well approximatted by Gaussian distributions with fixed variances, which allows to report the analysis on the closed system of moments. Thanks to a separation of time scales between ecology and evolution, I next derive a limit system of moments and analyse its stationary states. In particular, I identify the critical environmental speed for persistence, which reflects how both the existence of a refugium and the cost of dispersal impact extinction patterns. Additionally, the analysis provides key insights regarding the path towards this refugium. I show that there exists a critical environmental speed above which the species crosses a tipping point, resulting into an abrupt habitat switch from its native patch to the refugium. When selection for local adaptation is strong, this habitat switch passes through an evolutionary ‘‘death valley’’ that can promote extinction for lower environmental speeds than the critical one.