L4

Ken Brown reviews the history of the question in the title, and describes some recent progress towards answering it, including the identification of a "minimal criminal". The new material is joint work with Jason Bell (Waterloo) and Toby Stafford (Manchester).

A spatial graph is a graph whose nodes and edges carry spatial attributes. It is a smart modelling choice for capturing the skeleton of a shape, a blood vessel network, a porous tissue, and many other data objects with intrinsically complex geometry, often resulting in graphs with a high node and edge count. In this talk, we introduce a topological spatial graph coarsening approach based on a new framework that balances graph reduction against the preservation of topological characteristics, essential for faithfully representing the underlying shape. To capture the topological information required to calibrate the reduction level, we adapt the construction of classical topological descriptors made for point clouds (the so-called persistence diagrams) to spatial graphs. This relies on a new filtration called triangle-aware graph filtration. Our coarsening approach is parameter-free and we prove that it is equivariant under rotations, translations, and scaling of the initial spatial graph. We evaluate the performance of our method on synthetic and real spatial graphs and show that it significantly reduces the graph sizes while preserving the relevant topological information.

Recently a new inhabitant entered the zoo of convexity notions for vectorial variational problems: functional convexity. I would like to report of progress in understanding the corresponding integrands, but also new insight into fine properties of most general class of related integrands: It turns out that rank-one convex functions share surprisingly many pointwise differentiablity properties with ordinary convex functions.

We provide a derivation of the fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider on the discretized circle the rate equation for the process where pairs of particles sitting on the same side jump simultaneously to the two neighboring sites, and the reverse jump where a pair of particles sitting on a common site jump simultaneously to the side in the middle. Depending on the rates, in the vanishing mesh size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. We identify the limiting gradient structure to be driven by entropy with respect to a generalization of the diffusive transport type with nonlinear mobility via EDP convergence. Furthermore, the DLSS equation with nonlinear mobility of the power type shares qualitative similarities with the fast diffusion and porous medium equations, since we find traveling wave solutions with algebraic tails and polynomial compact support, respectively.    
       

Joint work with Alexander Mielke and Artur Stephan arXiv:2510.07149. The DLSS part is based on joints works with Daniel Matthes, Eva-Maria Rott and Giuseppe Savaré.

More than seventy years after its publication, Turing’s article “The Chemical Basis of Morphogenesis” is still able to surprise its reader, in particular for the power and the depth of its vision. If we know from his biographer, Andrew Hodges, that Turing became interested in embryology and morphogenesis because he wanted to build or, better, to grow a brain, many questions still arise for the reader of the original article: why did Turing – a mathematician, a logician, a cryptographer, one of the fathers of computer science – not use any informational metaphor associated with the notion of “genetic program” in his work on morphogenesis, preferring instead to develop a modelling approach based on a system of partial differential equations ? Where did he draw his modelling inspiration from, both from the point of view of the mathematics and from the point of view of references to biology ? In my presentation I will address these questions by highlighting the morphological connotations of Turing’s work in biology, that can be related to Turing’s interest, in D’Arcy Wentworth Thompson’s classic On Growth and Form (1917). The 1952 article is rather sparse in indications in this regard, which are, however, provided by Turing’s other writings, unpublished during his lifetime, in which he situates his work in continuity with Thompson’s morphological questions. I will also suggest that, as in a virtuous circle, Turing masterfully brings to life a synergy between a morphological look at the living (that implies that his work has a connotation in theoretical biology) and a mathematical exploration of the non-linear, helped by an appropriate and meaningful use of numerical calculus. 

Gravitational instantons are complete 4-dimensional hyperkähler manifolds with square-integrable curvature tensor. I will address the question whether all gravitational instantons (of type ALG) can be obtained as Hitchin moduli spaces. In particular, I will explain how to compute the (hyperkähler) Torelli map for (weakly) parabolic Higgs bundles on the 4-punctured sphere. This is based on recent joint work with Fredrickson, Mazzeo and Swoboda.

Stochastic search is ubiquitous in biology and ecology, from synaptic transmission and intracellular signaling to predators seeking prey and the spread of disease. In dynamic systems like these, the number of 'searchers' is rarely constant: new agents may be recruited while others can abandon the search. Despite the ubiquity of these dynamics, their combined influence on search times remains largely unexplored. In this talk we will introduce a general framework for stochastic search in which agents progressively join and leave the process, a mechanism we term 'dynamic redundancy and mortality'. Under minimal assumptions on the underlying search dynamics, our framework yields the exact distribution of the first-passage time to a target region and further reveals surprising connections to stochastic search with stochastic resetting, wherein a single searcher is randomly 'reset' to its initial state. We will then treat the target region as a queue, which we show has interarrival times governed by a thinned nonhomogeneous Poisson process. Altogether this work provides a rigorous foundation for studying stochastic search processes with a fluctuating number of searchers. This work is in collaboration with Dr. Aanjaneya Kumar (Santa Fe Institute) and José Giral-Barajas (Imperial College London).