L6

Understanding the molecular mechanisms that control the biology of health and disease requires development of models that traverse multiple scales of organisation in order to encapsulate the relationships between genes and linking to observable phenotypes. Measuring, parameterising and simulating the entire system that determines these phenotypes in exhaustive detail is typically impossible due to the underlying biological complexity, our limited knowledge and the paucity of available data. For example, approximately one third of human genes are poorly characterised and most genes perform multiple functions, which manifest according to the surrounding biochemical context. Indeed, new functions continue to emerge even for deeply studied genes. Therefore, simplifying abstractions in concert with empirical analysis of matched genome-scale and descriptive data are valuable strategies to fill knowledge gaps relevant to a focused biomedical question or hypothesis.

Epithelial plasticity is a key driver of cancer progression and is associated with the most life-threatening phenotypes; specifically, metastasis and drug resistance. Computational methods developed in my group enable modelling the molecular control of important cancer phenotypes. We applied a machine learning approach for genome-wide context-specific biochemical interaction network inference (CoSNI) to map gene function for the Epithelial to Mesenchymal Transition cell programme (EMT_MAP), predicting new mechanisms in control of cancer invasion. Analysis of patient data with EMT_MAP and our NetNC algorithm [Cancers 2020;12:2823; https://github.com/overton-group/NetNC] enabled discovery of candidate renal cancer prognostic markers with clear advantages over standard statistical approaches. NetNC recovers the network-defined signal in noisy data, for example distinguishing functional EMT Transcription Factor targets from ‘neutral’ binding sites and defining biologically coherent modules in renal cancer drug response time course data. These and other approaches, including SynLeGG (Nucleic Acids Research 2021;49:W613-8, www.overton-lab.uk/synlegg) and an information-theoretic approach to causality (GABI) offer mechanistic insights and opportunity to predict candidate cancer Achilles’ heels for drug discovery. Computational results were validated in follow-up experiments, towards new clinical tools for precision oncology.

Dynamical systems models are a powerful tool for analyzing interactions in ecosystems and their intrinsic properties such as stability and resilience. The human intestinal microbiome is a complex ecosystem of hundreds of microbial species, critical to our health, and when in a disrupted state termed dysbiosis, is involved in a variety of diseases.  Although dysbiosis remains incompletely understood, it is not caused by single pathogens, but instead involves broader disruptions to the microbial ecosystem.  Dynamical systems models would thus seem a natural approach for analyzing dysbiosis, but have been hampered by the scale of the human gut microbiome, which constitutes hundreds of thousands of potential ecological interactions, and is profiled using sparse and noisy measurements. Here we introduce a combined experimental and statistical machine learning approach that overcomes these challenges to provide the first comprehensive and predictive model of microbial dynamics at ecosystem-scale. We show that dysbiosis is characterized by competitive cycles of interactions among microbial species, in contrast to the healthy microbiome, which is stabilized by chains of positive interactions initiated by resistant starch-degrading bacteria. To achieve these results, we created cohorts of “humanized” gnotobiotic mice via fecal transplantation from healthy and dysbiotic human donors, and subjected mice to dietary and antibiotic perturbations, in the densest temporal interventional study to date. We demonstrate that our probabilistic machine learning method achieves scalability while maintaining interpretability on these data, by inferring a small number of modules of bacterial taxa that share common interactions and responses to perturbations. Our findings provide new insights into the mechanisms of microbial dysbiosis, have potential implications for therapies to restore the microbiome to treat disease, and moreover offer a powerful framework for analyzing other complex ecosystems.

In commutative algebra localizing a ring and its modules is a fundamental technique. In the general case of a Grothendieck abelian category or even a triangulated category with small direct sums this is replaced by forming the quotient category by a localizing subcategory. Therefore the classification of these localizing subcategories becomes an important problem. I will begin by recalling the case of the (derived) module category of a commutative noetherian ring due to Gabriel and Hopkins/Neeman, respectively, in order to give an idea how such a classification can look like.

The case of interest in this talk is the derived category D(G) of smooth representation in characteristic p of a p-adic Lie group G. This is motivated by the emerging p-adic Langlands program. In joint work with C. Heyer we have some modest initial results if G is compact pro-p or abelian. which I will present.

Stably density stratified shear flows arise widely in geophysical settings. Instabilities of these flows occur on scales that are too small to be directly resolved in numerical simulations, e.g., of the oceans and atmosphere, yet drive diabatic mixing events that often exert a controlling influence on much larger-scale processes. In the limit of strong stratification, the flows are characterized by the emergence of highly anisotropic layer-like structures with much larger horizontal than vertical scales. Owing to their relative horizontal motion, these structures are susceptible to stratified shear instabilities that drive spectrally non-local energy transfers. To efficiently describe the dynamics of this ``layered anisotropic stratified turbulence'' regime, a multiple-scales asymptotic analysis of the non-rotating Boussinesq equations is performed. The resulting asymptotically-reduced equations are shown to have a generalized quasi-linear (GQL) form that captures the essential physics of strongly stratified shear turbulence. The model is used to investigate the mixing efficiency of certain exact coherent states (ECS) arising in strongly stratified Kolmogorov flow. The ECS are computed using a new methodology for numerically integrating slow--fast GQL systems that obviates the need to explicitly resolve the fast dynamics associated with the stratified shear instabilities by exploiting an emergent marginal stability constraint.

Spectral sequences are computational tools to find the (co-)homology of mathematical objects and are used across various fields. In this talk I will focus on the LHS spectral sequence, which we associate to an extension of groups to compute group cohomology. The first part of the talk will serve as introduction to both group cohomology and general spectral sequences, where I hope to provide and intuition and some reduced formalism. As main example, and core of this talk, we will look at the LHS spectral sequence associated to the group extension $(\mathbb{Z}/3\mathbb{Z})^3 \rightarrow S \rightarrow \mathbb{Z}/3\mathbb{Z}$, where $S$ is a Sylow-3-subgroup of $SL_2(\mathbb{Z}/9\mathbb{Z})$. In particular I will present arguments that all differentials on the $E^2$ page vanish.

I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.

Non-shearing congruences of null geodesics on four-dimensional Lorentzian manifolds are fundamental objects of mathematical relativity. Their prominence in exact solutions to the Einstein field equations is supported by major results such as the Robinson, Goldberg-Sachs and Kerr theorems. Conceptually, they lie at the crossroad between Lorentzian conformal geometry and Cauchy-Riemann geometry, and are one of the original ingredients of twistor theory.
 
Identified as involutive totally null complex distributions of maximal rank, such congruences generalise to any even dimensions, under the name of Robinson structures. Nurowski and Trautman aptly described them as Lorentzian analogues of Hermitian structures. In this talk, I will give a survey of old and new results in the field.

In this talk I will discuss my extension of the Koszul duality theory of Beilinson, Ginzburg, and Soergel to the more general setting of quasi-abelian categories. In particular, I will define the notions of Koszul monoids, and quadratic monoids and their duals. Schneiders' embedding of a quasi-abelian category into an abelian category, its left heart, allows us to prove an equivalence of derived categories for certain categories of modules over Koszul monoids and their duals. The key examples of categories for which this theory works are the categories of complete bornological spaces and the categories of inductive limits of Banach spaces. These categories frequently appear in derived analytic geometry.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome