We study the model of randomly perturbed dense graphs, which is the union of any $n$-vertex graph $G_\alpha$ with minimum degree at least $\alpha n$ and the binomial random graph $G(n,p)$. In this talk, we shall examine the following central question in this area: to determine when $G_\alpha \cup G(n,p)$ contains $H$-factors, i.e. spanning subgraphs consisting of vertex disjoint copies of the graph $H$. We offer several new sharp and stability results.
This is joint work with Julia Böttcher, Olaf Parczyk, and Jozef Skokan.

 Given a solution of the Einstein equations, a fundamental question is whether one can extend the solution or whether the solution is maximal. If the solution is inextendible in a certain regularity class due to the geometry becoming singular, a further question is whether the strength of the singularity is such that it terminates classical time-evolution. The latter question, as will be explained in the talk, is intimately tied to the strong cosmic censorship conjecture in general relativity which states in the language of partial differential equations that global uniqueness holds generically for the initial value problem for the Einstein equations. I will then focus in the talk on recent results showing the locally Lipschitz inextendibility of FLRW models with particle horizons and spherically symmetric weak null singularities. The latter in particular apply to the spherically symmetric spacetimes constructed by Luk and Oh, improving their C^2-formulation of strong cosmic censorship to a locally Lipschitz formulation.

How to calculate the minimal number of summands of a nonzero polynomial in a given polynomial ideal? In this talk, we first discuss the roots of this question in computational algebra. Afterwards, we switch to the viewpoint of a commutative algebraist. In particular, we see that classical tools from this field, such as primary decomposition or the Castelnuovo–Mumford regularity, fail to provide a solution to this problem. Finally, we discuss a concrete example: A standard determinantal ideal generated by $t$-minors does not contain any polynomials with fewer than $t!/2$ terms.

Neural controlled differential equations (Neural CDEs) are a continuous-time extension of recurrent neural networks (RNNs). They are considered SOTA for modelling functions on irregular time series, outperforming other ODE benchmarks (ODE-RNN, GRU-ODE-Bayes) in offline prediction tasks. However, current implementations are not suitable to be used in online prediction tasks, severely restricting the domains of applicability of this powerful modeling framework. We identify such limitations with previous implementations and show how said limitations may be addressed, most notably to allow for online predictions. We benchmark our online Neural CDE model on three continuous monitoring tasks from the MIMIC-IV ICU database, demonstrating improved performance on two of the three tasks against state-of-the-art (SOTA) non-ODE benchmarks, and improved performance on all tasks against our ODE benchmark.

Joint work with Patrick Kidger, Lingyi Yang, and Terry Lyons.